MODULAR CURVES AND RAMANUJAN’S CONTINUED FRACTION

BRYDEN CAIS AND BRIAN CONRAD

Abstract. We use arithmetic models of modular curves to establish some properties of Ramanujan’s continued fraction. In particular, we give a new geometric proof that its singular values are algebraic units that generate specific abelian extensions of imaginary quadratic fields, and we use a mixture of geometric and analytic methods to construct and study an infinite family of two-variable polynomials over Z that are related to Ramanujan’s function in the same way that the classical modular polynomials are related to the classical j-function. We also prove that a singular value on the imaginary axis, necessarily real, lies in a radical tower in R only if all odd prime factors of its degree over Q are Fermat primes; by computing some ray class groups, we give examples where this necessary condition is not satisfied.

1. Introduction

Let C be an algebraic closure of R. For τ ∈ C − R, let q_{τ} = e^{2πi_{τ} τ} with i_{τ}^{2} = −1 and i_{τ} in the connected component of τ in C − R, so |q_{τ} | < 1 and q_{τ} = q_{−τ}.

The Ramanujan continued fraction F : C − R → C is

F (τ) := \frac{q_{τ}^{1/5}}{1+} \frac{q_{τ}}{1+} \frac{q_{τ}^{2}}{1+} \frac{q_{τ}^{3}}{1+} · · · .

where q_{τ}^{r} means e^{2πi_{τ} rτ} for r ∈ Q. Note that F (τ) = F (−τ). In his 1916 letter to Hardy, Ramanujan stated the remarkable identities

(1.1)  F (i) = \frac{e^{−2π/5}}{1+} \frac{e^{−2π}}{1+} \frac{e^{−4π}}{1+} \frac{e^{−6π}}{1+} \frac{e^{−8π}}{1+} ⋯ = \sqrt{\frac{5 + \sqrt{5}}{2}} − \frac{\sqrt{5} + 1}{2}

and

(1.2)  F (\frac{5}{2} + i) = \frac{− e^{−π/5}}{1+} \frac{− e^{−π}}{1+} \frac{e^{−2π}}{1+} \frac{− e^{−3π}}{1+} ⋯ = −\sqrt{\frac{5 − \sqrt{5}}{2}} + \frac{\sqrt{5} − 1}{2}

for any i ∈ C satisfying i^{2} + 1 = 0, and he asserted that if τ ∈ C − R satisfies τ^{2} ∈ Q, then “F (τ) can be exactly found.”

Watson [21] proved (1.1) and (1.2) as consequences of the identity

(1.3)  \frac{1}{F (τ)} − 1 − F (τ) = \frac{η(τ/5)}{η(5τ)},

Date: April 21, 2004. 1991 Mathematics Subject Classification. Primary 11F03; Secondary 14G35. Key words and phrases. q-series, modular curve, Ramanujan continued fraction. The second author was supported by the NSF and the Sloan Foundation. The authors thank K. Conrad for providing us

with a proof of Lemma D.1 and thank William Stein for his publically available computer cluster Meccah that was used for the numerical computations. We are also greatly indebted to James Parson for many enlightening and insightful discussions concerning Appendix C..

where η(τ) =^{def} q_{τ}^{1/24} ∏_{n=1}^{∞}(1−q_{τ}^{n}) is the Dedekind η-function, and he proved that F (τ) is an algebraic integer when τ ∈ C − R is quadratic over Q. Using Watson’s identity and an integrality result of Stark’s on Lfunctions at s = 1, in [1, Thm. 6.2] it is shown that F (τ) is an algebraic integral unit for such τ. Ramanujan formulated other modular equations satisfied by F, and there has been a lot of research (for example, see [1], [9], [23], and [24], as well as [2] for an overall survey) investigating these (and other) explicit equations and using them to explicitly compute F at specific imaginary quadratic points in C − R, making creative use of Watson’s identity and its variants.

In [8], Watson’s identity is used to show that F is a level-5 modular function, and by determining the minimal polynomial of F over Q(j), it is proved that Q(F) is the field of modular functions of level-5 having Fourier expansion at ∞ with rational coefficients (this was already known to Klein [7, vol. 2, p. 383], who described F as a ratio of theta-functions). They employ Shimura reciprocity to describe the Galois conjugates of singular values F (τ) at quadratic imaginary τ ∈ C − R and thereby provide an algorithm to compute the minimal polynomial (over Q) of any such singular value. Using standard Kummer theory (and the fact that F (τ) lies in an abelian—hence solvable—extension of K = Q(τ)), they show how to compute radical formulae for F (τ), thereby settling (in the affirmative) Ramanujan’s claim that that the singular values of F on the imaginary axis can be “exactly found.”

In this paper, we work with the function j_{5} = 1/F; as is well-known, this is a level-5 modular function with a unique simple pole at the cusp ∞ ∈ X(5), so it defines an isomorphism j_{5} : X(5) ≃ CP^{1}. At the end of §4 we will recall a proof of this fact which does not rest on the crutch of Watson’s identity. Our aim is to deduce additional properties of j_{5} (and hence of F) by means of the good behavior of j_{5} with respect to certain standard canonical integral models for X(5) over Z[1/5] and Z[ζ_{5}] that we describe in §2; since Z[1/5] ∩ Z[ζ_{5}] = Z, we will also be able to obtain results over Z.

In §3–§4 we quickly review Shimura’s canonical models for modular curves and some properties of Klein forms, and in §5 we use Klein forms and arithmetic models of modular curves to construct an integral model J_{5} for j_{5} on the model X(5)_{Z[ζ_{5}]} and to show that the rational function J_{5} has both its zero and polar loci supported in the subscheme of cusps over Z[ζ_{5}] (a contrast with the classical j-function on X(1) over Z). From this we obtain a new geometric proof that the values of j_{5} (or Ramanujan’s F) at CM-points are algebraic integral units (see Corollary 5.6). An easy application of Shimura’s reciprocity law on canonical models implies that for an arbitrary τ ∈ C − R that is quadratic over Q, the extension Q(τ, j_{5}(τ))/Q(τ) is an abelian extension of K = Q(τ) that is unramified away from 5; we give the associated open subgroup in the idele class group A_{K}^{×}/K^{×} in Corollary 5.7. (Such a description is not included in [8].) For example, we shall see that if the elliptic curve C^{×}/q_{τ}^{Z} has CM-order 𝒪_{τ} ⊆ 𝒪_{K} and τ is 5-integral, then K(j_{5}(τ)) is contained in the ray class field of conductor 5 · [𝒪_{K} : 𝒪_{τ}] for K, and if moreover τ is a 5-unit and 𝒪_{τ} = 𝒪_{K} then K(j_{5}(τ)) is the ray class field of conductor 5 for K. In particular, this latter ray class field is always generated by a singular value of Ramanujan’s function F.

A further application of the algebro-geometric link with modular curves is pursued in §6: for all n relatively prime to 5, we construct primitive polynomials F_{n} ∈ Z[X, Y] that are absolutely irreducible over Q and satisfy F_{n}(j_{5}(τ), j_{5}(nτ)) = 0. These F_{n}’s for very small n have appeared in the literature on a case-by-case basis, and their existence in general was known to Klein long ago, but there does not seem to have been a systematic construction given before for all n relatively prime to 5 in a manner that is welladapted to a study of algebraic properties (such as absolute irreducibility over Q). We also establish an analogue of Kronecker’s congruence for F_{p} mod p, but the shape of the congruence depends on p mod 5. It seems worth emphasizing that geometry can establish such congruences only up to a unit scaling factor; to eliminate the unit ambiguity, it is essential to bring in the analytic perspective via q-expansions. Similarly, the symmetry Φ_{n}(Y, X) = Φ_{n}(X, Y) for the classical level-n modular polynomial has an analogue for F_{n} when gcd(n, 5) = 1, as we show in §6, and geometric methods prove the result up to a factor in Z^{×} = {±1}; we need q-expansions to determine the sign. Explicitly, if n ≡ ±1 mod 5 then F_{n} is symmetric, but if n ≡ ±2 mod 5 then X^{deg_{Y} F_{n}}F_{n}(Y, −1/X) = ε_{n}F_{n}(X, Y) for a sign ε_{n} that depends on n in a slightly complicated manner; for example, if n = p is an odd prime congruent to ±2 mod 5 then ε_{n} = 1, and ε_{n} = −1 for n = 2, 32, 72, … (see Theorem 6.5).

One virtue of computing the F_{n}’s for many n’s is that we thereby noticed that F_{n} has remarkably small coefficients; we give a list of F_{n}’s for n ≤ 27 in Appendix C. The reason for such smallness of coefficients is that the q-expansion of j_{5} has small coefficients, and in §7 we provide an estimate on the coefficients of j_{5} that is obtained by adapting the circle-method arguments of Rademacher for the usual j-function. In [4], P. Cohen established asyptotic estimates on the maximal absolute value of a coefficient of the classical level-n modular polynomial Φ_{n} (as n → ∞), and in §7 we also give a variant on this method that applies to the F_{n}’s when gcd(n, 5) = 1. One consequence of these estimates is that, as n → ∞, the largest coefficient for F_{n} (in absolute value) is approximately the 60th root of the largest coefficient for Φ_{n}. This explains the apparent smallness of the coefficients of the F_{n}’s in examples with (n, 5) = 1.

We conclude in §8 by addressing the question of whether a singular value j_{5}(τ) can be expressed in radicals inside of R when τ^{2} ∈ Q. For example, one may wish to interpret Ramanujan’s assertion that j_{5}(τ) can always be “exactly found” when τ^{2} ∈ Q as saying that j_{5}(τ) ∈ R lies in a radical tower of subfields of R for such τ. All previously published explicit radical formulas for such real singular values do satisfy this condition, but we will show (in Theorem 8.5) that these many examples are exceptions to the rule: if τ^{2} ∈ Q and j_{5}(τ) lies in a radical tower in R then all odd prime factors of [Q(j_{5}(τ)) : Q] are Fermat primes (we can only prove the converse in the easy case when [Q(j_{5}(τ)) : Q] is a power of 2, and the casus irreducibilis suggests that the converse is probably not true in general). This is a very restrictive necessary condition for the existence of a radical formula in R, and it applies (with the same proof) to singular values of the usual j-function on the imaginary axis. In all published examples of radical formulas for singular values of j_{5} on the imaginary axis, the degree of the singular value over Q has been of the form 2^{e}, 2^{e} · 3, or 2^{e} · 5; this Fermat criterion is not satisfied for j_{5}(\sqrt{−101}) (see Example 8.4), so the stronger interpretation of Ramanujan’s claim using real radical towers is false. For τ^{2} = q ∈ Q with the height of q growing to ∞, it seems certain that in the asymptotic sense the necessary Fermat criterion fails to be satisfied 100% of the time.

Notation. The notation we use is standard, but we record one mild abuse of notation: we write G rather than the customary \underline{G} to denote the constant group scheme associated to an abstract group G (when working over a base scheme S that will always be understood from context). This is most commonly used in the case G = Z/nZ for a positive integer n.

2. Arithmetic models and analytic models of modular curves

In this section we summarize some basic definitions and theorems in the arithmetic theory of modular curves, and we link it up with the analytic models that are obtained as quotients of C − R. Everything we say in this section is well-known.

Pick a positive integer N. We will begin our work with level-N moduli problems over the ring of integers Z[ζ_{N}] of a splitting field Q(ζ_{N}) of the Nth cyclotomic polynomial, with ζ_{N} a choice of root of this polynomial. In [10], Katz and Mazur use Drinfeld level structures to develop a systematic theory of elliptic-curve moduli problems over integer rings (so they can study N-torsion moduli even if N is not a unit on the base scheme); unfortunately, this work omits a treatment of the modular interpretation of the cusps. The earlier work of Deligne and Rapoport [5] required the level to be invertible on the base (i.e., to study moduli of N-torsion level structures, they required the base scheme to live over Spec Z[1/N]) but it introduced the theory of generalized elliptic curves to provide a modular interpretation along the cusps.

Example 2.1. Let n be a positive integer and S a scheme. Consider a set of n copies of P_{S}^{1} indexed by Z/nZ. Let C_{n} be the S-scheme obtained by gluing 0 on the ith copy of P_{S}^{1} to ∞ on the (i + 1)th copy of P_{S}^{1}; when n = 1 this is the nodal plane cubic Y^{2} + XY = X^{3}. We call C_{n} the standard Néron n-gon over S.

The S-smooth locus C_{n}^{sm} is naturally identified with G_{m}×(Z/nZ), and with this evident S-group structure there is a unique action of C_{n}^{sm} on C_{n} extending the group law on C_{n}^{sm}. Equipped with this data, C_{n} is a generalized elliptic curve. Observe that the n-torsion in C_{n}^{sm} is naturally isomorphic to μ_{n} × (Z/nZ).

In [5, II, 1.15], it is proved that a generalized elliptic curve over an algebraically closed field must be isomorphic to either a smooth elliptic curve or to a standard Néron polygon.

The works of Deligne–Rapoport and Katz–Mazur can be combined, and hence it makes sense to consider the moduli functor that classifies full level-N structures on generalized elliptic curves: these are pairs (E, ι) where E is a generalized elliptic curve over an arbitrary scheme S and ι : (Z/NZ)^{2} → E^{sm}(S)[N] is a Drinfeld (Z/NZ)^{2}-structure on the smooth separated S-group E^{sm}. Such an ι is called S-ample (or ample) if its image meets all irreducible components of all geometric fibers of E over S. If E admits a full level-N structure, then its non-smooth geometric fibers must have number of sides divisible by N and E^{sm}[N] must be a finite locally free S-group with order N^{2}. In this case, there is a functorial alternating μ_{N}-valued self-duality e_{N} on the N-torsion, and for any ι as above it is automatic (essentially by [10, 10.4.1]) that ζ = e_{N}(ι(1, 0), ι(0, 1)) ∈ μ_{N}(S) is a root of the Nth cyclotomic polynomial. We say that (E, ι) is of type ζ.

The synthesis of the above work of Katz–Mazur and Deligne–Rapoport is (partly) summarized in the next two theorems (see [10, 10.9.1, 10.9.6-7] for proofs, and note that the theory of level structures in [10, Ch. 1] applies to the smooth locus of a generalized elliptic curve):

Theorem 2.2. Let N be a positive integer. There exists a coarse moduli scheme X(N)^{can} over Spec Z[ζ_{N}] for the moduli functor that classifies generalized elliptic curves equipped with an ample full level-N structure of type ζ_{N}. This moduli scheme is normal, proper, and flat over Spec Z[ζ_{N}], and its fibers are geometrically connected with pure dimension 1. It is smooth away from the supersingular geometric points in characteristics dividing N, and when N ≥ 3 it is a fine moduli scheme over Z[1/N].

The second theorem we require from [5] and [10] concerns the formal structure along the subscheme of cusps in X(N)^{can}. There is an elegant abstract technique for defining such a subscheme [5, II, 1.15], but for our expository purposes we will use an ad hoc trick as in [10]: there is a finite flat map j : X(N)^{can} → P_{Z[ζ_{N}]}^{1} induced by formation of j-invariants, and the reduced subscheme underlying the preimage of the section ∞ ∈ P^{1}(Z[ζ_{N}]) is called the cuspidal subscheme X(N)_{∞}^{can}. This is finite and flat over Z[ζ_{N}]. The key fact is:

Theorem 2.3. The closed subscheme X(N)_{∞}^{can} is a finite disjoint union of copies of Spec Z[ζ_{N}], and the formal completion of X(N)^{can} along this closed subscheme is canonically isomorphic to a finite disjoint union of copies of Spf Z[ζ_{N}][[q^{1/N}]].

Let us now define the cuspidal section ∞ ∈ X(N)^{can}(Z[ζ_{N}]) algebraically. This definition rests on the standard Néron N-gon C_{N} over Spec Z[ζ_{N}]. Since the smooth locus C_{N}^{sm} is canonically identified with G_{m} × Z/NZ as a group scheme, we have C_{N}^{sm}[N] = μ_{N} × Z/NZ. There is a canonical full level-N structure of type ζ_{N} defined by the map

ι : Z/NZ × Z/NZ → μ_{N} × Z/NZ

that carries (1, 0) to (ζ_{N}, 0) and carries (0, 1) to (1, 1). This defines a section

∞ : Spec Z[ζ_{N}] ↪ X(N)^{can}

that is one of the components of X(N)_{∞}^{can}.

To explain the link between the preceding algebraic theory and the analytic theory, fix a positive integer N and a primitive Nth root of unity ζ in C. We avoid picking a preferred choice of ζ (such as e^{±2π\sqrt{−1}/N}) because the algebraic theory must treat all choices on an equal footing, and we want to argue in a manner that translates most easily into the algebraic theory. Consider the moduli functor \overline{ℳ} _{ζ}(N) that classifies analytic families of full level-N structures of type ζ. For N ≥ 3, these structures admit no non-trivial automorphisms and there is a universal analytic family over an open modular curve Y_{ζ}(N) that is the complement of finitely many points (the cusps) in a compact connected Riemann surface X_{ζ}(N). For N ≥ 3 we may identify X_{ζ}(N) as a fine moduli space for the moduli functor that classifies analytic families of full level-N structures of type ζ on generalized elliptic curves [5, VI, §5; VII, §4], and this is isomorphic to the analytification of the complex fiber of X(N)^{can} via the map Z[ζ_{N}] → C carrying ζ_{N} to ζ.

Remark 2.4. The relative algebraic theory of the Tate curve, as developed in [5, VII], ensures that the q^{1/N}-parameter in Theorem 2.3 coincides with the canonical local coordinate at the cusp ∞ in the analytic theory when we use the identification of completed algebraic and analytic local rings at cusps on a complex fiber of X(N)^{can}. In what follows we will make essential use of this compatibility between the analytic and algebraic theories of q-expansion.

Fix N ≥ 3. There is a very well-known uniformization of the modular curve Y_{ζ}(N) via the analytic map

π_{ζ} : C − R ↠ Y_{ζ}(N)

that sends τ ∈ C − R to the point in Y_{ζ}(N) classifying the pair (E_{τ} , ι_{τ,ζ}), where

(2.1)  E_{τ} = C^{×}/q_{τ}^{Z}, ι_{τ,ζ}(1, 0) = ζ mod q_{τ}^{Z}, ι_{τ,ζ}(0, 1) = q_{τ}^{1/N} mod q_{τ}^{Z}.

There is a functorial left action of SL_{2}(Z/NZ) on X_{ζ}(N) defined by

(2.2)  [γ]_{ζ}(E, ι) = (E, ι ∘ γ′)

with γ′ = (\begin{matrix}d & b \\ c & a \end{matrix}) for γ = (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ SL_{2}(Z/NZ), but the lift of [γ]_{ζ} (via π_{ζ}) to an action on C−R depends

on ζ. On each connected component ℌ of C − R, the lift of [γ]_{ζ} is induced by the standard linear-fractional

action of any γ_{ℌ} ∈ SL_{2}(Z) that lifts (\begin{matrix}a & u_{ℌ}b \\ u_{ℌ}^{−1}c & d \end{matrix}) ∈ SL_{2}(Z/NZ), where ζ = e^{2πi_{ℌ}u_{ℌ}/N} for i_{ℌ} = \sqrt{−1} ∈ ℌ

and u_{ℌ} ∈ (Z/NZ)^{×}. Since u_{−ℌ} = −u_{ℌ}, we see that the action of γ_{−ℌ} on −ℌ is intertwined with the action of γ_{ℌ} on ℌ by means of negation on C − R. Observe also that the analytic action [·]_{ζ} of SL_{2}(Z/NZ) on X_{ζ}(N) arises from an algebraic action of SL_{2}(Z/NZ) on X(N)^{can} over Z[ζ_{N}] that is defined by means of the same moduli-theoretic definition (E, ι) ↦ (E, ι ∘ γ′).

Remark 2.5. For any two primitive Nth roots of unity ζ and ζ′ in C, we may identify \overline{ℳ} _{ζ}(N) and \overline{ℳ} _{ζ′}(N) by composing ι with the automorphism of (Z/NZ)^{2} defined by (n, m) ↦ (en, m) for the unique e ∈ (Z/NZ)^{×} such that ζ′ = ζ^{e}. This sets up an abstract isomorphism α_{ζ′,ζ} : X_{ζ}(N) ≃ X_{ζ′}(N) that carries π_{ζ} to π_{ζ′}. The isomorphism α_{ζ′,ζ} is generally not SL_{2}(Z/NZ)-equivariant: if ζ′ = ζ^{e} for e ∈ (Z/NZ)^{×} then

α_{ζ′,ζ} ∘ [γ]_{ζ} = [γ_{e}^{−1}γγ_{e}]_{ζ′} ∘ α_{ζ′,ζ} with γ_{e} = (\begin{matrix}e & 0 \\ 0 & 1 \end{matrix}). Note that α_{ζ′,ζ} is equivariant for the action of (Z/NZ)^{×}

via the diagonal embedding c ↦ (\begin{matrix}c & 0 \\ 0 & c^{−1} \end{matrix}) into SL_{2}(Z/NZ).

The map π_{ζ} presents Y_{ζ}(N) as the quotient of C − R by the group of automorphisms generated by negation and the standard linear fractional action of the principal congruence subgroup

Γ(N) = ker(SL_{2}(Z) → SL_{2}(Z/NZ)).

The uniformization π_{ζ} gives rise to a canonical cusp ∞_{ζ} ∈ X_{ζ}(N), and the parameter q_{τ}^{1/N} goes over to a canonical local coordinate q_{∞_{ζ}} around ∞_{ζ} ∈ X_{ζ}(N). Under the identification of X_{ζ}(N) with the analytic fiber of X(N)^{can}, ∞_{ζ} arises from ∞ ∈ X(N)^{can}(Z[ζ_{N}]) and q_{∞_{ζ}} corresponds to q^{1/N}.

For N = 5, something remarkable happens: there is another canonical cusp 0_{ζ} (this is not π_{ζ}(0)): we

define 0_{ζ} on X_{ζ}(5) to be the image of ∞_{ζ} under the action of (\begin{matrix}c & 0 \\ 0 & c^{−1} \end{matrix}) ∈ SL_{2}(Z/5Z) on X_{ζ}(5) for a

generator c of (Z/5Z)^{×}; this c is unique up to a sign, and so the choice of c does not matter. Explicitly,

since (\begin{matrix}2 & 5 \\ 5 & 13 \end{matrix}) ∈ SL_{2}(Z) lifts such a matrix (with c = 2), we see that if ℌ is a connected component of

C − R containing i = \sqrt{−1} then for ζ = e^{2πi/5} we may extend π_{ζ}|_{ℌ} by continuity to points of P^{1}(Q) and π_{ζ}(2/5) = 0_{ζ}. In particular, π_{ζ}(0) ≠ 0_{ζ}. When working algebraically with X(5)^{can} over Spec Z[ζ_{5}], we define the cusp 0 ∈ X(5)^{can}(Z[ζ_{5}]) in terms of ∞ exactly as we just defined 0_{ζ} in terms of ∞_{ζ}. The isomorphism α_{ζ′,ζ} : X_{ζ}(N) ≃ X_{ζ′}(N) carries ∞_{ζ} to ∞_{ζ′}, q_{∞_{ζ}} to q_{∞_{ζ′}}, and (for N = 5) 0_{ζ} to 0_{ζ′}.

By means of π_{ζ}, we may identify the meromorphic function field of X_{ζ}(N) with the field of level-N modular functions; i.e., the meromorphic functions on C − R that are invariant under the actions of negation and of Γ(N), and that are meromorphic at the points of P^{1}(Q) when such points are approached through neighborhoods in the horocycle topology on either connected component of C − R. The compact connected Riemann surface X_{ζ}(N) has genus zero for N ≤ 5, and so for such N the meromorphic function field of X_{ζ}(N) is a rational function field C(j_{N,ζ}) where j_{N,ζ} : X_{ζ}(N) ≃ CP^{1} is an isomorphism onto the Riemann sphere. To uniquely define j_{N,ζ} for N ≤ 5, we need to impose some additional normalization conditions at cusps as follows.

For N ≤ 5, we can define the modular function j_{N,ζ} up to an additive constant by requiring that j_{N,ζ}(∞_{ζ}) = [1 : 0] and that the Laurent expansion of j_{N,ζ} at ∞_{ζ} have leading coefficient 1 with respect to the local coordinate q_{∞_{ζ}}. We eliminate the additive constant for N = 5 by demanding j_{5,ζ}(0_{ζ}) = 0. That is,

(2.3)  div(j_{5,ζ}) = (0_{ζ}) − (∞_{ζ}).

For N = 5, the isomorphism α_{ζ′,ζ} : X_{ζ}(5) ≃ X_{ζ′}(5) must therefore carry j_{5,ζ} to j_{5,ζ′}, so j_{5,ζ}∘π_{ζ} is independent of ζ:

Definition 2.6. The common function j_{5,ζ} ∘ π_{ζ} on C − R is denoted j_{5}.

At the end of §4 we will review the proof of the well-known fact that 1/j_{5} is Ramanujan’s continued fraction F.

Our ability to get integral algebraic results for j_{5} will rest on the integral model X(5)^{can} over Z[ζ_{5}], but to get results over Z we require a model over Q. More specifically, we shall now construct a model over Z[1/5]; the equality Z[1/5] ∩ Z[ζ_{5}] = Z will ensure that we are able to get results over Z and not only over Z[ζ_{5}].

Fix N ≥ 3. We shall define a twisted modular curve X_{μ}(N) over Z[1/N] that naturally descends the Z[ζ_{N}, 1/N]-scheme X(N)^{can}; in particular, X_{μ}(N) must be proper and smooth over Spec Z[1/N] with geometrically connected fibers of dimension 1. The scheme X_{μ}(N) is defined to represent the Γ_{μ}(N)-moduli functor that classifies pairs (E, ι) where E is a generalized elliptic curve over a variable Z[1/N]-scheme S and

ι : μ_{N} × Z/NZ → E^{sm}[N]

is an isomorphism of finite étale S-groups such that ι is ample on E and intertwines the Weil pairing on E^{sm}[N] with the evident μ_{N}-valued symplectic form on μ_{N} ×Z/NZ. On the category of Z[ζ_{N}][1/N]-schemes, this moduli functor is naturally isomorphic to the one that is represented by the Z[ζ_{N}]-scheme X(N)^{can} upon inverting N, and hence the existence of X_{μ}(N) is established by using finite étale descent with a suitable action of Gal(Q(ζ_{N})/Q) on X(N)^{can}. We likewise get Y_{μ}(N) by descending the open affine complement Y (N)^{can} of the cuspidal subscheme in X(N)^{can}.

3. Canonical models

For our purposes, we need to know that the modular curves X_{μ}(N)_{Q} over Q are both moduli spaces (as explained above) as well as canonical models in the sense of Shimura (and hence have well-understood Galois-theoretic behavior at CM points). The link with Shimura’s theory rests on a q-expansion principle for rational functions:

Lemma 3.1. Let ζ = e^{±2πi/N} with i ∈ C satisfying i^{2} + 1 = 0. Using π_{ζ} : C − R → Y_{ζ}(N) to identify the meromorphic function field of X_{ζ}(N) with the level-N modular functions on C − R, the function field Q(X_{μ}(N)) consists of the level-N modular functions f such that the q-expansion of f at ∞ (in the parameter q_{τ}^{1/N}) has coefficients in Q.

Proof. The arithmetic theory of the Tate curve provides a canonical isomorphism of complete local rings 𝒪̂_{X_{μ}(N),∞} ≃ Q[[q^{1/N}]]. The analytic theory of the Tate curve and our two choices for ζ ensure that applying Cˆ⊗_{Q}(·) to this isomorphism yields the same calculation of the completed analytic local ring on X_{ζ}(N)(C) at ∞ as is obtained via π_{ζ} and the analytic parameter q_{τ}^{1/N}. Thus, the problem is purely algebraic: we must prove that if f ∈ C(X_{μ}(N)_{C}) then f ∈ Q(X_{μ}(N)) if the image of f in the local field C(X_{μ}(N)_{C})_{∞} ≃ C((q^{1/N})) lies in the subfield Q(X_{μ}(N))_{∞} ≃ Q((q^{1/N})). Such an f is invariant under the action of Aut(C/Q), so we just have to prove that the fixed field of Aut(C/Q) in C(X_{μ}(N)_{C}) is Q(X_{μ}(N)).

More generally, let L/k be an extension of fields and K, E ⊆ L intermediate extensions linearly disjoint over k such that k is perfect, k is algebraically closed in E, and K is algebraically closed (e.g., k = Q, K = C, E = Q(X_{μ}(N)), L = C(X_{μ}(N)_{C})). By linear disjointness, the ring K ⊗_{k} E is a domain and its fraction field is naturally isomorphic to the subfield KE ⊆ L, so KE admits a natural action of Aut(K/k). It is a standard fact in the theory of linearly disjoint extensions that the Aut(K/k)-invariants in KE must be precisely the elements of E, and this gives what we need. ■

To apply this lemma, we use Shimura’s canonical models for modular curves. Fix a connected component ℌ of C−R, and let i = \sqrt{−1} ∈ ℌ. We are now going to apply Shimura’s reciprocity law on canonical models, but we first need to explain why the moduli scheme X_{μ}(N)_{Q} equipped with its canonical isomorphism

X_{μ}(N)(C) ≃ Γ(N)\(ℌ ∪ P^{1}(Q))

is a canonical model in the sense of Shimura. This requires some preliminary notation, as follows.

Let G = GL_{2} and let Z = G_{m} be its center. We write A_{L} to denote the adele ring of a number field L, and A_{L}^{∞} = L ⊗_{Z} Ẑ to denote the ring of finite adeles for L. Let U be an open subgroup of G(A_{Q}) whose projection into G(R) lies in G(R)^{0} = GL_{2}^{+}(R) and such that U contains Z(Q)G(R)^{0} = Q^{×} GL_{2}^{+}(R) and G(A_{Q})/U is compact. Let Γ_{U} = U ∩ G(Q). This is a discrete subgroup in G(R)^{0} that is commensurable with Z(Q) · (G(Z) ∩ G(R)^{0}) = Q^{×} SL_{2}(Z). The analytic quotient Γ_{U}\ℌ is the complement of finitely many points in a unique compact connected Riemann surface X_{Γ_{U}} whose meromorphic function field is the field of modular functions on ℌ that are invariant under Γ_{U} for the standard action of Γ_{U} ⊆ G(R)^{0} on ℌ. By Shimura’s theory of canonical models (see [19, §6.7], especially [19, Prop. 6.27]), X_{Γ_{U}} admits a model X_{U} that is a smooth proper geometrically-connected curve defined over a number field k_{U} ⊆ C such that k_{U}/Q is abelian and, at CM-points of X_{U}, the arithmetic properties of special values of any h ∈ k_{U}(X_{U}) may be read off from Shimura’s reciprocity law [19, 6.31]. Let us see how these generalities work out in a special case.

The subgroups U of most interest to us are

(3.1)  V_{N} = Z(Q)G(R)^{0} · \{g ∈ G(Ẑ) | g ≡ (\begin{matrix}1 & 0 \\ 0 & ∗ \end{matrix}) mod N\} .

By [19, Prop. 6.9(3), Rem. 6.28], X_{V_{N}} is a canonical model for X_{e^{2πi/N}} (N) = X(N)_{C}^{can} via the embedding Z[ζ_{N}] → C that sends ζ_{N} to e^{2πi/N}, and k_{V_{N}} = Q and Q(X_{V_{N}}) is the field of level-N modular functions on ℌ whose q-expansion at ∞ (in the parameter q_{τ}^{1/N}) has coefficients in Q. Thus, by Lemma 3.1, X_{μ}(N)_{Q} equipped with its canonical isomorphism X_{μ}(N)(C) ≃ X_{e^{2πi/N}} (N) is Shimura’s canonical model X_{V_{N}} for X_{e^{2πi/N}} (N).

Let y ∈ X_{μ}(N)(C) be a CM-point (so we may identify y with a closed point, also denoted y, on X_{μ}(N) with residue field Q(y) embedded into C). Choose τ ∈ C − R with π_{e^{2πiτ /N}} (τ) = y, and consider the finite-dimensional Q-algebras K = Q(τ) and A = End_{Q}(K) as ring-schemes over Spec Q. Let ρ : K^{×} → A^{×} be the natural map of algebraic unit-groups induced by the natural ring-scheme map K → A; note that the determinant det ρ : K^{×} → Q^{×} is the norm. Use the ordered Q-basis {τ, 1} of K to identify A with a matrix-algebra over Q, and hence to identify A^{×} with GL_{2} as algebraic groups over Q. On A_{Q}-points, ρ therefore defines a continuous homomorphism

(3.2)  ρ_{τ} : A_{K}^{×} → GL_{2}(A_{Q})

that lands inside GL_{2}^{+}(A_{Q}) = GL_{2}^{+}(R) × GL_{2}(A_{Q}^{∞}) and carries K^{×} into Q^{×} and K_{∞}^{×} into GL_{2}^{+}(R). As a special case of [19, 6.33], we thereby obtain:

Theorem 3.2 (Shimura). Let \overline{Q} ⊆ C be the algebraic closure of Q. For any CM-point y ∈ X_{μ}(N)(\overline{Q}) ⊆ X_{μ}(N)(C) and τ ∈ C − R with π_{e^{2πiτ /N}} (τ) = y, the compositum Q(τ, y) ⊆ C of K = Q(τ) and Q(y) is the finite abelian extension of K that is the class field associated to the open subgroup

(K^{×} · {s ∈ A_{K}^{×} | ρ_{τ} (s) ∈ V_{N}})/K^{×} ⊆ A_{K}^{×}/K^{×},

with V_{N} as in (3.1). In particular, this extension is unramified at finite places away from N[𝒪_{K} : 𝒪_{τ}] where 𝒪_{τ} is the CM-order of C^{×}/q_{τ}^{Z}.

Let us record a useful easy corollary (to be applied with p = 5):

Corollary 3.3. Let τ ∈ C − R be imaginary quadratic, and assume N = p ≥ 3 is prime. Let K = Q(τ), y = π_{e^{2πiτ /p}}(τ) ∈ X_{μ}(N)(\overline{Q}) ⊆ X_{μ}(N)(C), and 𝒪_{τ} = End(C^{×}/q_{τ}^{Z}) ⊆ C the CM order for the fiber of the universal elliptic curve over y.

The finite abelian extension K(y)/K is associated to the open subgroup

K^{×} · (K_{∞}^{×} × U_{τ})/K^{×} ⊆ A_{K}^{×}/K^{×},

where U_{τ} ⊆ ∏_{v∤∞} 𝒪_{K,v}^{×} = ∏_{ℓ∤∞}(Z_{ℓ} ⊗_{Z} 𝒪_{K})^{×} is the group of finite ideles s = (s_{ℓ}) whose ℓ-component s_{ℓ} ∈ (Z_{ℓ} ⊗_{Z} 𝒪_{K})^{×} lies in (Z_{ℓ} ⊗_{Z} 𝒪_{τ})^{×} for ℓ ≠ p and whose p-component s_{p} ∈ (Z_{p} ⊗_{Z} 𝒪_{K})^{×} satisfies

s_{p} ≡ 1 mod \frac{p}{τ} · Z_{p} ⊗_{Z} 𝒪_{τ} , s_{p} ∈ Z_{p}^{×} + p(Z_{p} ⊗_{Z} 𝒪_{τ}).

In particular, K(y)/K is unramified away from p[𝒪_{K} : 𝒪_{τ}] and if τ is p-integral then K(y) is contained in the ray class field of conductor p[𝒪_{K} : 𝒪_{τ}] over K, with equality if moreover 𝒪_{τ} = 𝒪_{K} and τ is a p-unit.

4. Klein Forms and j_{5}

Let ζ be a primitive 5th root of unity in C. Recall that in §2 we defined the rational parameter j_{5,ζ} for X_{ζ}(5) by the requirements that div(j_{5,ζ}) = 0_{ζ} − ∞_{ζ} and that the Laurent expansion of j_{5,ζ} at ∞_{ζ} has leading coefficient 1 with respect to a canonical local coordinate q_{∞_{ζ}}. We also noted that the holomorphic function j_{5} = j_{5,ζ} ∘ π_{ζ} on C − R is independent of ζ. We need to give two explicit formulas for j_{5}, using Klein forms and a q-product; both will be useful in examples (e.g., the description via Klein forms makes it easy to compute the action of SL_{2}(Z/5Z) on j_{5}). Thus, we first shall give a rapid review of the basics of Klein forms, partly to give a convenient reference for formulas and partly to set the notation we shall use.

Let us begin with a review of some basic formulas from the theory of elliptic functions. Let Λ be a lattice in C. Recall that the Weirstrass σ-function on C is

σ(z, Λ) =^{def} z ∏_{ω∈Λ−\{0\}} (1 − \frac{z}{ω} ) e^{z/ω+ \frac{1}{2} (z/ω)^{2}},

and (see [20, I.5]) for any ω ∈ Λ we have

\frac{σ(z + ω, Λ)}{σ(z, Λ)} = ε_{Λ}(ω)e^{η_{Λ}(ω)(z+ω/2)},

where ε_{Λ}(ω) = 1 (resp. ε(ω) = −1) if and only if ω ∈ 2Λ (resp. ω ∈ Λ − 2Λ), and η_{Λ} : Λ → C is the additive quasi-period map that measures the failure of the Weierstrass function −℘_{Λ} to integrate to a Λ-periodic function on C. If we let

Λ_{τ} = Zτ ⊕ Z

for τ ∈ C − R, then (z, τ) ↦ ℘_{Λ_{τ}} (z) is holomorphic, and so τ ↦ η_{Λ_{τ}} (1) and τ ↦ η_{Λ_{τ}} (τ) are holomorphic. Let η(·, Λ) : C = R ⊗_{Z} Λ → C be the R-linear extension of η_{Λ}; this is not holomorphic, but clearly

τ ↦ η(a_{1}τ + a_{2}, Zτ ⊕ Z) = a_{1}η_{Λ_{τ}} (τ) + a_{2}η_{Λ_{τ}} (1)

is holomorphic on C − R for a fixed a = (a_{1}, a_{2}) ∈ R^{2}.

Let Isom_{R}(R^{2}, C) denote the set of R-linear isomorphisms R^{2} ≃ C. For τ ∈ C−R we define W_{τ} : R^{2} ≃ C by W_{τ} (0, 1) = 1 and W_{τ} (1, 0) = τ; these are exactly the W ∈ Isom_{R}(R^{2}, C) such that W (0, 1) = 1, and W_{τ} (Z^{2}) = Λ_{τ}.

Definition 4.1. Choose a ∈ Q^{2} and W ∈ Isom_{R}(R^{2}, C), and let z = W (a) ∈ C and Λ = W (Z^{2}). The Klein form κ_{a} : Isom_{R}(R^{2}, C) → C is

κ_{a}(W) =^{def} e^{−η(W (a),W (Z^{2}))·W (a)/2}σ(W (a), W (Z^{2})) = e^{−η(z,Λ)z/2}σ(z, Λ). (4.1)

We shall write κ_{a}(τ) to denote κ_{a}(W_{τ}), so τ ↦ κ_{a}(τ) is holomorphic on C − R. The transformation law κ_{(a_{1},a_{2})}(−τ) = −κ_{(a_{1},−a_{2})}(τ) is immediate from the definitions, and this lets us pass between connected components of C − R when working with Klein forms.

The Klein forms satisfy several additional well-known obvious properties that we shall use, and so for ease of reference we summarize the properties that we will need:

• They are homogenous of degree 1:

κ_{a}(λW) = λκ_{a}(W) (4.2)

for any λ ∈ C^{×}. This allows us to systematically pass between κ_{a}(τ)’s and κ_{a}(W)’s.

• For a ∈ Q^{2}, W ∈ Isom_{R}(R^{2}, C), and b = (b_{1}, b_{2}) ∈ Z^{2}, we have

κ_{a+b}(W) = ϵ_{W} (a, b)κ_{a}(W)

for an explicit root of unity ϵ_{W} (a, b) (determined by Legendre’s period relation for η_{Λ}). If W = W_{τ} and a = (a_{1}, a_{2}) then

κ_{a+b}(τ) = ϵ_{τ} (a, b)κ_{a}(τ) (4.3)

for b = (b_{1}, b_{2}) ∈ Z^{2}, where

ϵ_{τ} (a, b) = (−1)^{b_{1}+b_{2}+b_{1}b_{2}} exp(πi_{τ} (a_{1}b_{2} − b_{1}a_{2})) (4.4)

with i_{τ}^{2} = −1 and i_{τ} in the connected component of τ in C − R.

• For γ ∈ Aut(Z^{2}) = GL_{2}(Z), we have κ_{a}(W ∘ γ) = κ_{γ(a)}(W). An equivalent formulation is

κ_{a}(γ(τ)) = j(γ, τ)κ_{γ(a)}(τ), (4.5)

where j(γ, τ) = uτ + v is the standard automorphy factor for γ = ( \begin{matrix}r & s \\ u & v \end{matrix} ).
• Fix τ ∈ C − R and a = (a_{1}, a_{2}) ∈ Q^{2}. Define q = e^{2πi_{τ} z} with z = W_{τ} (a) ∈ C. We have

κ_{a}(τ) = −\frac{q_{τ}^{(1/2)(a_{1}^{2}−a_{1})}}{2πi_{τ}} e^{πi_{τ} a_{2}(a_{1}−1)}(1 − q) ∏_{n=1}^{∞} \frac{(1 − q_{τ}^{n}q)(1 − q_{τ}^{n}/q)}{(1 − q_{τ}^{n})^{2}}. (4.6)

This follows from the product formula in [14, §18.2, Thm. 4].

We will need to explicitly construct level-N modular functions for N = 5, and this is most easily done by using Klein forms and the criterion in:

Lemma 4.2. Fix a positive integer N ≥ 1 and a finite subset 𝒜 ⊆ (N^{−1}Z)^{2}. Let m : 𝒜 → Z be a function. Define the mermorphic function

f = ∏_{α∈𝒜} κ_{a}^{m(a)}

on C − R. Let Q : (NZ)^{2} → Z be the quadratic form

Q(x) = ∑_{a∈𝒜} m(a)⟨a, x⟩^{2},

where ⟨·, ·⟩ : (N^{−1}Z)^{2} × (NZ)^{2} → Z is the evident duality pairing.

The function f is an automorphic form on Γ(N) of weight − ∑_{a∈𝒜} m(a) if and only if Q vanishes modulo N/ gcd(N, 2). If 𝒜 is stable under (a_{1}, a_{2}) ↦ (a_{1}, −a_{2}), then f is invariant under τ ↦ −τ if m(a_{1}, a_{2}) = m(a_{1}, −a_{2}) for all a = (a_{1}, a_{2}) ∈ 𝒜.

Proof. The vanishing of the quadratic form amounts to a vanishing condition on coefficients, and the equivalence of f being an automorphic form on Γ(N) and such vanishing of coefficients is proved in [13, p. 68]. The final part concerning invariance under τ ↦ −τ is trivial. ■

Theorem 4.3. On C − R, we have

(4.7)  j_{5} = \frac{κ_{( \frac{2}{5} ,0)}κ_{( \frac{2}{5} , \frac{1}{5} )}κ_{( \frac{2}{5} , \frac{2}{5} )}κ_{( \frac{2}{5} ,− \frac{2}{5} )}κ_{( \frac{2}{5} ,− \frac{1}{5} )}}{κ_{( \frac{1}{5} ,0)}κ_{( \frac{1}{5} , \frac{1}{5} )}κ_{( \frac{1}{5} , \frac{2}{5} )}κ_{( \frac{1}{5} ,− \frac{2}{5} )}κ_{( \frac{1}{5} ,− \frac{1}{5} )}}

and

(4.8)  j_{5}(τ) = q_{τ}^{−1/5} ∏_{n=1}^{∞} \frac{(1 − q_{τ}^{5n−2})(1 − q_{τ}^{5n−3})}{(1 − q_{τ}^{5n−4})(1 − q_{τ}^{5n−1})}.

Proof. Let us define 𝒥_{5} to be the ratio of products of Klein forms in (4.7); we will prove that 𝒥_{5} satisfies the properties that uniquely characterize j_{5}. The invariance 𝒥_{5}(−τ) = 𝒥_{5}(τ) follows from the identity κ_{(a_{1},a_{2})}(−τ) = κ_{(a_{1},−a_{2})}(τ). The product formula (4.8) for 𝒥_{5} follows easily from (4.6). By Lemma 4.2 with N = 5, taking

𝒜 = {(ℓ/5, k/5) : −2 ≤ k ≤ 2, ℓ = 1, 2}, m(ℓ/5, k/5) = (−1)^{ℓ},

we deduce that 𝒥_{5} is a level-5 modular function. The canonical surjection π_{ζ} : C − R ↠ Y_{ζ}(5) lets us consider 𝒥_{5} as a meromorphic function 𝒥_{5,ζ} on X_{ζ}(5). By construction, we have 𝒥_{5,ζ} ∘ π_{ζ} = 𝒥_{5}, and so the isomorphism α_{ζ′,ζ} : X_{ζ}(5) ≃ X_{ζ′}(5) as in Remark 2.5 carries 𝒥_{5,ζ} to 𝒥_{5,ζ′}.

The product expansion (4.8) makes it clear that 𝒥_{5,ζ} has no poles on Y_{ζ}(5). Thus, the only possible poles are at the cusps. From the q-expansion (4.8) we see that

ord_{∞_{ζ}} 𝒥_{5,ζ} = −1

and that the q-expansion of 𝒥_{5,ζ} at ∞_{ζ} begins with 1/q_{∞_{ζ}} = 1/q^{1/5}. It remains to show that 𝒥_{5,ζ} has no poles at the other cusps, and that its unique zero is at the cusp 0_{ζ}. Moreover, it is enough to consider a single ζ. We shall choose a connected component ℌ of C − R and work with ζ = e^{2πi/5} and τ ∈ ℌ where i ∈ ℌ is the unique point satisfying i^{2} = −1; this choice yields the standard formulas on ℌ when (via π_{ζ}) we lift the action of SL_{2}(Z/5Z) on X_{ζ}(5).

The transformation formulas for Klein forms under SL_{2}(Z) make it easy to express SL_{2}(Z/5Z)-conjugates of 𝒥_{5} in terms of Klein forms. In this way, the transitivity of SL_{2}(Z/5Z) on the 12 cusps of X_{ζ}(5) enables us to compute the q-expansion of 𝒥_{5} at each cusp. This computation shows that 𝒥_{5} has no poles at cusps away from ∞_{ζ} and it only vanishes at the cusp π_{ζ}(2/5) for ζ = e^{2πi/5}. In §2 we showed π_{ζ}(2/5) = 0_{ζ} for any ζ. ■

Using the well-known Rogers–Ramanujan identities, it can be shown [17, p. 155] that

F (τ) = q_{τ}^{1/5} ∏_{n=1}^{∞} \frac{(1 − q_{τ}^{5n−1})(1 − q_{τ}^{5n−4})}{(1 − q_{τ}^{5n−2})(1 − q_{τ}^{5n−3})}.

By (4.8), this proves the identity F = 1/j_{5}. Unfortunately, we do not know a more direct way to verify that Ramanujan’s F is a (level-5) modular function.

5. Application of arithmetic models

By construction, j_{5} induces an isomorphism of Riemann surfaces j_{5,ζ} : X_{ζ}(5) ≃ CP^{1} for each primitive 5th root of unity ζ in C. By Lemma 3.1, this descends to an algebraic isomorphism j_{5,μ} : X_{μ}(5)_{Q} ≃ P_{Q}^{1}. Since X_{μ}(5) is a proper smooth Z[1/5]-scheme with geometrically connected fibers of genus 0, the isomorphism j_{5,μ} must uniquely extend to an isomorphism X_{μ}(5) ≃ P_{Z[1/5]}^{1} over Z[1/5] that we also denote j_{5,μ}. To deduce arithmetic properties of j_{5} over Z, it is essential to remove the denominators at 5 in this latter isomorphism. For this purpose, we will now work over Z[ζ_{5}] by using Theorem 2.2 with N = 5.

Let X denote the Z[ζ_{5}]-scheme X(5)^{can}, and let j_{X} ∈ Q(ζ_{5})(X) be the rational function obtained from j_{5,μ} by extension of scalars (or by descent of any j_{5,ζ} over C). All fibers of X over Spec Z[ζ_{5}] are geometrically connected (by Stein factorization), as the C-fiber is connected.

The Z[ζ_{5}]-scheme X is smooth (and hence regular) near the cusps, so the ideal sheaf of the section ∞ is invertible, and hence the inverse sheaf 𝒪(∞) makes sense on X. The key to integrality results for j_{5} is:

Lemma 5.1. The rational function j_{X} on the Z[ζ_{5}]-scheme X is a regular funtion on X −{∞}, and {j_{X}, 1} is a pair of generating sections of the line bundle 𝒪(∞) on X.

Proof. The q-expansion of j_{5} at ∞ has integral coefficients and a simple pole with leading coefficient 1, so the rational function j_{X} on X induces a local generator of 𝒪(∞) along ∞ ∈ X(Z[ζ_{5}]). It therefore remains to show that j_{X} is a regular function on X − {∞}. Since X − {∞} is a connected normal noetherian scheme, it suffices to check that j_{X} is defined in codimension ≤ 1 on X − {∞}. The situation along the generic fiber is clear via the analytic theory (see (2.3)), and so we only need to study j_{X} at the generic points of the geometrically connected fiber X_{s} of X over each closed point s = 𝔪 ∈ Spec Z[ζ_{5}]. Note that if s is not the unique point of residue characteristic 5, then the connected X_{s} is smooth and hence irreducible. Since ∞ ∈ X(Z[ζ_{5}]) is supported in the smooth locus of X over Spec Z[ζ_{5}], the section ∞_{s} ∈ X_{s}(F_{5}) in the unique fiber X_{s} of characteristic 5 lies in a unique irreducible component of X_{s}.

Fix a choice of closed point s ∈ Spec Z[ζ_{5}]. The local ring 𝒪_{X,η} at each generic point η of X_{s} is a discrete valuation ring with uniformizer given by a local parameter in Z[ζ_{5}]_{𝔪}. Thus, the integral structure of the q-expansion of j_{5} at ∞ ensures that for every closed point s of Spec Z[ζ_{5}], j_{X} is a local unit at the generic point of the unique irreducible component of X_{s} that contains ∞_{s}. It remains to work near the generic points η of the unique fiber in characteristic 5 such that \overline{\{η\}} does not meet ∞. It suffices to find a section σ_{η} ∈ X(Z[ζ_{5}]) supported in the smooth locus and passing through the chosen mod-5 component \overline{\{η\}} such that j_{X} lies in the coordinate ring of the formal completion of X along σ_{η}.

Finiteness of X over the usual j-line implies that \overline{\{η\}} must contain some cusp, so it suffices to check that the q-expansion of j_{5} at each cusp has coefficients in Z[ζ_{5}]. This is a purely analytic problem on X_{ζ}(5) for any primitive 5th root of unity ζ ∈ C, and it suffices to consider a single choice of ζ. We choose ζ = e^{2πi/5} for i = \sqrt{−1} in a chosen connected component ℌ of C − R, as this makes the action of SL_{2}(Z) on X_{ζ}(5) lift (via π_{ζ}) to the standard action on ℌ via linear fractional transformations. The action of SL_{2}(Z) on X_{ζ}(5) is transitive on the set of cusps, so it suffices to prove that the coefficients of the q-expansion of j_{5} ∘ γ at ∞ lies in Z[ζ] for all γ ∈ SL_{2}(Z). By (4.7) and (4.8), this desired integrality follows immediately from the product formula (4.6) and the transformation law (4.5) for the standard action of SL_{2}(Z) on C − R. ■

Let (𝒪(1); s_{0}, s_{1}) be the universal line bundle on P^{1} equipped with an ordered pair of generating sections s_{0} and s_{1}. By Lemma 5.1 and the universal property of the projective line, there is a unique morphism

J_{5} : X(5)^{can} → P_{Z[ζ_{5}]}^{1}

over Z[ζ_{5}] such that there is an isomorphism J_{5}^{∗}(𝒪(1)) ≃ 𝒪(∞) carrying J_{5}^{∗}(s_{0}) to j_{X} and J_{5}^{∗}(s_{1}) to 1. In particular, J_{5}^{−1}([1 : 0]) = ∞ as subschemes of X(5)^{can}.

Theorem 5.2. The map J_{5} is an isomorphism over Z[ζ_{5}][1/5], and on the unique characteristic-5 fiber it contracts all irreducible components except for the unique fibral irreducible component C_{∞} containing ∞. The reduced irreducible component C_{∞} contains the cusp 0 as its only other cusp and it is mapped isomorphically onto P_{F_{5}}^{1} under J_{5}. The other 5 irreducible components of the mod-(1 − ζ_{5}) fiber of X(5)^{can} map onto a common point in F_{5}^{×} ∈ P^{1}(F_{5}).

Remark 5.3. One immediate consequence of this theorem is the numerical fact that the q-expansion of j_{5,ζ} at each cusp other than ∞_{ζ} and 0_{ζ} has all higher-degree coefficients divisible by 1 − ζ_{5}. Another immediate consequence is that both the zero and polar schemes of J_{5} lie entirely in the cuspidal subscheme. This second consequence generalizes (2.3).

Proof. By passing to the complex-analytic fiber relative to an embedding Z[ζ_{5}] → C defined by some primitive 5th root of unity ζ ∈ C, the map J_{5} induces the map j_{5,ζ} : X_{ζ}(5) ≃ CP^{1}. Thus, the analytic isomorphism property for j_{5,ζ} over C implies that J_{5} is an algebraic isomorphism on Q(ζ_{5})-fibers, and so by properness the map J_{5} is surjective. If we work over Z[ζ_{5}][1/5], then J_{5} is therefore a surjective birational map between proper smooth curves over Z[ζ_{5}][1/5], and these curves have (geometrically) connected fibers. Thus, after inverting 5 we see that the proper J_{5} must become quasi-finite and hence finite, so (by normality) it is an isomorphism over Z[ζ_{5}][1/5].

There is a unique supersingular j-value in characteristic 5, so [10, 13.2.2] implies that the mod-(1−ζ_{5}) fiber X_{F_{5}} of X = X(5)^{can} is reduced and is scheme-theoretically constructed by gluing some P_{F_{5}}^{1}’s transversally at a single F_{5}-point. It follows from [10, 13.8.4] that there are exactly 6 = #P^{1}(F_{5}) irreducible components in X_{F_{5}} and that these are in natural bijection with the lines in (Z/5Z)^{2} in such a way that the action of SL_{2}(Z/5Z) on X(5)^{can} is compatible with the natural transitive action of SL_{2}(Z/5Z) on the set of such lines. In particular, this action on X(5)^{can} is transitive on the set of irreducible components of X_{F_{5}}, so there are 12/6 = 2 cusps on each component. The geometric points (E, ι) of X(5)^{can} satisfying ι(1, 0) = 0 define one of these components. This component contains the cusps ∞ and 0, and so it is C_{∞}.

Let us show that the map J_{5} : C_{∞} → P_{F_{5}}^{1} between smooth proper connected curves is an isomorphism when C_{∞} is given its reduced structure. Since J_{5}^{−1}([1 : 0]) = ∞ with ramification degree e_{∞|[1:0]} = 1 (due to the structure of the q-expansion of j_{5} at ∞), we conclude that the map C_{∞} → P_{F_{5}}^{1} between integral proper curves is birational with generic degree 1. Thus, it is an isomorphism.

Let C be an irreducible component of X_{F_{5}} distinct from C_{∞}. The map J_{5} : C → P_{F_{5}}^{1} is not surjective (it cannot hit [1 : 0]), so J_{5}(C) is a single closed point in A_{F_{5}}^{1}. Since all C’s pass through a common F_{5}-point in X_{F_{5}} (namely, a supersingular point), it follows that the J_{5}(C)’s for C ≠ C_{∞} are equal to a common point in A^{1}(F_{5}). It remains to show that J_{5}(C) ≠ 0 for C ≠ C_{∞}.

Let γ = (\begin{matrix}c & 0 \\ 0 & c^{−1} \end{matrix}) ∈ SL_{2}(Z/5Z) with c = ±2, so [γ](∞) = 0 by definition. Hence, [γ] must preserve C_{∞}.

It follows that via the contraction map J_{5}, the involution [γ] of X(5)^{can} is intertwined with an involution of P_{Z[ζ_{5}]}^{1} that switches [1 : 0] and [0 : 1]. Such an involution must be t ↦ u/t for u ∈ Z[ζ_{5}]^{×}. That is, J_{5} ∘ [γ] = u/J_{5}. It follows that J_{5}^{−1}([0 : 1]) = [γ](J_{5}^{−1}([1 : 0])) = {0}. This is disjoint from all C ≠ C_{∞}, so J_{5}(C) ≠ 0 for C ≠ C_{∞}. ■

Remark 5.4. For irreducible components C of X_{F_{5}} distinct from C_{∞}, the common point J_{5}(C) ∈ F_{5}^{×} is equal to −2. To verify this fact (which we will not use), it suffices to compute j_{5,ζ} on X_{ζ}(5) at any cusp other than 0_{ζ} and ∞_{ζ} (for a single choice of ζ). In Table B.1 we list of values of j_{5,ζ} at the cusps when using ζ = e^{2πi/5} and working with τ in the connected component of C − R that contains i = \sqrt{−1}. By inspection of the table we see that these values have reduction −2 in F_{5} except at the cusps 0_{ζ} and ∞_{ζ}.

We have noted in Remark 5.3 that the zero and polar schemes for J_{5} are equal to the cuspidal sections 0 and ∞ respectively. These two sections are switched by the involution w of X(5)^{can} that is induced by

± (\begin{matrix}2 & 0 \\ 0 & 2^{−1} \end{matrix}) ∈ SL_{2}(Z/5Z), and a key fact is that J_{5} intertwines w with an involution of P_{Z[ζ_{5}]}^{1}:

Corollary 5.5. The identity J_{5} ∘ w = −1/J_{5} holds.

Proof. By Theorem 5.2 and Remark 5.3, J_{5} ∘ w = u/J_{5} for some u ∈ Z[ζ_{5}]^{×}. Just as X(5)^{can} and J_{5} descend to X_{μ}(5) and j_{5,μ} over Z[1/5], it is clear that the action on X(5)^{can} by diagonal matrices in SL_{2}(Z/5Z) also descends over Z[1/5]. In particular, w descends, and so u ∈ Q. Thus, u = ±1. To compute u, we use analysis

as follows. For the lift γ̃ = (\begin{matrix}2 & 5 \\ 5 & 13 \end{matrix}) ∈ SL_{2}(Z) of (\begin{matrix}2 & 0 \\ 0 & 2^{−1} \end{matrix}) ∈ SL_{2}(Z/5Z), we must have j_{5} ∘ γ̃ = u/j_{5}.

However, via (4.6) and (4.7), it is a simple analytic calculation with the transformation law (4.5) to check that j_{5} ∘ γ̃ has q-expansion with initial term −q_{τ}^{1/5}. Thus, u = −1. ■

We now obtain a geometric proof of [1, Thm. 6.2]:

Corollary 5.6. If τ ∈ C−R is quadratic over Q, then F (τ) is an algebraic integral unit; equivalently, j_{5}(τ) is an algebraic integral unit.

Proof. Fix a primitive 5th root of unity ζ ∈ C, and use this to consider C as a Z[ζ_{5}]-algebra. The point τ maps to a point x_{τ} = π_{ζ}(τ) ∈ X_{ζ}(5) = X(5)^{can}(C), and j_{5}(τ) = J_{5}(x_{τ}). Since τ is imaginary quadratic, C^{×}/q_{τ}^{Z} is a CM elliptic curve. Thus, x_{τ} must be an algebraic point. Let K ⊆ C be a number field containing Q(ζ) such that x_{τ} is a K-point of X(5)^{can}, and hence J_{5}(x_{τ}) ∈ P^{1}(K) ⊆ P^{1}(C). We wish to investigate the properties of J_{5}(x_{τ}).

The ring of integers 𝒪_{K} of K is Dedekind, so we may use the valuative criterion for properness to uniquely extend x_{τ} to a map x̃_{τ} : Spec 𝒪_{K} → X(5)^{can}. We want to prove that J_{5}(x̃_{τ}) ∈ P^{1}(𝒪_{K}) is disjoint from the sections [0 : 1] and [1 : 0] in P^{1}. By construction J_{5}^{−1}([1 : 0]) = {∞}. By Corollary 5.5, J_{5}^{−1}([0 : 1]) = {0}. Thus, to prove that j_{5}(τ) is an algebraic integral unit it suffices to prove that (the image of) x̃_{τ} is disjoint from the cuspidal subscheme. This disjointness is clear, since specialization into a cusp forces multiplicative reduction, yet CM elliptic curves have potentially good reduction at all places. ■

We would like to determine the field Q(τ, j_{5}(τ)) ⊆ C for any imaginary quadratic τ ∈ C − R. Since Q(X_{μ}(5)) = Q(j_{5,μ}) and j_{5,μ} = j_{5,ζ} ∘ π_{ζ} for ζ = e^{±2πi/5}, Theorem 3.2 and Corollary 3.3 yield:

Corollary 5.7. Let τ ∈ C − R be quadratic over Q. Let K = Q(τ), and let ρ_{τ} : A_{K}^{×} → GL_{2}(A_{Q}) be the canonical representation in (3.2). Let 𝒪_{τ} ⊆ C be the CM-order of C^{×}/q_{τ}^{Z}, and let

V = Q^{×} · \{g = (g_{∞}, g^{∞}) ∈ GL_{2}(R) × GL_{2}(A_{Q}^{∞}) | det g_{∞} > 0, g^{∞} ∈ GL_{2}(Ẑ), g^{∞} ≡ (\begin{matrix}1 & 0 \\ 0 & ∗ \end{matrix}) mod 5\} .

The extension K(j_{5}(τ))/K is the abelian extension of K with associated open subgroup

(5.1)  (K^{×} · {s ∈ A_{K}^{×} | ρ_{τ} (s) ∈ V })/K^{×} ⊆ A_{K}^{×}/K^{×}.

In particular, K(j_{5}(τ))/K is unramified away from 5[𝒪_{K} : 𝒪_{τ}].

If τ is 5-integral then K(j_{5}(τ)) is contained in the ray class field of conductor 5[𝒪_{K} : 𝒪_{τ}] for K, and K(j_{5}(τ)) is equal to this ray class field if moreover 𝒪_{τ} = 𝒪_{K} and τ is a 5-unit.

6. Modular Equations

In this section we use the integral models from §2 and our work in §5 to show that for any positive integer n relatively prime to 5, j_{5}(τ) and j_{5}(nτ) satisfy a “modular equation” F_{n}(j_{5}(τ), j_{5}(nτ)) = 0 for a suitable primitive F_{n} ∈ Z[X, Y] that is absolutely irreducible over Q. We then establish a Kronecker congruence for such F_{n}’s and we analyze the case n = 5. This is motivated by the algebro-geometric treatment of the classical modular polynomials, except that there are two complications: X_{ζ}(5) has more than one cusp, and we need to use two moduli schemes to get a result over Z.

The modular polynomials F_{n} are going to be constructed by geometric methods, as this provides the clearest way to analyze the properties of F_{n} (such as absolute irreducibility over Q and congruences modulo p). This construction rests on the interaction of Γ_{μ}(N)-structures and Γ_{0}(n)-structures for N = 5 and gcd(n, N) = 1, so let us begin by defining a moduli functor that mixes both kinds of structures. For N ≥ 3, consider the enhanced moduli functor ℳ that classifies triples (E, ι, C) where E is an elliptic curve, ι is a Γ_{μ}(N)-structure, and C ↪ E is a Γ_{0}(n)-structure in the sense of [10, §3.4] (so if n is a unit on the base then C is an order-n finite étale subgroup of E such that C is cyclic on geometric fibers). The Γ_{0}(n)-moduli problem on elliptic curves is relatively representable and finite flat [10, §4.5, 6.6.1]; applying this to the universal object over Y_{μ}(N) yields the existence of a fine moduli scheme Y (Γ_{μ}(N), Γ_{0}(n)) for ℳ on the category of Z[1/N]-schemes, with a forgetful map

π_{n} : Y (Γ_{μ}(N), Γ_{0}(n)) → Y (Γ_{μ}(N)) = Y_{μ}(N)

that is finite and flat. By [10, 6.6.1], Y (Γ_{μ}(N), Γ_{0}(n)) is an affine curve over Spec Z[1/N] and it is regular.

The finite flat j-maps to A_{Z[1/N]}^{1} allow us to define compactifications X(Γ_{μ}(N), Γ_{0}(n)) and X_{μ}(N) by normalizing P_{Z[1/N]}^{1} in the function fields of the affine normal modular curves Y (Γ_{μ}(N), Γ_{0}(n)) and Y_{μ}(N) (this recovers the same X_{μ}(N) that we defined in §2 via moduli-theoretic methods). Thus, we get a natural finite map

\overline{π}_{n} : X(Γ_{μ}(N), Γ_{0}(n)) → X_{μ}(N)

between normal proper Z[1/N]-curves, and over the open subscheme Y_{μ}(N) ⊆ X_{μ}(N) this recovers π_{n}. Since gcd(n, N) = 1, there is a natural automorphism w_{n} on Y (Γ_{μ}(N), Γ_{0}(n)) whose action on moduli is

w_{n} : (E, ι, C) ↦ (E/C, ι′, E[n]/C)

with ι′ defined as the composite isomorphism

ι′ : μ_{N} × Z/NZ →^{n^{−1}×1} μ_{N} × Z/NZ →^{ι} E[N] ≃ (E/C)[N],

where the intervention of n^{−1} in the first step ensures that ι′ is a symplectic isomorphism. Observe that we have a commutative diagram

(6.1)  \xymatrix{Y (Γ_{μ}(N), Γ_{0}(n)) \ar[r]^{w_{n}^{2}} \ar[d]_{π_{n}} & Y (Γ_{μ}(N), Γ_{0}(n)) \ar[d]^{π_{n}} \\ Y_{μ}(N) \ar[r]_{[δ_{n}]} & Y_{μ}(N)}

with

(6.2)  δ_{n} = (\begin{matrix}n & 0 \\ 0 & n^{−1} \end{matrix}) ∈ SL_{2}(Z/NZ).

In terms of moduli, the finite flat covering π′_{n} = π_{n} ∘ w_{n} sends (E, ι, C) to (E/C, ι′), where ι′ is defined as above. Since the cuspidal locus is quasi-finite over the base Spec Z[1/N], the involution w_{n} of the normal Z[1/N]-curve Y (Γ_{μ}(N), Γ_{0}(n)) uniquely extends to an involution \overline{w}_{n} of the normal proper Z[1/N]-curve X(Γ_{μ}(N), Γ_{0}(n)). Thus, we may define the finite map \overline{π}′_{n} = \overline{π}_{n} ∘ \overline{w}_{n} extending π′_{n}. The compatibility (6.1) extends to compactified modular curves in the evident manner. The maps \overline{π}_{n} and \overline{π}′_{n} are analogues of the classical degeneracy maps for N = 1.

Consider the proper morphism

\overline{π}_{n} × \overline{π}′_{n} : X(Γ_{μ}(N), Γ_{0}(n)) → X_{μ}(N) ×_{Z[1/N]} X_{μ}(N).

This map is quasi-finite, since there are at most finitely many cyclic n-isogenies between a pair of elliptic curves over an algebraically closed field, and so it is finite. It is generically injective (since a generic pair of elliptic curves admit at most one cyclic isogeny of a given degree, up to sign), and hence it is generically a closed immersion because the irreducible X(Γ_{μ}(N), Γ_{0}(n)) has generic characteristic zero. Thus, the regular X(Γ_{μ}(N), Γ_{0}(n)) maps birationally onto its image under \overline{π}_{n} × \overline{π}′_{n}, but this image generally has singularities.

Definition 6.1. The Kroneckerian model Z_{N,n} of X(Γ_{μ}(N), Γ_{0}(n)) is the scheme-theoretic image of the finite map \overline{π}_{n} × \overline{π}′_{n}.

Since X_{μ}(N) has a moduli-theoretic interpretation for N ≥ 3, the rigidity of generalized elliptic curves (i.e., the triviality of the deformation theory of morphisms) ensures that the map X(Γ_{μ}(N), Γ_{0}(n)) → Z_{N,n} is formally unramified, and so Z_{N,n} has much milder singularities than in the classical case N = 1.

The Krockerian model Z_{N,n} is a proper flat Z[1/N]-scheme that is reduced. The map X(Γ_{μ}(N), Γ_{0}(n))_{Q} → Z_{N,n,Q} between integral proper Q-curves expresses X(Γ_{μ}(N), Γ_{0}(n))_{Q} as the normalization of Z_{N,n,Q}. In particular, Z_{N,n,Q} is geometrically irreducible, and hence is geometrically connected. The fibers of the proper flat map Z_{N,n} → Spec Z[1/N] are therefore geometrically connected curves (and are reducible in characteristics dividing n). Motivated by the classical case N = 1, we will construct the modular polynomials F_{n} for j_{5} by studying Z_{5,n,Q} as an irreducible curve on the surface

X_{μ}(5)_{Q} × X_{μ}(5)_{Q} = P_{Q}^{1} × P_{Q}^{1}.

Lemma 6.2. For N ≥ 3 and gcd(N, n) = 1, the following properties hold.

(1) The projections Z_{N,n,Q} ⇉ X_{μ}(N)_{Q} are finite with generic degree [Γ(1) : Γ_{0}(n)] = n ∏_{p|n}(1 + 1/p).

(2) Let σ be the involution of X_{μ}(N) × X_{μ}(N) that switches the factors. Using (6.2), the self-map σ ∘ (1 × [δ_{n^{−1}}]) on X_{μ}(N) × X_{μ}(N) restricts to the identity on Z_{N,n}. In particular, if n ≡ ±1 mod N then σ acts as the identity on Z_{N,n}.

Proof. Since X(Γ_{μ}(N), Γ_{0}(n))_{Q} → Z_{N,n,Q} is the normalization, and so is a finite birational isomorphism, (1) follows from the fact that \overline{π}_{n} and \overline{π}′_{n} = \overline{π}_{n} ∘ \overline{w}_{n} have common degree equal to the degree of the Γ_{0}(n)-moduli problem: [Γ(1) : Γ_{0}(n)].

To establish (2), we first note that Z_{N,n} is equal to the scheme-theoretic image of (\overline{π}_{n} × \overline{π}′_{n}) ∘ ϕ for any automorphism ϕ of X(Γ_{μ}(N), Γ_{0}(n)). Taking ϕ = \overline{w}_{n}, we see that Z_{N,n} is the scheme-theoretic image of

(\overline{π}_{n} × \overline{π}′_{n}) ∘ \overline{w}_{n} = \overline{π}′_{n} × (\overline{π}_{n} ∘ \overline{w}_{n}^{2}) = \overline{π}′_{n} × ([δ_{n}] ∘ \overline{π}_{n}) = (1 × [δ_{n}]) ∘ (\overline{π}′_{n} × \overline{π}_{n}).

Thus, if σ denotes the involution of X_{μ}(N) × X_{μ}(N) that switches the factors, then σ ∘ (1 × [δ_{n^{−1}}]) is the identity on Z_{N,n}. ■

Now specialize to the case N = 5. By Theorem 5.2, j_{5,μ} defines an isomorphism X_{μ}(5) ≃ P_{Z[1/5]}^{1} carrying ∞ to [1 : 0]. Thus, for any n relatively prime to 5 we may consider the Kroneckerian model Z_{5,n} as a reduced closed subscheme in P_{Z[1/5]}^{1} × P_{Z[1/5]}^{1} that is proper and flat over Z[1/5] with geometrically connected fibers of dimension 1, and with geometrically integral generic fiber over Q. Let 𝒵_{n} be the closure of Z_{5,n} in P_{Z}^{1} × P_{Z}^{1}, so this is a proper flat curve over Spec Z. In particular, it must be the closure of its irreducible generic fiber Z_{5,n,Q} ⊆ P_{Q}^{1} × P_{Q}^{1}. By Lemma 6.2(1), the projections

Z_{5,n,Q} ⇉ P_{Q}^{1}

are finite with generic degree equal to deg π_{n}. Thus, Z_{5,n,Q} ⊆ P_{Q}^{1} × P_{Q}^{1} is the zero-scheme of an irreducible bihomogeneous absolutely irreducible polynomial F̃_{n}(X_{0}, X_{1}; Y_{0}, Y_{1}) with degree deg π_{n} in both the X’s and the Y ’s. Moreover, both X_{0} and Y_{0} must arise in F̃_{n} (since Z_{N,n,Q} meets Y_{μ}(N)_{Q} × Y_{μ}(N)_{Q}) and so by irreducibility it follows that the dehomogenization F̃_{n}(X, 1; Y, 1) has degree deg π_{n} in each of X and Y.

Let F_{n}(X, Y) be a primitive polynomial over Z that is a scalar multiple of F̃_{n}(X, 1; Y, 1) (this determines F_{n} up to sign), so F_{n} ∈ Z[X, Y] is geometrically irreducible over Q with degree deg π_{n} in each of X and Y. We claim that the modular equation

F_{n}(j_{5}(τ), j_{5}(nτ)) = 0

holds. In the complex-analytic theory, the map π′_{n} corresponds to τ ↦ nτ, and hence the functions j_{5,μ} ∘ \overline{π}_{n} and j_{5,μ} ∘\overline{π}′_{n} on X(Γ_{μ}(N), Γ_{0}(n)) correspond to the functions j_{5}(τ) and j_{5}(nτ) on C−R. Thus, the modular equation follows from the definition of F_{n}. To remove the sign ambiguity in the definition of F_{n}, we first must prove:

Theorem 6.3. There are unique monomial terms X^{r_{n}}Y^{deg π_{n}} and Y^{s_{n}}X^{deg π_{n}} in F_{n} with respective Y -degree and X-degree deg π_{n}, and both occur in F_{n} with coefficient in Z^{×} = {±1}.

Proof. We may (and do) work over Z[ζ_{5}], and we may work with the Γ(5)^{can}-moduli functor instead of the Γ_{μ}(5)-moduli functor. Consider the finite flat maps

\overline{π}_{n}, \overline{π}′_{n} : X(Γ(5)^{can}, Γ_{0}(n)) ⇉ X(5)^{can}

over Spec Z[ζ_{5}] and the automorphism \overline{w}_{n} of X(Γ(5)^{can}, Γ_{0}(n)); these are defined just as we define the maps \overline{π}_{n}, \overline{w}_{n}, and \overline{π}′_{n} over Z[1/5].

The key fact is that \overline{w}_{n} preserves \overline{π}_{n}^{−1}(\{0, ∞\}). To verify this property, we first note that the cuspidal subscheme X(5)_{∞}^{can} is a disjoint union of copies of Spec Z[ζ_{5}], and \overline{π}_{n}^{−1}(X(5)_{∞}^{can}) is the cuspidal subscheme of X(Γ(5)^{can}, Γ_{0}(n)); also, \overline{w}_{n} restricts to an automorphism of the cuspidal subscheme. Since \overline{π}_{n} is a finite flat surjection, it is therefore enough to verify that \overline{w}_{n} preserves \overline{π}_{n}^{−1}(\{0, ∞\}) on fibers over a single point of Spec Z[ζ_{5}]. We will work at the unique point of characteristic 5.

As we noted in the proof of Theorem 5.2, the fiber X(5)_{F_{5}}^{can} consists of 6 copies of P_{F_{5}}^{1} glued transversally at a unique (supersingular) F_{5}-point, and there are exactly 2 cusps on each of these irreducible components, with one of the irreducible components C_{∞} containing ∞ and 0 as its cusps. This component C_{∞} is characterized by the property that its geometric points (E, ι) have ι(1, 0) = 0.

We similarly find (using the methods as in the proof of [10, 13.7.6]) that X(Γ(5)^{can}, Γ_{0}(n))_{F_{5}} is a reduced curve with 6 irreducible components that are glued at supersingular points, and that the finite flat projection

\overline{π}_{n} : X(Γ(5)^{can}, Γ_{0}(n)) → X(5)^{can}

sets up a bijection between these 6 irreducible components and the 6 irreducible components of X(5)_{F_{5}}^{can}. Let C′_{∞} be the irreducible component of X(Γ(5)^{can}, Γ_{0}(n))_{F_{5}} whose non-cuspidal geometric points (E, ι, C) have ι(1, 0) = 0, so C′_{∞} = \overline{π}_{n}^{−1}(C_{∞}). An inspection of the definition of w_{n} shows that \overline{w}_{n} preserves the vanishing property for the 5-torsion point ι(1, 0), so \overline{w}_{n} carries C′_{∞} to itself. Thus, \overline{w}_{n} preserves \overline{π}_{n}^{−1}(\{0, ∞\}) in characteristic 5, and hence over Z[ζ_{5}].

By Theorem 5.2, j_{5,ζ_{5}} identifies P_{Z[ζ_{5}]}^{1} with the contraction of X(5)^{can} along the mod-(1−ζ_{5}) components

distinct from C_{∞}. Let X̃ denote the normal proper flat Z[ζ_{5}]-curve obtained by contracting X(Γ(5)^{can}, Γ_{0}(n)) along the mod-(1 − ζ_{5}) components distinct from C′_{∞} (see [3, 6.7/3] for the existence of such a contraction, using some connected components of the cuspidal divisor to construct the divisor in the hypothesis in [3, 6.7/3]). Since the automorphism \overline{w}_{n} preserves C′_{∞}, it uniquely factors through the contraction to define an automorphism w̃_{n} of X̃. Likewise, \overline{π}_{n} uniquely factors through the contraction to define a proper surjective map π̃_{n} : X̃ → P_{Z[ζ_{5}]}^{1} that must be quasi-finite, and hence finite, as well as flat (since it is a finite map from a normal surface to a regular surface; see [16, 23.1]). We define π′_{n} = π̃_{n} ∘ w̃_{n}, and let U be the common preimage of G_{m} = P^{1} − {[0 : 1], [1 : 0]} under π̃_{n} and π′_{n}.

Using the structure map π̃_{n}, all regular functions on U are integral over over Z[ζ_{5}][j_{5}, 1/j_{5}]. Using the structure map π′_{n}, all regular functions on U are integral over Z[ζ_{5}][j_{5}(nτ), 1/j_{5}(nτ)], where we adopt the usual abuse of notation by writing j_{5}(nτ) to denote the function τ ↦ j_{5}(nτ). We conclude that the irreducible minimal monic polynomial for the function j_{5}(τ) over the field Q(j_{5}(nτ)) has coefficients in Z[ζ_{5}][j_{5}(nτ), 1/j_{5}(nτ)], and similarly with the roles of the functions j_{5}(τ) and j_{5}(nτ) reversed. These minimal polynomials are exactly the Q(Y)^{×}- and Q(X)^{×}-multiples of F_{n}(X, Y) that are respectively monic in X and monic in Y, so it follows that the irreducible F_{n}(X, Y) ∈ (Z[X])[Y] has its dominant Y -term (with Y -degree deg π_{n}) with a coefficient that is a unit in Z[X, 1/X]. That is, there is a unique monomial term X^{r_{n}}Y^{deg π_{n}} in F_{n} and its Z-coefficient is a unit; by the same argument, there is a unique monomial term Y^{s_{n}}X^{deg π_{n}} in F_{n} with X-degree deg π_{n} and its Z-coefficient is a unit. ■

The two unit coefficients in Theorem 6.3 need not be equal. We now make an arbitrary choice (that is meaningful because of Theorem 6.3):

Definition 6.4. The primitive polynomial F_{n} ∈ Z[X, Y] is the unique one that is absolutely irreducible over Q, satisfies F_{n}(j_{5}(τ), j_{5}(nτ)) = 0, and has its unique monomial X^{r_{n}}Y^{deg π_{n}} with maximal Y -degree occur with coefficient equal to 1.

An important symmetry property of F_{n} is:

Theorem 6.5. Let n > 1 be an integer. If n ≡ ±1 mod 5 then F_{n}(Y, X) = F_{n}(X, Y). If n ≡ ±2 mod 5 then X^{deg π_{n}}F_{n}(Y, −1/X) = ε_{n}F_{n}(X, Y) with ε_{n} = ±1, where ε_{n} = −1 if and only if one of the following mutually exclusive conditions holds:

(1) n = 2m^{2} with m ≥ 1 and the odd part m_{odd} satisfying m_{odd} ≡ ±1 mod 5 when m is odd, m_{odd} ≡ ±2 mod 5 when m is even and not a power of 2, and m = 4^{a} with a ≥ 1 when m > 1 and m is a power of 2,

(2) n = 4p^{a} with a ≡ 3 mod 4 and an odd prime p ≡ ±2 mod 5,

(3) n = 4^{2b+1}p^{a}m^{2} with odd m ≥ 1 and an odd prime p ∤ m such that a ≡ 1 mod 4, p ≡ ±2 mod 5, and m > 1 or b > 0.

Remark 6.6. The least n > 2 such that n ≡ ±2 mod 5 and ε_{n} = −1 is n = 32; the second-smallest such n is n = 72. Note that the sign ε_{n} is invariant under replacing F_{n} with −F_{n}, and so it is independent of our arbitrary choice of sign-convention in the definition of F_{n}.

Proof. Since [δ_{n}] as in (6.1) acts on P_{Q}^{1} = X_{μ}(5)_{Q} via the identity for n ≡ ±1 mod 5 and via t ↦ −1/t for n ≡ ±2 mod 5 (by Corollary 5.5), Lemma 6.2(2) implies that the zero schemes of F_{n}(Y, X) and F_{n}(X, Y) (say over Q) coincide for n ≡ ±1 mod 5, and the zero schemes of X^{deg π_{n}}F_{n}(Y, −1/X) and F_{n}(X, Y) (say over Q) coincide for n ≡ ±2 mod 5. Thus, this gives the result up to a Q^{×}-multiple. By primitivity, there is in fact only an ambiguity of sign. We cannot have F_{n}(X, Y) = −F_{n}(Y, X), as then F_{n}(X, X) = 0 and this contradicts the fact that the irreducible F_{n}(X, Y) cannot be divisible by X − Y (F_{n} has degree deg π_{n} in each variable, and deg π_{n} > 1 since n > 1).

It remains to compute the sign ε_{n} such that X^{deg π_{n}}F_{n}(Y, −1/X) = ε_{n}F_{n}(X, Y) for n ≡ ±2 mod 5. Recall that we defined F_{n} by the property that the unique monomial in F_{n} with Y -degree deg π_{n} has coefficient of 1. That is, F_{n} modulo multiplication by X^{Z} is equivalent to a monic polynomial in Y in Z[X, 1/X][Y], and this monic polynomial agrees with ε_{n}F_{n}(Y, −1/X) up to multiplication by X^{Z}. The unique monomial-term in F_{n}(X, Y) with X-degree deg π_{n} has the form u_{n}Y^{s_{n}}X^{deg π_{n}} for some u_{n} ∈ {±1}, so F_{n}(Y, −1/X) has its top-degree Y -term equal to (−1)^{s_{n}}u_{n}Y^{deg π_{n}} modulo X^{Z}. Thus, ε_{n} = (−1)^{s_{n}}u_{n}. Since j_{5}(τ) = q_{τ}^{−1/5} + …, if we write F_{n}(j_{5}, Y) mod^{×}j_{5}^{Z} as a Y -monic element in Z[j_{5}, Y] ⊆ Z((q_{τ}^{1/5}))[Y] then the unique monomial (q_{τ}^{−1/5})^{α_{n}}Y^{β_{n}} with the highest q-pole has Y -degree s_{n} and coefficient u_{n} = (−1)^{s_{n}}ε_{n} (note that generally α_{n} ≠ deg π_{n} because we have multiplied F (j_{5}, Y) by a power of j_{5} to make it monic in Y). We shall use this viewpoint to compute ε_{n} by studying F_{n}(j_{5}, Y) over Q(ζ)((q_{τ}^{1/5})).

Let Δ′_{n} = Γ(5) (\begin{matrix}n & 0 \\ 0 & 1 \end{matrix}) Γ(5), so Γ(5)\Δ′_{n} has size deg π_{n} = n ∏_{p|n}(1 + 1/p). Since gcd(n, 5) = 1, modulo

multiplication by j_{5}^{Z} we have

(6.3)  F_{n}(j_{5}, Y) ≡ ∏_{α∈Γ(5)\Δ′_{n}}(Y − j_{5} ∘ α) mod^{×}j_{5}^{Z}

(where we are considering multiplicative congruence). To analyze j_{5} ∘ α, we need to find a convenient set of coset representatives for Γ(5)\Δ′_{n}. To compute such a set, recall (as in [19, Prop. 3.36] in much greater generality) that a set of representatives for Γ(5)\{α ∈ End_{Z}(Z^{2}) | det α = n} is

(6.4)  \{σ_{a} · (\begin{matrix}a & 5b \\ 0 & d \end{matrix}) | ad = n, 0 ≤ b < d, a > 0\} ,

where σ_{a} ∈ SL_{2}(Z) depends only on a mod 5 and is a lift of (\begin{matrix}a^{−1} & 0 \\ 0 & a \end{matrix}) ∈ SL_{2}(Z/5Z). For example, for

definitiveness we can take

(6.5)  σ_{±1} = ± (\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}) , σ_{±2} = ± (\begin{matrix}2 & 5 \\ 5 & 13 \end{matrix}) .

We are interested in only the subset Γ(5)\Δ′_{n}, and obviously the representatives for such left-cosets must be primitive matrices and so a set of representatives for Γ(5)\Δ′_{n} is contained in

(6.6)  \{σ_{a} · (\begin{matrix}a & 5b \\ 0 & d \end{matrix}) | ad = n, 0 ≤ b < d, a > 0, gcd(a, b, d) = 1\};

an elementary count (via decomposition along the prime factorization of n) shows that the set (6.6) has size n · ∏_{p|n}(1 + 1/p), and so (6.6) is a set of representatives for Γ(5)\Δ′_{n}. Note that the preceding calculations are valid for any n relatively prime to 5 without requiring n ≡ ±2 mod 5 (and so we may and will use the above conclusions in our later analysis of the size of the coefficients of F_{n} for any large n).

For α_{a,b,d} = σ_{a} (\begin{matrix}a & 5b \\ 0 & d \end{matrix}) in the set (6.4), we have

j_{5} ∘ α_{a,b,d} = (j_{5} ∘ σ_{a})((aτ + 5b)/d)

with j_{5} ∘ σ_{a} only depending on ±a mod 5. Since a/d > 0 and

−j_{5} ∘ σ_{±1} = −q_{τ}^{−1/5} + …, −j_{5} ∘ σ_{±2} = q_{τ}^{1/5} + …,

we see that −j_{5} ∘ α_{a,b,d} has a pole if and only if a ≡ ±1 mod 5. Thus, there is a unique set of α_{a,b,d}’s in (6.6) such that the product of the corresponding −j_{5} ∘ α_{a,b,d}’s has a maximal-order pole, namely the product for those α_{a,b,d}’s with a ≡ ±1 mod 5.

We now restrict attention to α_{a,b,d}’s with gcd(a, b, d) = 1; these are the α_{a,b,d}’s in the set (6.6). For a fixed factorization n = ad with a > 0 and a ≡ ±1 mod 5, we have

(6.7)  ∏_{b∈B_{d}} −j_{5} ∘ α_{a,b,d} = ∏_{b∈B_{d}} −(e^{−2πi(aτ+5b)/d} + …) = (−1)^{|B_{d}|}ζ_{d}^{∑_{b∈B_{d}} −5b}q_{τ}^{−a/d} + …,

with B_{d} = {b | 0 ≤ b < d, (b, n/d, d) = 1}. To compute the power of ζ_{d} in (6.7), we apply the following lemma to the primitive dth root of unity ζ_{d}^{5}:

Lemma 6.7. For positive integers d and k with k|d, and ζ_{d} a primitive dth root of unity,

ζ_{d}^{∑_{0≤b<d,(b,k)=1} −b} = (−1)^{−1+d/k+δ_{k,2}}

where δ_{k,2} = 1 for k = 2 and δ_{k,2} = 0 otherwise.

Proof. The case k = 1 is straightforward, so we may assume k > 1. Thus, for 0 ≤ b < d we may use the division algorithm to write b = kq + r with 0 ≤ q < d/k and 0 ≤ r < k with (k, r) = 1 (or more loosely, r ∈ (Z/kZ)^{×}). Since each such r shows up d/k times, if we define ζ_{k} = ζ_{d}^{d/k} and ζ_{d/k} = ζ_{d}^{k} then

ζ_{d}^{∑_{0≤b<d,(b,k)=1} −b} = ζ_{d/k}^{∑_{0≤q<d/k} q} · ζ_{k}^{∑_{r∈(Z/kZ)^{×}} r} = (−1)^{d/k−1} · ζ_{k}^{∑_{r∈(Z/kZ)^{×}} r} = (−1)^{d/k−1}(−1)^{φ(k)}Φ_{k}(0)

where Φ_{k}(0) is the constant term of the kth cyclotomic polynomial Φ_{k} that has degree φ(k). Since Φ_{k} = ∏_{e|k}(X^{e} − 1)^{μ(k/e)}, we have Φ_{k}(0) = (−1)^{∑_{e|k} μ(k/e)} = 1 since k > 1. Since φ(k) is even for k > 2 whereas φ(2) = 1, we are done. ■

By the lemma, for a|n and d = n/a with a > 0 and a ≡ ±1 mod 5, we get

(6.8)  ∏_{0≤b<d,(b,(a,d))=1} −j_{5} ∘ α_{a,b,d} = (−1)^{|B_{d}|−1+d/(a,d)+δ_{(a,d),2}}q_{τ}^{−a/d} + ….

The product of the terms in (6.8) over all a’s gives the unique most polar contribution in the coefficients of F_{n}, so the product of the sign-coefficients on the right side of 6.8 over all factorizations n = ad with a > 0 and a ≡ ±1 mod 5 is

(6.9)  ∏_{d|n,d≡±2 mod 5}(−1)^{d/(n/d,d)+δ_{(n/d,d),2}} = u_{n}

(with the product taken over positive divisors), and this dominant term arises from a product of t_{n} distinct j_{5} ∘ α’s with

(6.10)  t_{n} = ∑_{d|n,d≡±2 mod 5} |B_{d}|

(with the sum taken over positive divisors), so the most polar term occurs against Y^{deg π_{n}−t_{n}} in (6.3). Hence, s_{n} = deg π_{n} − t_{n}. The case n = 2 is trivial, so we may assume n > 2, and so

deg π_{n} = [Γ(1) : Γ_{0}(n)] = n ∏_{p|n}(1 + 1/p)

is even. Thus, s_{n} ≡ t_{n} mod 2, and so (−1)^{s_{n}} = (−1)^{t_{n}}. Using (6.9) and (6.10), we thereby get (6.11)

ε_{n} = (−1)^{s_{n}}u_{n} = (−1)^{t_{n}}u_{n} = ∏_{d|n,d≡±2 mod 5}(−1)^{−1+d/(n/d,d)+δ_{(n/d,d),2}} = ∏_{d|n} (\frac{d}{5})^{−1+d/(n/d,d)+δ_{(n/d,d),2}} .

The exponent −1 + d/(n/d, d) + δ_{(n/d,d),2} is even when d is odd, so ε_{n} = 1 for odd n. This shows that if ε_{n} = −1 then n must be even; we will proceed by analyzing the cases ord_{2}(n) = 1, ord_{2}(n) = 2, and ord_{2}(n) > 2.

Now we may suppose n = 2^{e}n′ with odd n′ relatively prime to 5 and arbitrary e ≥ 1. If e = 1, then n′ ≡ ±1 mod 5 and n′ > 1 (since n > 2). The contribution at d = 2d′ in (6.11) is (d|5) and hence pairing up the contributions at 2d′ and 2n′/d′ (except if d′ = n′/d′) gives a product of (n′|5) = 1 for each such pair of terms. Thus, the total product ε_{n} is equal to 1 if n′ is not a square and is equal to (m|5) if n′ = m^{2}. This shows that if ord_{2}(n) = 1 and n > 2 then ε_{n} = −1 if and only if n = 2m^{2} with m ≡ ±1 mod 5, as desired.

If e = 2, then n′ ≡ ±2 mod 5 (so n′ is not a square) and the only possible non-trivial contributions to the product for ε_{n} in (6.11) are (d|5) = −(d′|5) at d = 2d′ with (d′, n′/d′) = 1 and (d|5) = (d′|5) at d = 4d′ without restriction on the divisor d′ of n′. For the product of the first collection of such terms (at all d = 2d′ with (d′, n′/d′) = 1) we can pair up 2d′ and 2n′/d′ to get (n′|5) = −1 appearing half as many times as there are subsets of the set of prime factors of n′; any such power set has even size since n′ > 1 (as then n′ has a non-empty set of prime factors), and has size divisible by 4 if and only if n′ has at least two prime factors. Thus, the divisors d = 2d′ with (d′, n′/d′) = 1 contribute a total product of −1 to ε_{n} in (6.11) when n′ > 1 is an odd prime power (with the prime necessarily ≡ ±2 mod 5), and otherwise these divisors contribute a total product of 1 to ε_{n}; similarly, the total contribution to ε_{n} for the set of divisors d = 4d′ is

∏_{d′|n′} (\frac{d′}{5} ) = (−1)^{σ_{0}(n′)/2}.

Thus, when n = 4n′ with odd n′ we see that ε_{n} = −1 if and only if one of the following mutually exclusive conditions holds:

• n = 4p^{a} for an odd prime p ≡ ±2 mod 5 and a ≡ 3 mod 4 (this ensures that n ≡ ±2 mod 5 and that −(−1)^{σ_{0}(p^{a})/2} = (−1)^{(a+3)/2} is equal to −1),

• n = 4p^{a}m^{2} with an odd prime p ≡ ±2 mod 5 and an odd m > 1 not divisible by p such that a ≡ 1 mod 4 (this ensures that n ≡ ±2 mod 5 and that (−1)^{σ_{0}(p^{a}m^{2})/2} = (−1)^{(a+1)/2} is equal to −1).

It remains to consider n = 2^{e}n′ with n′ odd and e ≥ 3. Writing an even divisor d of n in the form 2^{e′}d′ with e′ ≥ 1 and d′|n′, we get a contribution of (d|5) at d in (6.11) in each of the following mutually exclusive cases:

• (d, n/d) = 1 (that is, e′ = e and (d′, n′/d′) = 1),
• (d, n/d) = 2 with d/2 odd (that is, e′ = 1 and (d′, n′/d′) = 1),

• (d, n/d) > 2 with d/(n/d, d) even (that is, e/2 < e′ ≤ e − 1 with the extra condition (d′, n′/d′) ≠ 1

when e′ = e − 1).

Note in particular that terms with e/2 < e′ < e − 1 (vacuous for e = 3) contribute (d|5) in (6.11) without restriction on d′. All other d’s not mentioned in the preceding list make trivial contribution to ε_{n} (due to evenness on the exponent of (d|5) in (6.11) for such d).

We first consider the case when e is even (so e ≥ 4), so n′ is odd and n′ ≡ ±2 mod 5 (so n′ is a nonsquare, and hence σ_{0}(n′) is even and n′ > 1). For each d′|n′, we get a contribution of (2d′|5) = −(d′|5) at terms d = 2d′ in (6.11) if (d′, n′/d′) = 1 and we get a contribution of (2^{e−1}d′|5) = −(d′|5) at terms d = 2^{e−1}d′ in (6.11) if (d′, n′/d′) ≠ 1. Thus, the combined contributions in (6.11) for e′ = 1 and e′ = e − 1 is the product of −(d′|5)’s with each divisor d′ of n′ appearing exactly once. Since n′ is a nonsquare, the number of such divisors is even (namely, σ_{0}(n′)) and we may pair up d′ with n′/d′ (contributing a product of (n′|5) = −1) to arrive at a total product of (−1)^{σ_{0}(n′)/2} by the terms with e′ = 1 and e′ = e − 1. At the terms d = 2^{e′}d′ with e/2 < e′ < e − 1 and e′ = e the contribution in (6.11) is (d|5) = (−1)^{e′}(d′|5) without restriction on d′, so taking the product of these contributions with fixed e′ gives (−1)^{σ_{0}(n′)/2} since the sign (−1)^{e′} appears an even number of times. Hence, we conclude that

ε_{n} = (−1)^{(e/2)σ_{0}(n′)/2}

when e = ord_{2}(n) is even. This is equal to −1 if and only if e/2 is odd and σ_{0}(n′)/2 is odd, and this gives rise to exactly the listed cases when ord_{2}(n) is both even and larger than 2.

Finally, suppose e > 2 is odd, so n′ ≡ ±1 mod 5. Since now (n′|5) = 1, a computation as for the case of even e shows that the total contribution of all terms d = 2^{e′}d′ in (6.11) with a fixed e′ satisfying (e + 1)/2 ≤ e′ < e − 1 or e′ = e is 1 when n′ is not a square and is (−1)^{e′}(m|5) when n′ = m^{2}. There are (e − 1)/2 such terms, with total product 1 when n′ is not a square and total product (−1)^{e}(−1)^{(e−3)/2}(m|5)^{(e−1)/2} = (−(m|5))^{(e−1)/2} when n′ = m^{2}; when n′ = 1 this is (−1)^{(e−1)/2}. We also have to account for contributions in (6.11) at terms d = 2^{e′}d′ with e′ = 1 and e′ = e − 1. For e′ = 1 we get contributions (2d′|5) = −(d′|5) when (d′, n′/d′) = 1, and for e′ = e − 1 we get contributions (2^{e−1}d′|5) = (d′|5) when (d′, n′/d′) ≠ 1. In particular, when n′ = 1 we find that ε_{n} = (−1)^{(e+1)/2}; this gives the asserted list of possibilities when n is a power of 2 that is divisible by 8. Assuming now that n′ > 1 (i.e., n is not a power of 2), so n′ has a non-empty set of prime factors, the number of divisors d′|n′ with (d′, n′/d′) = 1 is even. Thus, the product of the contributions for e′ = 1 and e′ = e − 1 is equal to ∏_{d′|n′}(d′|5), and by pairing d′ with n′/d′ (when d′ ≠ n′/d′) and recalling that (n′|5) = 1 we see that this product is 1 when n′ is a nonsquare and is (m|5) when n′ = m^{2}. Thus, when n′ > 1 is a nonsquare we get ε_{n} = 1 and when n′ = m^{2} > 1 we get ε_{n} = −(−(m|5))^{(e+1)/2}. Setting this equal to −1 gives the elements on the desired list of possibilities that have not yet been obtained. ■

We next wish to establish an analogue of Kronecker’s congruence.

Theorem 6.8 (Kronecker’s congruence). Let p ≠ 5 be prime. If p ≡ ±1 mod 5 then

(6.12)  F_{p}(X, Y) ≡ (Y − X^{p})(Y^{p} − X) mod pZ[X, Y],

and if p ≡ ±2 mod 5 then

(6.13)  F_{p}(X, Y) ≡ (Y − X^{p})(XY^{p} + 1) mod pZ[X, Y].

These congruences are sensitive to the arbitrary choice of sign in Definition 6.4.

Proof. Since the map π_{p} has degree p + 1, by Theorem 6.3 we know a priori that F_{p} mod p has degree p + 1 in each of X and Y. Thus, if we can prove that the right sides of (6.12) and (6.13) divide F_{p} mod p then degree considerations force the congruence up to a nonzero scalar c_{p} ∈ F_{p}^{×}. Since F_{p} ∈ Z[X, Y] has a unique monomial term X^{r_{p}}Y^{p+1} and this term appears in F_{p} with coefficient equal to 1, the same holds under reduction modulo p (with the same r_{p}), and so the Y -monicity of the right sides of (6.12) and (6.13) shows that such a nonzero scalar ambiguity c_{p} ∈ F_{p}^{×} is necessarily trivial. That is, it suffices to prove that F_{p} mod p is divisible by the right sides of (6.12) and (6.13).

The symmetry properties of F_{p} in Theorem 6.5 reduce the divisibility claim to the assertion that Y − X^{p} divides F_{p} mod p. That is, we claim that one of the irreducible components of Spec F_{p}[X, Y]/(F_{p}) is defined by the vanishing of Y − X^{p}. We will construct such an irreducible component by studying irreducible components of the mod-p fiber of Y (Γ_{μ}(5), Γ_{0}(p)).

By the definition of F_{p}, the quasi-finite Kronecker map

j_{5,μ} × (j_{5,μ} ∘ w_{p}) : Y (Γ_{μ}(5), Γ_{0}(p)) → A_{Z[1/5]}^{2}

over Spec Z[1/5] factors through the zero-scheme of F_{p}(X, Y). Passing to mod-p fibers, the quasi-finite map

j_{5,μ} × (j_{5,μ} ∘ w_{p}) : Y (Γ_{μ}(5), Γ_{0}(p))_{F_{p}} → A_{F_{p}}^{2}

factors through Spec F_{p}[X, Y]/(F_{p}). Thus, to prove that F_{p} mod p is divisible by Y − X^{p}, it is enough to find an irreducible component Y′ of Y (Γ_{μ}(5), Γ_{0}(p))_{F_{p}} such that j_{5,μ}(E, ι)^{p} = j_{5,μ}(E/C, ι′) for geometric points y = (E, ι, C) of Y′, where w_{p}(y) = (E/C, ι′, E[p]/C); we may also ignore finitely many points y (such as supersingular points). There are two irreducible components of Y (Γ_{μ}(5), Γ_{0}(p))_{F_{p}}, corresponding to C being étale or multiplicative. Let Y′ be the component corresponding to multiplicative C, so for ordinary geometric points y ∈ Y′ we see that there is an isomorphism E/C ≃ E^{(p)} carrying the p-isogeny E → E/C to the relative Frobenius morphism E → E^{(p)}. If P = ι(1, 0) and Q = ι(0, 1) in the finite étale group E[5], then ι′(1, 0) = p^{−1}P^{(p)} and ι′(0, 1) = Q^{(p)} in E^{(p)}.

Let k be the field over which the geometric point y lives. By the definition of the Γ_{μ}(5)-moduli functor, the point w_{p}(y) = (E^{(p)}; p^{−1}P^{(p)}, Q^{(p)}) ∈ Y_{μ}(5)(k) is the image of y ∈ Y_{μ}(5)(k) under the Frobenius morphism of the F_{p}-scheme Y_{μ}(5). Since the isomorphism j_{5,μ} : Y_{μ}(5)_{F_{p}} ≃ P_{F_{p}}^{1} commutes with absolute Frobenius morphisms (as do all maps of F_{p}-schemes), and the absolute Frobenius morphism on P_{F_{p}}^{1} is given by t ↦ t^{p} in terms of the standard coordinate on P^{1}, we conclude that j_{5,μ}(w_{p}(y)) = j_{5,μ}(y)^{p} as desired. ■

To construct an analogue of F_{n} for all n, the key issue is to formulate a good enhanced moduli functor that allows n to be divisible by 5. We will explain the case n = 5, because in this case the polynomial F_{5} relating j_{5}(τ) and j_{5}(5τ) has a very special form. The general case is left to the interested reader (and will not be used later).

Theorem 6.9. There is a unique rational function h ∈ Q(x) such that j_{5}(τ/5)^{5} = h(j_{5}(τ)), and h is not a polynomial.

This theorem says that we may find an F_{5} with the form h_{1}(Y)X^{5}−h_{2}(Y) for relatively prime h_{1}, h_{2} ∈ Z[Y] with deg h_{1} > 0. Such explicit h_{1} and h_{2} are provided by a famous identity of Ramanujan [24, Theorem 3.2], but we prefer to suppress the explicit description of h = h_{2}/h_{1} until we have provided a geometric proof of its existence.

Proof. We begin by constructing a geometrically-connected finite cover of Y_{μ}(5)_{Q} that replaces the role of Y (Γ_{μ}(N), Γ_{0}(n)) in our study of F_{n} for n relatively prime to 5. Let us work more generally at the outset with any N ≥ 3 instead of with N = 5. Over the Q-scheme Y_{μ}(N) there is a universal elliptic curve E → Y_{μ}(N) equipped with an isomorphism

ι : μ_{N} × (Z/NZ) ≃ E[N]

as Y_{μ}(N)-groups. Let Q = ι(1, 1) ∈ E[N](Y_{μ}(N)), so the fiber [N]^{−1}(Q) → Y_{μ}(N) is an E[N]-torsor over Y_{μ}(N). Via ι, [N]^{−1}(Q) acquires a structure of torsor over μ_{N} × (Z/NZ). Passing to the quotient by the action of Z/NZ defines a μ_{N}-torsor Y′_{μ}(N) → Y_{μ}(N). This torsor is geometrically connected over Q because it is dominated by the geometrically-connected cover Y_{μ}(N^{2}). Our moduli-theoretic construction of Y′_{μ}(N) extends across the cusps, so we get a geometrically-connected proper smooth curve X′_{μ}(N) over Spec Q that is a μ_{N}-torsor over X_{μ}(N).

We shall need to carry out some calculations with analytic models for modular curves. Thus, we choose a connected component ℌ of C − R and (without loss of generality) take ζ = e^{2πi/N} with i = \sqrt{−1} ∈ ℌ. We will use only τ ∈ ℌ because the action of SL_{2}(Z) on X_{ζ}(N) lifts via π_{ζ} to the usual action on ℌ.

Define Γ^{0}(m) ⊆ Γ(1) to be the preimage of the subgroup of matrices in SL_{2}(Z/mZ) with vanishing upperright corner. The normal subgroup Γ′(N) = Γ^{0}(N^{2}) ∩ Γ(N) in Γ(N) has the property that the quotient Γ(N)/Γ′ is cyclic of order N, and this quotient of Γ(N)/Γ(N^{2}) is naturally identified with the geometric Galois group of the covering X′_{μ}(N) → X_{μ}(N). In terms of analytic models, C(X′_{μ}(N)) is identified with the field of level-N^{2} modular functions that are Γ′(N)-invariant, and a generator of the Galois group of C(X′_{μ}(N)) over C(X_{μ}(N)) is represented by the action induced by τ ↦ τ + N on ℌ. In particular, since

(\begin{matrix}1 & 0 \\ 0 & N \end{matrix}) Γ′(N) (\begin{matrix}1 & 0 \\ 0 & N \end{matrix})^{−1} ⊆ Γ(N),

for any f ∈ C(X_{μ}(N)) the function g_{f}(τ) =^{def} (f ∘ π_{ζ})(τ/N) lies in C(X′_{μ}(N)) and a generator of the Galois group of C(X′_{μ}(N)) over C(X_{μ}(N)) acts on g_{f} via the operation τ ↦ τ + N.

Let us specialize these considerations to the case N = 5. We conclude that the function g(τ) = j_{5}(τ/5) lies in a degree-5 extension K = Q(X′_{μ}(5)) of Q(X_{μ}(5)) = Q(j_{5}) with Q algebraically closed in K, and that K(ζ_{5})/Q(ζ_{5}, j_{5}) a Galois extension such that the Galois-orbit of g consists of the elements ζg for ζ ∈ μ_{5}. Thus, g^{5} is Galois-invariant, and so

g^{5} ∈ K ∩ Q(ζ_{5}, j_{5}) = Q(j_{5}).

This produces the desired h ∈ Q(x), except for the fact that h cannot be a polynomial. That is, we must prove that g^{5} ∉ Q[j_{5}]. Assuming otherwise, it would follow that g only has poles at cusps on X′_{μ}(5)_{Q} lying over the unique point ∞ ∈ X_{μ}(5)_{Q} where j_{5} has a pole. This is a contradiction since g(τ) = j_{5}(τ/5) has a pole at the cusp 1/2 that is not Γ(5)-equivalent to ∞. ■

It is natural to want to make the rational function h in Theorem 6.9 explicit. Ramanujan discovered the answer:

(6.14)  h(x) = x · \frac{x^{4} + 3x^{3} + 4x^{2} + 2x + 1}{x^{4} − 2x^{3} + 4x^{2} − 3x + 1}.

In Example B.3 in Appendix B we will give an easy derivation of this identity by using the definition of j_{5} in terms of Klein forms; h could also be determined by pole- and cusp-chasing via Table B.1.

7. Estimates on coefficients of F_{n}

Since the classical modular polynomials Φ_{n}(X, Y) are notoriously impossible to compute, due to the gigantic height of these polynomials, one might expect the F_{n}’s to be equally useless in practice. Remarkably, this is not so: an inspection of the table of F_{n}’s in Appendix C shows that (for gcd(n, 5) = 1) the coefficients are rather small!

Computation of F_{n} for larger values of n, however, reveals that the coefficients of F_{n} do grow quickly with n. Nonetheless, a comparison of the coefficients of F_{n} and the coefficients of the classical modular polynomial Φ_{n} of level n shows that the coefficients of F_{n} are dramatically smaller than those of Φ_{n}. This has the practical consequence that for moderate n we can work computationally with F_{n}.

In this section we use the methods of Cohen and Rademacher (that estimate the coefficients of the Φ_{n}’s and the q-expansion coefficients of j) to estimate the q-expansion coefficients of j_{5} and to prove that F_{n} has small coefficients (with respect to n) in comparison to Φ_{n} when gcd(n, 5) = 1.

For any nonzero P = ∑ a_{I}z^{I} ∈ C[z_{1}, …, z_{m}], we define the logarithmic height

h(P) = log(max_{I} |a_{I}|).

Observe that for positive c ∈ R^{×}, h(cP) = log(c) + h(P). In this section, we will prove

Theorem 7.1. For (n, 5) = 1,

h(F_{n}) = \frac{1}{10}[Γ(1) : Γ_{0}(n)] (log n − 2 ∑_{p|n} \frac{log p}{p} + O(1))

as n → ∞.

Before we prove Theorem 7.1, let us record a corollary that compares the coefficients of F_{n} against coefficients of Φ_{n} for n relatively prime to 5.

Corollary 7.2. For any congruence subgroup Γ ⊆ SL_{2}(Z), let \overline{Γ} denote the image of Γ in SL_{2}(Z)/\{±1\}. We have

lim_{\begin{matrix}n→∞ \\ (n,5)=1 \end{matrix}} \frac{h(Φ_{n})}{h(F_{n})} = [\overline{Γ}(1) : \overline{Γ}(5)] = 60.

Proof. This follows at once from Theorem 7.1 and the asymptotic formula for h(Φ_{n}) analogous to Theorem 7.1 as given in the main theorem in [4]. ■

Remark 7.3. Since h(F_{n}) is the logarithmic height, Corollary 7.2 shows that the largest coefficient (in absolute value) of Φ_{n} is comparable to the 60th power of the largest coefficient (in absolute value) of F_{n}.

We now prove Theorem 7.1. In what follows, it is understood that n denotes a positive integer relatively prime to 5. We also fix i = \sqrt{−1} ∈ C. Our proof closely follows Cohen’s paper [4] that establishes a similar estimate for the coefficients of the classical modular polynomial Φ_{n}.

Let t ∈ R. Since |j_{5}(it)| → ∞ as t → ∞, there exist s ∈ R and distinct positive t_{0}, t_{1} ∈ R with t_{0}, t_{1} ≥ 1 and s ≥ 1 such that |j_{5}(it_{0})| = s and |j_{5}(it_{1})| = 2s. Since Γ(5) acts without fixed points on C − R and j_{5} thereby realizes each connected component of C − R as a covering space of the complement of a finite subset of CP^{1}, it follows that the derivative j′_{5} is nonvanishing on C−R. Since j_{5} = q^{−1/5} +… has real coefficients, so j_{5} has real-analytic restriction to each connected component of the imaginary axis with deleted origin, the derivative of j_{5} on each such component (as parameterized by ±it with t ∈ (0, ∞)) is positive because j_{5} blows up to ∞ as q → 0. Thus, j_{5}(it) is strictly increasing in t ∈ (0, ∞), so t_{0} < t_{1} and j_{5}(it) is strictly increasing for t_{0} ≤ t ≤ t_{1}. We fix, once and for all, any such s and corresponding t_{0} < t_{1} as above.

Lemma 7.4. For t_{0} ≤ t ≤ t_{1} we have

h(F_{n}(j_{5}(it), Y)) = ∑_{ad=n, a>0} S_{d}(t) + O([Γ(1) : Γ_{0}(n)])

as n → ∞, where

(7.1)  S_{d}(t) = ∑_{\begin{matrix}0≤b<d \\ (a,b,d)=1 \end{matrix}} log max\{1, |j_{5}((ait + b)/d)|^{χ_{5}(a)}\},

and χ_{5} : Z − 5Z → {±1} is the unique quadratic Dirichlet character modulo 5. The implicit O-constant depends only on t_{0}, t_{1}.

Proof. It is well-known that the coefficients of a monic polynomial P (x) = (x − ω_{1}) ⋯ (x − ω_{d}) are bounded between 2^{−d}M and 2^{d}M where M = ∏_{j=1}^{d} max{1, |ω_{j}|}. Taking logarithms yields

h(P) = ∑_{j=1}^{d} log max{1, |ω_{i}|} + O(d)

with an implicit absolute O-constant that is independent of d and P. We now apply this general estimate to suitable specializations of F_{n} in the first variable.

By (6.3) and (6.6),

(7.2)  F_{n}(j_{5}(it), Y) = j_{5}(it)^{r_{n}} ∏_{\begin{matrix}ad=n, a>0 \\ 0≤b′<d \\ (a,b′,d)=1 \end{matrix}}(Y − j_{5} ∘ σ_{a}((ait + 5b′)/d))

with σ_{a} as in (6.5) and r_{n} defined by the condition that ±X^{r_{n}}Y^{deg π_{n}} is the unique monomial term appearing in F_{n}(X, Y) with Y -degree deg π_{n} (see Theorem 6.3). By Corollary 5.5, we have |j_{5} ∘ σ_{a}| = |j_{5}|^{χ_{5}(a)}, and since F_{n}(j_{5}(it), Y) has degree [Γ(1) : Γ_{0}(n)] in Y we obtain

h(F_{n}(j_{5}(it), Y)) = r_{n} log j_{5}(it) + ∑_{ad=n, a>0} ∑_{\begin{matrix}0≤b′<d \\ (a,b′,d)=1 \end{matrix}} log max\{1, |j_{5}((ait + 5b′)/d)|^{χ_{5}(a)}\} + O([Γ(1) : Γ_{0}(n)]).

Since log(2s) = log(j_{5}(it_{1})) ≥ log j_{5}(it) and 1 ≤ r_{n} ≤ deg π_{n} = [Γ(1) : Γ_{0}(n)], we can absorb the first term into O([Γ(1) : Γ_{0}(n)]). By the division algorithm, we may write 5b′ = dq + b with 0 ≤ b < d. From Theorem A.3, for any integer k we have j_{5}(z + k) = ζj_{5}(z) for some fifth root of unity ζ. It follows that the function |j_{5}(z)| is invariant under integer translations, so

|j_{5}((ait + 5b′)/d)| = |j_{5}((ait + b)/d + q)| = |j_{5}((ait + b)/d)|.

Finally, since (n, 5) = 1 and ad = n, multiplication by 5 on Z/dZ is an isomorphism. Thus, we have

∑_{\begin{matrix}0≤b′<d \\ (a,b′,d)=1 \end{matrix}} log max\{1, |j_{5}((ait + 5b′)/d)|^{χ_{5}(a)}\} = ∑_{\begin{matrix}0≤b<d \\ (a,b,d)=1 \end{matrix}} log max\{1, |j_{5}((ait + b)/d)|^{χ_{5}(a)}\} = S_{d}(t).

■

Our next goal is to estimate the sums S_{d}(t) for fixed t between t_{0} and t_{1}. In order to do this, we must bound |j_{5}((ait + b)/d)|^{χ_{5}(a)} for factorizations n = ad with a, d > 0, 0 ≤ b < d, and (a, b, d) = 1. This will be straightforward if at/d is sufficiently large, but slightly complicated when at/d is small. When at/d is close to 0, then (ait + b)/d is close to b/d. In these cases, we will use our knowledge of the behavior of j_{5} near the cusps of X(5) to provide an upper bound for |j_{5}((ait + b)/d)|^{χ_{5}(a)}.

Recall that the Farey sequence ℱ_{N} is the ordered list

ℱ_{N} = {h/k ∈ [0, 1] | h, k ∈ Z, gcd(h, k) = 1, 1 ≤ k ≤ N}

whose elements are enumerated in increasing order. We will estimate |j_{5}((ait+b)/d)|^{χ_{5}(a)} when b/d is “close” to h/k ∈ ℱ_{N} and at/d ≤ 1/2 by expanding j_{5} in a local parameter about the cusp h/k. Equivalently, we shall find some γ_{h/k} ∈ SL_{2}(Z) taking h/k to the cusp ∞ and use our knowledge of the q-expansion of j_{5} ∘γ_{h/k} to estimate |j_{5}((ait+b)/d)|^{χ_{5}(a)}. First, we need to make precise what we mean by “close” to h/k. To do this, put N = ⌊d/(nt)^{1/2}⌋; evidently N ≥ 1. We will partition the interval I(N) = [1/(N + 1), (N + 2)/(N + 1)) into disjoint intervals I_{N}(h/k) containing h/k for each h/k ∈ ℱ_{N} and consider b/d “close” to h/k when b/d ∈ I_{N}(h/k).

Let us enumerate the Farey fractions ℱ_{N} as λ_{0}, …, λ_{K}. Recall that for any consecutive Farey fractions h_{1}/k_{1} = λ_{i−1} and h_{2}/k_{2} = λ_{i} the mediant μ_{i} = (h_{1} + h_{2})/(k_{1} + k_{2}) satisfies λ_{i−1} < μ_{i} < λ_{i}. We set μ_{0} = 0 and define the interval

I_{N}(λ_{i}) = \begin{cases}[μ_{i}, μ_{i+1}) & i < K \\ [N/(N + 1), (N + 2)/(N + 1)) & i = K \end{cases} , (7.3)

so λ_{i} ∈ I_{N}(λ_{i}). Since μ_{1} = 1/(N + 1), we have the disjoint-union decomposition

I(N) = [1/(N + 1), (N + 2)/(N + 1)) = ⋃_{h/k∈ℱ_{N} −{0}} I_{N}(h/k). (7.4)

By [4, Lemma 3] , with λ_{i} = h/k ≠ 0,

\frac{1}{2Nk} ≤ λ_{i} − μ_{i−1} ≤ \frac{1}{(N + 1)k}

\frac{1}{2Nk} ≤ μ_{i} − λ_{i} ≤ \frac{1}{(N + 1)k} . (7.5)

We can now provide good estimates for j_{5}((ait+b)/d). For convenience, we adopt the notation of [4, §3.1] and put

g_{h,k}(x) = 2π · \frac{nt/(dk)^{2}}{(at/d)^{2} + (x − h/k)^{2}} .

Lemma 7.5. For σ, t ∈ R, let q = e^{2πi(σ+it)}. Assume g(σ + it) := ∑_{j≥0} a_{j}q^{j} is absolutely convergent and nonzero for t > 0. There is a constant C_{1} ≥ 0 depending only on g such that

|g(σ + it)| ≤ C_{1} (7.6)

for t ≥ 1/2 and all σ. If further a_{0} ≠ 0, then there is a positive constant C_{0} > 0 depending only on g such that

C_{0} ≤ |g(σ + it)| (7.7)

for t ≥ 1/2 and all σ. In particular, if at/d ≥ 1/2 then

(7.8)  log max\{1, |j_{5}((ait + b)/d)|^{χ_{5}(a)}\} = \begin{cases}(2π/5)(nt/d^{2}) + O(1) & if a ≡ ±1 mod 5 \\ O(1) & otherwise \end{cases} .

If at/d ≤ 1, put N = ⌊d/(nt)^{1/2}⌋ and let I_{N} be defined by (7.3). For b/d ∈ I_{N}(h/k),

(7.9)  log max\{1, |j_{5}((ait + b)/d)|^{χ_{5}(a)}\} = \begin{cases}\frac{1}{5}g_{h,k}(b/d) + O(1) & if (h, k) ≡ (±a, 0) mod 5 \\ O(1) & otherwise \end{cases} .

The implicit O-constants are independent of all parameters a, b, d, t, h, k.

Remark 7.6. Observe that there is an overlap in the range of applicability of (7.8) and (7.9), namely 1/2 ≤ at/d ≤ 1. For these values of t, both (7.8) and (7.9) give the same estimate of O(1), as is readily verified by (7.5). Indeed, it is not difficult to see from the definition of g_{h,k}(x) that (7.8) and (7.9) differ by at most a constant when at/d is in any compact interval bounded away from 0. Thus, for values of at/d in this range of overlap, we shall use whichever of these two estimates is more convenient.

Proof. Since the sum defining g(σ + it) is absolutely convergent for t > 0, we have g(σ + it) → a_{0} as t → ∞. We may therefore find T ≥ 1/2 such |a_{0}|/2 ≤ |g(σ + it)| ≤ |a_{0}| + 1 for all t > T. This proves (7.6) and (7.7) when t > T. To handle T ≤ t ≤ 1/2, we observe that g(σ + it) is periodic under σ ↦ σ + 1, so since the region K = {|σ| ≤ 1/2} ∩ {1/2 ≤ t ≤ T } is compact and g(z) is continuous and nonzero on K, we have C_{0} ≤ |g(z)| ≤ C_{1} for some absolute constants C_{0} > 0 and C_{1} as claimed. The estimate (7.8) now follows from (7.6) when a ≡ ±1 mod 5 and from (7.7) when a ≡ ±2 mod 5 upon taking g = (q^{1/5}j_{5})^{χ_{5}(a)}.

To prove (7.9), suppose that b/d ∈ I_{N}(h/k) and put

γ_{h/k} := (\begin{matrix}v & u \\ −k & h \end{matrix}) ,

where uk + hv = 1, and we require 5|u if 5|k (so γ_{h,k} and σ_{h}^{−1} have the same image in SL_{2}(Z/5Z) when 5|k). The usual calculation shows that

Im (γ_{h/k}((ait + b)/d)) = \frac{nt/(dk)^{2}}{(at/d)^{2} + (b/d − h/k)^{2}} = \frac{1}{2π} g_{h,k}(b/d). (7.10)

Now by (7.5) we have |b/d − h/k| ≤ 1/(N + 1)k, and since N = ⌊d/(nt)^{1/2}⌋ satisfies 1/(N + 1) ≤ (nt)^{1/2}/d, it follows (using an easy direct check when h/k = 0) that

(7.11)  |b/d − h/k| ≤ 1/(N + 1)k ≤ (nt)^{1/2}/(dk).

Since h/k ∈ ℱ_{N} we have k ≤ N ≤ d/(nt)^{1/2}, which implies 1 ≤ d/(k(nt)^{1/2}), whence multiplying by nt/d^{2} and recalling that ad = n yields

at/d = nt/d^{2} ≤ (nt)^{1/2}/(dk). (7.12)

Combining (7.10) with (7.11), and (7.12), we obtain

Im (γ_{h/k}((ait + b)/d)) = \frac{nt/(dk)^{2}}{(at/d)^{2} + (b/d − h/k)^{2}} ≥ \frac{nt/(dk)^{2}}{nt/(dk)^{2} + nt/(dk)^{2}} = 1/2.

If k ≢ 0 mod 5, then from our calculations in Table B.1 we have

j_{5} ∘ γ_{h/k}^{−1} = c(h/k) + O(q),

where c(h/k) ≠ 0 is constant. Since j_{5}((ait+b)/d) = j_{5}∘γ_{h/k}^{−1}(γ_{h/k}((ait+b)/d)) and Im (γ_{h/k}((ait + b)/d)) ≥ 1/2, we apply (7.6) and (7.7) to g = j_{5} ∘ γ_{h/k}^{−1} to find that |j_{5}((ait + b)/d)|^{χ_{5}(a)} = O(1) for all b/d ∈ I_{N}(h/k).

It remains to estimate |j_{5}((ait + b)/d)|^{χ_{5}(a)} when k ≡ 0 mod 5. In these cases, γ_{h,k} and σ_{h}^{−1} have the same image in SL_{2}(Z/5Z), so

|j_{5}((ait + b)/d)|^{χ_{5}(a)} = |j_{5}(γ_{h,k}(ait + b)/d)|^{χ_{5}(ah)}.

If h ≢ ±a mod 5, then ah ≢ ±1 mod 5, and since Im (γ_{h/k}((ait + b)/d)) ≥ 1/2, we can apply (7.8) to obtain |j_{5}((ait + b)/d)|^{χ_{5}(a)} = O(1). On the other hand, if h ≡ ±a mod 5 then ah ≡ ±1 mod 5, so another application of (7.8) and (7.10) yields (7.9). ■

Using Lemma 7.5, we can estimate the sums S_{d}(t):

Lemma 7.7. If d < (nt)^{1/2} then

(7.13)  S_{d} = O(n/d)

and if d ≥ (nt)^{1/2} then

(7.14)  S_{d} = \frac{1}{10} \frac{d}{(a, d)}φ((a, d)) log(d^{2}/n) + O(σ_{1}(d/(a, d))) + O(dσ_{1}((a, d))/(a, d)),

where the implicit O-constants depends only on t_{0}, t_{1}.

Proof. The estimate (7.13) follows immediately from (7.8). To obtain (7.14), we set N = ⌊d/(nt)^{1/2}⌋ and observe that N ≥ 1. Recall the definition of I(N) given in (7.4). By Theorem A.3, we have |j_{5}(z + m)| = |j_{5}(z)| for any integer m, so we may reindex the sum in (7.1) via

b ↦ \begin{cases}b & if b/d ∈ [1/(N + 1), 1] \\ b + d & if b/d ∈ [0, 1/(N + 1)) \end{cases} ,

so that our original sum S_{d}(t) over all b/d ∈ [0, 1] is now a sum over all

b/d ∈ I(N) = [1/(N + 1), (N + 2)/(N + 1)).

We therefore obtain

S_{d}(t) = ∑_{\begin{matrix}0≤b<d \\ (a,b,d)=1 \end{matrix}} log max\{1, |j_{5}((ait + b)/d)|^{χ_{5}(a)}\} = ∑_{\begin{matrix}b/d∈I(N) \\ (a,b,d)=1 \end{matrix}} log max\{1, |j_{5}((ait + b)/d)|^{χ_{5}(a)}\}.

Since (7.4) gives a partition of I(N) into the disjoint intervals I_{N}(h/k) for h/k ∈ ℱ_{N} − {0}, we have

S_{d}(t) = ∑_{h/k∈ℱ_{N} −\{0\}} ∑_{\begin{matrix}b/d∈I_{N} (h/k) \\ (a,b,d)=1 \end{matrix}} log max\{1, |j_{5}((ait + b)/d)|^{χ_{5}(a)}\}

= ∑_{k=1}^{N} ∑_{\begin{matrix}h=1 \\ (h,k)=1 \end{matrix}}^{k} ∑_{\begin{matrix}b/d∈I_{N} (h/k) \\ (a,b,d)=1 \end{matrix}} log max\{1, |j_{5}((ait + b)/d)|^{χ_{5}(a)}\}.

We split this sum up into sums over (h, k) ≡ ±(a, 0) mod 5 and over (h, k) ≢ ±(a, 0) mod 5, and use (7.9) to estimate the terms occurring in each sum:

S_{d}(t) = ∑_{\begin{matrix}1≤h≤k≤N \\ (h,k)=1 \\ (h,k)≡(±a,0) mod 5 \end{matrix}} ∑_{\begin{matrix}b/d∈I_{N} (h/k) \\ (a,b,d)=1 \end{matrix}} \frac{1}{5} (g_{h,k}(b/d) + O(1)) + ∑_{\begin{matrix}1≤h≤k≤N \\ (h,k)=1 \\ (h,k)≢(±a,0) mod 5 \end{matrix}} ∑_{\begin{matrix}b/d∈I_{N} (h/k) \\ (a,b,d)=1 \end{matrix}} O(1)

Since the intervals I_{N}(h/k) for h/k ∈ ℱ_{N} − {0} partition I(N), the second double sum above is at most ∑_{b/d∈I(N)} O(1); since this sum has at most d terms, we get a contribution of O(d). Exactly the same reasoning shows that the O(1)-term in the first double sum contributes O(d), for a total error term of O(d), with an implicit constant that depends only on t_{0} and t_{1}. Thus,

(7.15)  S_{d}(t) = \frac{1}{5} ∑_{\begin{matrix}1≤h≤k≤N \\ (h,k)=1 \\ (h,k)≡(±a,0) mod 5 \end{matrix}} ∑_{\begin{matrix}b/d∈I_{N} (h/k) \\ (a,b,d)=1 \end{matrix}} g_{h,k}(b/d) + O(d),

where the O(d)-term lies outside of the double summation. Now by [4, Lemma 6], we have

∑_{\begin{matrix}b/d∈I_{N} (h/k) \\ (a,b,d)=1 \end{matrix}} g_{h,k}(b/d) = k^{−2} ∑_{f|(a,d)} μ(f)F_{f}(dh/fk) + O (n^{1/2}σ_{1}((a, d))/(k(a, d))) ,

where

(7.16)  F_{f}(θ) = \frac{2π^{2}d}{f} ∑_{v∈Z} e^{−2π|v|nt/df}e^{2πivθ}

and the O-constant depends only on t_{0}, t_{1}. Therefore,

S_{d}(t) = \frac{1}{5} ∑_{\begin{matrix}1≤h≤k≤N \\ (h,k)=1 \\ (h,k)≡(±a,0) mod 5 \end{matrix}} (k^{−2} ∑_{f|(a,d)} μ(f)F_{f}(dh/fk) + O (n^{1/2}σ_{1}((a, d))/(k(a, d)))) + O(d),

where again the O(d)-term lies outside of the double summation.

The sum of the error terms in the double summation is bounded above by

\frac{Cn^{1/2}σ_{1}((a, d))}{(a, d)} · ∑_{\begin{matrix}1≤h≤k≤N \\ (h,k)=1 \end{matrix}} \frac{1}{k} ≤ \frac{Cn^{1/2}σ_{1}((a, d))}{(a, d)} · ∑_{1≤k≤N} \frac{φ(k)}{k} ≤ \frac{Cn^{1/2}σ_{1}((a, d))}{(a, d)}N

≤ \frac{Cn^{1/2}σ_{1}((a, d))}{(a, d)} \frac{d}{(nt)^{1/2}} ≤ Cdσ_{1}((a, d))/(a, d)

with some constant C depending only on t_{0}, t_{1}; in the last line we have used the fact that 1 ≤ t_{0} ≤ t. This estimate also absorbs the additional O(d)-term, so we obtain

(7.17)  S_{d}(t) = \frac{1}{5} ∑_{f|(a,d)} μ(f) ∑_{\begin{matrix}1≤h≤k≤N \\ (h,k)=1 \\ (h,k)≡(±a,0) mod 5 \end{matrix}} k^{−2}F_{f}(dh/fk) + O(dσ_{1}((a, d))/(a, d)).

with the error term outside of the double summation. We still need to estimate the sum

∑_{\begin{matrix}1≤h≤k≤N \\ (h,k)=1 \\ (h,k)≡(±a,0) mod 5 \end{matrix}} k^{−2}F_{f}(dh/fk),

which by (7.16) is

\frac{2π^{2}d}{f} ∑_{v∈Z} C_{N}(d|v|/f)e^{−2π|v|nt/df}, (7.18)

where

C_{N}(m) = ∑_{\begin{matrix}1≤k≤N \\ k≡0 mod 5 \end{matrix}} k^{−2}c_{k}(m)  and  c_{k}(m) = ∑_{\begin{matrix}1≤h≤k \\ (h,k)=1 \\ h≡±a mod 5 \end{matrix}} e^{2πihm/k}.

By Lemma D.3, we have |c_{k}(m)| ≤ (k, m) when m ≠ 0, and so

|C_{N}(m)| ≤ ∑_{k=1}^{∞} k^{−2}(k, m) ≤ ∑_{d|m} d ∑_{j=1}^{∞}(jd)^{−2} = O(σ_{1}(|m|)/|m|) (7.19)

for an absolute implicit O-constant. When m = 0, Lemma D.3 tells us that

C_{N}(0) = ∑_{\begin{matrix}1≤k≤N \\ k≡0 mod 5 \end{matrix}} k^{−2} ∑_{\begin{matrix}1≤h≤k \\ (h,k)=1 \\ h≡±a mod 5 \end{matrix}} 1

= ∑_{\begin{matrix}1≤k≤N \\ k≡0 mod 5 \end{matrix}} k^{−2} (\frac{1}{2}φ(k)) ,

which by Lemma D.1 is

= \frac{1}{2π^{2}} log N + O(1), (7.20)

with an absolute implicit O-constant.

Putting (7.19) and (7.20) together and using (7.18), we see that

(7.21)  ∑_{\begin{matrix}1≤h≤k≤N \\ (h,k)=1 \\ (h,k)≡(±a,0) mod 5 \end{matrix}} k^{−2}F_{f}(dh/fk) = \frac{d}{f} log N + O(d/f) + O ( ∑_{v∈Z−\{0\}} f \frac{σ_{1}(d|v|/f)}{d|v|}e^{−2π|v|nt/df} )

(7.22)

with absolute implicit O-constants. Now since f|(a, d) and (a, d)|(n/d) we clearly have df ≤ n so that

e^{−2π|v|nt/df} ≤ e^{−2π|v|t} ≤ e^{−2π|v|}

since 1 ≤ t_{0} ≤ t. Therefore, putting C_{1} = ∑_{v∈Z−\{0\}} \frac{σ_{1}(|v|)}{|v|}e^{−2π(|v|−1)} and using the fact that σ_{1}(d|v|/f) ≤ σ_{1}(d/f)σ_{1}(|v|), we have

∑_{v∈Z−\{0\}} f \frac{σ_{1}(d|v|/f)}{d|v|}e^{−2π|v|nt/df} ≤ C_{1}(f/d)σ_{1}(d/f)e^{−2πnt/df} ≤ C_{1}σ_{1}(d/f)e^{−2πn/df},

where we have again used the inequality t ≥ t_{0} ≥ 1 and the fact that f|d (so f/d < 1). Thus,

∑_{v∈Z−\{0\}} f \frac{σ_{1}(d|v|/f)}{d|v|}e^{−2π|v|nt/df} = O(σ_{1}(d/f)e^{−2πn/df}) (7.23)

with an absolute implicit O-constant, and since N = ⌊\sqrt{d^{2}/nt}⌋ we see that log N = \frac{1}{2} log(d^{2}/n) + O(1) with an implicit O-constant that depends only on t_{0} and t_{1} since 1 ≤ t_{0} ≤ t ≤ t_{1}.

Incorporating the estimate (7.23) into (7.21) therefore gives

(7.24)  ∑_{\begin{matrix}1≤h≤k≤N \\ (h,k)=1 \\ (h,k)≡(±a,0) mod 5 \end{matrix}} k^{−2}F_{f}(dh/fk) = \frac{d}{2f} log(d^{2}/n) + O(d/f) + O (σ_{1}(d/f)e^{−2πn/df}) ,

where the implicit O-constants depend only on t_{0}, t_{1}. Combining (7.24) with (7.17), we have

(7.25) S_{d}(t) = \frac{1}{5} ∑_{f|(a,d)} μ(f) (\frac{d}{2f} log(d^{2}/n) + O(d/f) + O (σ_{1}(d/f)e^{−2πn/df})) + O(dσ_{1}((a, d))/(a, d)).

Now,

∑_{f|(a,d)} μ(f)σ_{1}(d/f)e^{−2πn/df} ≤ ∑_{f|(a,d)} σ_{1}(d/f)e^{−2πn/df} ≤ ∑_{f|(a,d)} σ_{1}(df/(a, d))e^{−2πnf/(d(a,d))},

and since σ_{1}(df/(a, d)) ≤ σ_{1}(d/(a, d))σ_{1}(f) and e^{−2πnf/(a,d)} ≤ e^{−2πf} we have

∑_{f|(a,d)} μ(f)σ_{1}(d/f)e^{−2πn/df} ≤ σ_{1}(d/(a, d)) ∑_{f=1}^{∞} σ_{1}(f)e^{−2πf}. (7.26)

Finally, using the well-known identity

∑_{f|(a,d)} μ(f)(d/f) = (d/(a, d)) ∑_{f|(a,d)} μ(f)(a, d)/f = d/(a, d)φ((a, d))

we see from (7.25) and (7.26) that

S_{d}(t) = \frac{1}{10} \frac{d}{(a, d)}φ((a, d)) log(d^{2}/n) + O(dφ(a, d)/(a, d)) + O (σ_{1}(d/(a, d))) + O(dσ_{1}((a, d))/(a, d)),

where all implicit O-constants depend only on t_{0} and t_{1}. The trivial inequality φ((a, d)) ≤ σ_{1}((a, d)) therefore completes the proof. ■

Lemma 7.8. For 1 ≤ t_{0} ≤ t ≤ t_{1},

h(F_{n}(j_{5}(it), Y)) = \frac{1}{10}[Γ(1) : Γ_{0}(n)] (log n − 2 ∑_{p|n} \frac{log p}{p} + O(1))

as n → ∞, where the O-constant depends only on t_{0}, t_{1}.

Proof. Using Lemmas 7.7 and 7.4, we see that

h(F_{n}(j_{5}(it), Y)) = ∑_{ad=n, a>0} S_{d}(t) + O([Γ(1) : Γ_{0}(n)])
= H_{1} + H_{2} + O([Γ(1) : Γ_{0}(n)]),

where

H_{1} = ∑_{\begin{matrix}d|n \\ d<(nt)^{1/2} \end{matrix}} O(n/d) = O(∑_{d|n} n/d) = O(σ_{1}(n)) = O([Γ(1) : Γ_{0}(n)])

from [4, A2], and

H_{2} = ∑_{\begin{matrix}d|n \\ d≥(nt)^{1/2} \end{matrix}} (\frac{1}{10} \frac{d}{(a, d)}φ((a, d)) log(d^{2}/n) + O(σ_{1}(d/(a, d))) + O(dσ_{1}((a, d))/(a, d)))

= ∑_{\begin{matrix}d|n \\ d≥(nt)^{1/2} \end{matrix}} \frac{1}{10} \frac{d}{(a, d)}φ((a, d)) log(d^{2}/n) + O([Γ(1) : Γ_{0}(n)])

by (A2) and (A3) of [4]. We can rewrite this as

H_{2} = \frac{1}{10} ∑_{d|n} \frac{d}{(a, d)}φ((a, d))(log n − 2 log(n/d)) + O([Γ(1) : Γ_{0}(n)]).

Finally, applying (A1) and Lemma A1 of [4], we obtain

H_{2} = \frac{1}{10}[Γ(1) : Γ_{0}(n)] (log n − 2 ∑_{p|n} \frac{log p}{p} + O(1)) ,

as desired. ■

We have thus estimated h(F_{n}(j_{5}(it), Y)). Using the next lemma, this estimate will enable us to obtain Theorem 7.1.

Lemma 7.9. Let P ∈ C[Y] be any nonzero polynomial of degree ≤ D. For any θ > 0 there exists an absolute constant c_{θ} > 0 depending only on θ such that

|h(P) − log sup_{θ≤y≤2θ} |P (y)|| ≤ c_{θ}D.

Proof. This follows from the proof of [4, Lemma 9] which establishes the result for θ = 1728 but is in fact completely general. ■

We now prove Theorem 7.1. For convenience, we define h(0) = −∞ and use the usual conventions with the symbol −∞ in estimates below. Let D = [Γ(1) : Γ_{0}(n)] and write

F_{n}(X, Y) = P_{0}(X)Y^{D} + P_{1}(X)Y^{D−1} + ⋯ + P_{D}(X)

with P_{j} ∈ Z[X] and P_{0} ≠ 0. In what follows, we implicitly omit any P_{j}’s that equal zero.

Certainly, h(F_{n}) = max_{0≤j≤D} h(P_{j}). Since the degree of P_{j} is at most D, Lemma 7.9 with θ = s ≥ 1 yields

h(F_{n}) = max_{0≤j≤D} log sup_{s≤x≤2s} |P_{j}(x)| + O(D) = sup_{s≤x≤2s} max_{0≤j≤D} log |P_{j}(x)| + O(D)

with an O(D)-term that lies outside of the supremum and has an O-constant depending only on s. Since max_{0≤j≤D} log |P_{j}(x)| = h(F_{n}(x, Y)) we get h(F_{n}) = sup_{s≤x≤2s} h(F_{n}(x, Y)) + O(D). By the choice of s, the interval [t_{0}, t_{1}] corresponds bijectively under t ↦ j_{5}(it) to the interval [s, 2s], so any x ∈ [s, 2s] satisfies x = j_{5}(it) for some t ∈ [t_{0}, t_{1}]. Lemma 7.8 now completes the proof as n → ∞ since D = [Γ(1) : Γ_{0}(n)] and

log n − 2 ∑_{p|n} \frac{log p}{p} → ∞

as n → ∞.

Remark 7.10. Let us illustrate Theorem 7.1. We have the following table, in which Φ_{n} denotes the classical modular polynomial of level n and H is the non-logarithmic height H(∑_{I} a_{I}X^{I}) = max_{I} |a_{I}|:

\begin{matrix}Level & H(Φ_{n}) \\ 32 & 2^{12} · 3^{144} · 5^{144} · 11^{72} · 17^{18} · 23^{36} · 29^{36} · 47^{27} · 53^{18} · 59^{18} · 71^{9} · 83^{18} · 89^{18} \\ 41 & 2^{684} · 3^{126} · 5^{126} · 11^{36} · 17^{18} · 23^{27} · 29^{36} · 41^{3} · 47^{18} · 59^{9} · 71^{18} · 107^{9} \\ 47 & 2^{774} · 3^{144} · 5^{153} · 11^{72} · 17^{36} · 23^{27} · 29^{27} · 41^{9} · 47^{3} · 89^{18} · 113^{18} · 137^{9} \\ 53 & 2^{900} · 3^{162} · 5^{162} · 11^{54} · 17^{54} · 23^{18} · 29^{18} · 41^{18} · 47^{18} · 53^{3} · 59^{9} · 83^{18} · 107^{18} · 131^{18} \\ 59 & 2^{972} · 3^{180} · 5^{198} · 11^{81} · 17^{18} · 23^{36} · 29^{36} · 41^{18} · 47^{18} · 53^{18} · 59^{3} · 101^{18} · 113^{9} · 149^{18} · 173^{9} \end{matrix}

Level  H(F_{n})  log(H(Φ_{n}))/ log(H(F_{n}))
32  2 · 5 · 937 · 1997 · 5381  51.4514292315 …
41  2^{9} · 3^{4} · 5^{3} · 41 · 1459  52.7001592098 …
47  3^{4} · 5 · 47 · 311 · 337 · 4129  55.3569927370 …
53  2^{3} · 53 · 843701 · 2543873  55.1097204607 …
59 2^{2} · 59 · 127 · 22369 · 231573773  54.3504335762 …

Remark 7.11. One might be interested in comparing the q-series coefficients of j_{5} with those of j. Indeed, we have

j_{5} = q^{−1/5} (1 + q − q^{3} + q^{5} + q^{6} − q^{7} − 2q^{8} + 2q^{10} + 2q^{11} − q^{12} − 3q^{13} − q^{14} + 3q^{15} + 3q^{16} − 2q^{17} + ⋯) ,

while

j = q^{−1} + 744 + 196884q + 21493760q^{2} + 864299970q^{3} + 20245856256q^{4} + 333202640600q^{5} + ⋯,

so it seems that the q-coefficients of j_{5} are very small in comparison to those of j. Let us write q^{1/5}j_{5} = ∑_{n=0}^{∞} c_{n}q^{n} and j = q^{−1} + ∑_{n=0}^{∞} b_{n}q^{n} with b_{n}, c_{n} ∈ Z. One can adapt Rademacher’s improvement of the Hardy-Littlewood circle method to obtain the (divergent) asymptotic expansion

(7.27)  c_{n} = \frac{2π}{5\sqrt{5n − 1}} ∑_{k=1}^{∞} \frac{A_{k}(n)}{k}I_{1} (\frac{4π}{25k} \sqrt{5n − 1})

as n → ∞, where

I_{1}(u) = ∫_{1−i∞}^{1+i∞} t^{−2}e^{t+u^{2}/4t} dt

is the I Bessel-function of order 1 and

A_{k}(n) = ∑_{\begin{matrix}h∈(Z/5kZ)^{×} \\ h≡±1 mod 5 \end{matrix}} e^{\frac{2πi}{25k} (h+h̄−5nh)}

with hh̄ ≡ 1 mod 25k. By “divergent” expansion we mean that at any finite truncation the series (7.27) gives a genuine asymptotic expansion for c_{n} as n → ∞, and that we can prescribe an optimal cut-off point (depending on n) for any particular n.

Using the well-known estimate

I_{1}(u) ∼ \frac{e^{u}}{(2πu)^{1/2}}

as u → ∞ and taking the truncation at k = 1 in (7.27) yields

c_{n} ∼ \frac{\sqrt{2}}{(5n − 1)^{3/4}} cos (\frac{2π}{25} (5n − 2)) e^{\sqrt[\frac{4π}{25}]{5n−1}}. (7.28)

It is remarkable how accurately this asymptotic on the first-term truncation approximates c_{n}. Indeed, we have the following table:

\begin{matrix}n & c_{n} & \frac{\sqrt{2}}{(5n−1)^{3/4}} cos (\frac{2π}{25} (5n − 2)) e^{\sqrt[\frac{4π}{25}]{5n−1}} \\ 10 & 2 & 1.98558 \\ 21 & 5 & 4.90972 \\ 32 & -7 & -7.13225 \\ 43 & -37 & -37.055 \\ 54 & -15 & -14.4614 \\ 65 & 131 & 131.995 \\ 76 & 204 & 204.887 \\ 87 & -216 & -215.274 \\ 98 & -875 & -875.131 \\ 109 & -279 & -280.932 \\ ⋮ & ⋮ & ⋮ \\ 150 & 7939 & 7932.7 \\ ⋮ & ⋮ & ⋮ \\ 197 & -23562 & -23555.2 \end{matrix}

It is worth comparing the asymptotic formula (7.28) for c_{n} with that for b_{n}:

c_{n} ∼ \frac{\sqrt{2}}{(5n)^{3/4}} cos (\frac{2π}{25} (5n − 2)) e^{\sqrt[\frac{4π}{5\sqrt{5}}]{n}},
b_{n} ∼ \frac{1}{\sqrt{2}n^{3/4}} e^{4π\sqrt{n}},

so

lim_{n→∞} \frac{log b_{n}}{log c_{n}} = 5\sqrt{5} = 11.1803398874989 ….

This explains why the coefficients of the q-expansion of j seem to be so much larger than the coefficients of the q-expansion of j_{5}.

8. Radical formulas for singular values in R

An interesting question in the context of radical formulas for singular values of j_{5} (or of Ramanujan’s F) is whether such formulas can be given inside of R when the singular value lies in R.

Definition 8.1. An finite extension of subfields E ⊆ E′ inside of R is solvable in real radicals if there exists a finite tower

E = E_{1} ⊆ E_{2} ⊆ ⋯ ⊆ E_{n}

in R with E′ ⊆ E_{n} and E_{i} = E_{i−1}(a_{i}) with a_{i}^{d_{i}} ∈ E_{i−1} for 1 < i ≤ n and some positive integers d_{i}. A real algebraic number α ∈ R is solvable in real radicals if Q(α)/Q is solvable in real radicals.

Remark 8.2. Despite appearances, this definition is a property of the abstract extension E′/E: it is independent of how E′ is embedded in R. Indeed, due to the existence and (non-canonical) uniqueness of real closures of real fields, and the evident fact that E′/E is an algebraic extension of real fields, we see that Definition 8.1 says exactly that E′ admits an E-embedding into a radical tower in a real closure of E.

Pick τ ∈ C − R that is quadratic over Q, and assume j_{5}(τ) ∈ R (or equivalently, assume τ is Γ(5)- equivalent to −\overline{τ}; the typical example is a purely imaginary τ, which is to say τ^{2} ∈ Q). Let K = Q(τ), so K(j_{5}(τ))/K is an abelian extension and clearly

(8.1)  [Q(j_{5}(τ)) : Q] = [K(j_{5}(τ)) : K].

The question we wish to address is whether or not j_{5}(τ) is solvable in real radicals. Our analysis will only be applicable when K(j_{5}(τ)) is Galois over Q, and the description of the associated open subgroup in A_{K}^{×}/K^{×} provided by Corollary 3.3 shows that this condition holds when τ^{2} ∈ Q. The solvability of such real singular values in real radicals is quite rare, as the following theorem makes clear:

Theorem 8.3. Let τ ∈ C − R be quadratic over Q and assume τ is Γ(5)-equivalent to −\overline{τ}. If j_{5}(τ) ∈ R is solvable in real radicals then all odd prime factors of [Q(j_{5}(τ)) : Q] are Fermat primes, and if [Q(j_{5}(τ)) : Q] is a power of 2 then j_{5}(τ) lies in a tower of quadratic extensions over Q inside R.

In particular, if τ is a 5-unit and C^{×}/q_{τ}^{Z} has CM by 𝒪_{K}, then j_{5}(τ) is solvable in real radicals only if the size of the the ray class group of conductor 5 for Q(τ) is of the form 2^{e} ∏_{j} p_{j}^{e_{j}} where the p_{j}’s are Fermat primes.

The final part of the theorem follows from the rest because of (8.1) and the fact that the given additional conditions on τ imply that Q(τ, j_{5}(τ)) is the ray class field of conductor 5 for Q(τ) (by Corollary 5.7). In fact, the Fermat criterion on the size of the ray class group of conductor 5 may be replaced with the Fermat criterion for the usual class group because the ratio of the size of the two groups has only 2, 3, and 5 as possible prime factors. We also remark that the theorem is true for the classical j-function (with 5 replaced by 1 everywhere), using the same proof.

Example 8.4. A simple example where the necessary criterion of Theorem 8.3 is violated is τ = \sqrt{−101}. Indeed, the ring of integers of Q(\sqrt{−101}) is Z[\sqrt{−101}] and \sqrt{−101} is a 5-unit, so [Q(j_{5}(\sqrt{−101})) : Q] is the order of the the ray class group of conductor 5 for Q(\sqrt{−101}). This group is Z/4Z × Z/28Z, and the prime 7 is not a Fermat prime. Thus, j_{5}(\sqrt{−101}) ∈ R is not solvable in real radicals. Of course, it is solvable in complex radicals.

Other examples of the form τ = \sqrt{−n} with squarefree n ≡ 1 mod 4 and 5 ∤ n (so Q(τ) has ring of integers Z[τ] with τ a 5-unit) are n = 149, 173, 341, 349 with the prime 7 intervening, n = 269, 389, 829, 1021 with the prime 11 intervening, and n = 941, 1181, 2837, 3401 with the prime 23 intervening.

In view of the preceding remarks, Theorem 8.3 is a consequence of the following general result (with K^{+} = Q, K = Q(τ), L^{+} = Q(j_{5}(τ)), L = Q(τ, j_{5}(τ))):

Theorem 8.5. Let K^{+} be a number field in R that is solvable in real radicals over Q, and let K be a quadratic extension of K^{+} in C with K not contained in R. Let L/K be a finite abelian extension inside of C such that L/K^{+} is Galois. Let L^{+} = L ∩ R, so [L : K] = [L^{+} : K^{+}] and L = K ⊗_{K^{+}} L^{+}.

If [L : K] is a power of 2, then L^{+}/K^{+} is a tower of quadratic extensions. If the real number field L^{+} is solvable in real radicals over K^{+} (or, equivalently, over Q), then all odd prime factors of [L : K] are Fermat primes.

This purely algebraic theorem can be formulated and proved in terms of a choice of real closure of a real number field K^{+}, but we prefer to work with the more concrete (but operationally equivalent) language of subfields of R and C.

Proof. Let Γ = Gal(L/K). We may assume Γ ≠ {1}. Since we have a short exact sequence

1 → Γ → Gal(L/K^{+}) → Gal(K/K^{+}) → 1

with Γ abelian, we may view Γ as a finite-length nonzero Z[Gal(K/K^{+})]-module. Since Gal(K/K^{+}) has order 2, by decomposing Γ into its primary components it is clear that we can find a Gal(K/K^{+})-stable filtration by subgroups

1 = Γ_{0} ⊆ Γ_{1} ⊆ ⋯ ⊆ Γ_{n} = Γ

such that [Γ_{i} : Γ_{i−1}] = q_{i} is prime for all 1 ≤ i ≤ n. We may clearly arrange for q_{1} to be any prime factor q of the order of the abelian group Γ.

Let L_{i} be the fixed field of Γ_{i} in L, so the L_{i}’s are a decreasing chain of subfields of L containing the subfield K ⊈ R and each L_{i} is Galois over K^{+}. Thus, L_{i}^{+} = L_{i} ∩ R is a decreasing chain of subfields of L^{+} that contain K^{+}, and clearly K ⊗_{K^{+}} L_{i}^{+} → L_{i} is an isomorphism. Thus, [L_{i−1}^{+} : L_{i}^{+}] = q_{i} for all i and L_{i} is Galois over L_{i′}^{+} for i′ ≥ i. If [L : K] is a power of 2 then q_{i} = 2 for all i and we have therefore expressed L^{+} as a tower of quadratic extensions of K^{+}. This settles the existence result when [L : K] is a power of 2. For the proof that when such a formula exists then all odd prime factors of [L : K] are Fermat primes, we may now assume that q_{1} = q is any choice of odd prime factor of [L : K] and we want to show that it is a Fermat prime. We can rename L_{1}^{+} as K^{+} (note that L_{1}^{+}/Q is solvable in real radicals!) and L_{1} as K to reduce to the case when [L : K] = [L^{+} : K^{+}] is equal to q.

Let

K^{+} = E_{0} ⊆ E_{1} ⊆ ⋯ ⊆ E_{m}

be a tower of subfields of R such that L^{+} ⊆ E_{m} and E_{i} = E_{i−1}(a_{i}) with a_{i}^{d_{i}} = b_{i} ∈ E_{i−1} for 1 ≤ i ≤ m and positive integers d_{i}. Obviously we can assume that each d_{i} = p_{i} is prime and that [E_{i} : E_{i−1}] > 1 for all i (so b_{i} ≠ 0 for all i). We can also assume that L^{+} is not contained in E_{m−1} (the case L^{+} = K^{+} is trivial). For each i, we claim that [E_{i} : E_{i−1}] = p_{i}. This is obvious if p_{i} = 2, so suppose p_{i} is odd. If [E_{i} : E_{i−1}] < p_{i} then T^{p_{i}} − b_{i} ∈ E_{i−1}[T] is reducible, and so b_{i} has a p_{i}th root β ∈ E_{i−1} (see [15, Thm. 9.1, Ch. VI]). However, a_{i} ∈ E_{i} − E_{i−1} is also a p_{i}th root of b_{i}, so the ratio a_{i}/β ∈ E_{i} is a nontrivial p_{i}th root of unity. Since p_{i} is odd, this contradicts the fact that E_{i} is a subfield of R. Thus, [E_{i} : E_{i−1}] = p_{i} for all i.

Since [L^{+} : K^{+}] = q is prime and L^{+} is not contained in E_{m−1}, the inclusion K^{+} ⊆ L^{+} ∩ E_{m−1} must be an equality. Consider the composite L^{+}E_{m−1} inside of E_{m}. This is an intermediate extension over E_{m−1} and is strictly larger than E_{m−1}, so since [E_{m} : E_{m−1}] = p_{m} is prime we obtain L^{+}E_{m−1} = E_{m}. Thus, [L^{+}E_{m−1} : E_{m−1}] = p_{m}. However, we claim that L^{+} ⊗_{K^{+}} E_{m−1} → L^{+}E_{m−1} is an isomorphism, and hence p_{m} = [L^{+} : K^{+}] = q. To prove this isomorphism assertion it is enough to check after applying K ⊗_{K^{+}} (·) to both sides. We have K ⊗_{K^{+}} L^{+} = L since K/K^{+} is a non-real quadratic extension, and similarly K ⊗_{K^{+}} (L^{+}E_{m−1}) = KL^{+}E_{m−1} = LE_{m−1} (composites formed inside C). Thus, we wish to show that the surjection

L ⊗_{K^{+}} E_{m−1} ↠ LE_{m−1}

is an isomorphism, and since L/K^{+} is a Galois extension satisfying

L ∩ E_{m−1} = (L ∩ R) ∩ E_{m−1} = L^{+} ∩ E_{m−1} = K^{+}

it follows (see Lemma 8.7 below) that L ⊗_{K^{+}} E_{m−1} is a field. This establishes the isomorphism claim. We have proved that E_{m} = L^{+} ⊗_{K^{+}} E_{m−1}, so by applying K ⊗_{K^{+}} (·) we see that

KE_{m} = K ⊗_{K^{+}} E_{m} = L ⊗_{K^{+}} E_{m−1} = L ⊗_{K} (K ⊗_{K^{+}} E_{m−1}) = L ⊗_{K} (KE_{m−1})

is a degree-q extension of KE_{m−1} that is Galois (since L/K is Galois) and generated by adjoining a qth root of a nonzero element b_{m} ∈ E_{m−1}^{×} ⊆ KE_{m−1} (since E_{m} = E_{m−1}(a_{m})). Hence, KE_{m} must contain a primitive qth root of unity! Thus, KE_{m} ∩ R = E_{m} must contain Q(ζ_{q})^{+}. However, our initial hypothesis that K^{+} be solvable in real radicals over Q implies the same for E_{m}, and hence Q(ζ_{q})^{+} is solvable in real radicals over Q. Since [Q(ζ_{q})^{+} : Q] = (q − 1)/2, it is therefore enough to show that a Galois extension of Q inside of R that is solvable in real radicals must have degree over Q equal to a power of 2. This is a special case of the next theorem (with E = Q and K = Q(ζ_{q})^{+}). ■

Theorem 8.6. Let E be a field, and let K/E be a finite Galois extension. Assume that [K : E] has order divisible by a prime p. If F is a radical tower over E into which K admits an E-embedding, then either F contains a root of unity of order p (so the characteristic is distinct from p) or F contains a primitive root of unity of odd prime order. In particular, if E is a real field and [K : E] is odd and > 1, then F is not a real field.

Some aspects of the proof of this theorem will be similar to steps in the proof of Theorem 8.5. However, the nature of the hypotheses are sufficiently different that it seems simpler to just repeat the similar steps and not to axiomatize the situation too much.

Proof. Since Gal(K/E) = [K : E] has order divisible by p, there must be a subgroup of order p. Thus, there is an intermediate field E′ between K and E with K/E′ Galois of degree p. By using an E-embedding of K into F, we may rename E′ as E to reduce to the case [K : E] = p.

We may express the radical tower F/E in steps

E = E_{0} ⊆ E_{1} ⊆ ⋯ ⊆ E_{n} = F

where E_{i+1} = E_{i}(a_{i}) with a_{i}^{p_{i}} ∈ E_{i} for primes p_{i}. We may certainly assume [E_{i+1} : E_{i}] > 1 for all i without loss of generality. Fix an E-embedding of K into F. Consider the intersection K ∩ E_{n−1}, a field between K and E. Since [K : E] is prime, either K ∩ E_{n−1} = K or K ∩ E_{n−1} = E. If the former, then K ⊆ E_{n−1} so we can rename E_{n−1} as F and induct on n (once we handle the case n = 1!). On the other hand, if K∩E_{n−1} = E then consider the composite field KE_{n−1} inside of F. This is finite Galois over E_{n−1} of degree > 1. The following well-known lemma (using k = E, F_{0} = E_{n−1}) ensures that [KE_{n−1} : E_{n−1}] = [K : E] = p.

Lemma 8.7. Let F/k be an extension of fields and let K/k be a subextension that is finite Galois. Let F_{0}/k be another intermediate extension with K ∩ F_{0} = k. The natural map K ⊗_{k} F_{0} → KF_{0} is an isomorphism.

Proof. Let y be a primitive element for K/k, say with minimal polynomial f ∈ k[T]. Note that K/k is a splitting field of f over k. Since KF_{0}/F_{0} is generated by a root of f, it is necessary and sufficient to prove that f ∈ F_{0}[T] is irreducible. Suppose f = gh is a monic factorization of f over F_{0}; it suffices to show that this factorization is trivial. Since f splits over K and hence over F, when we consider the factorization f = gh in F [T] we see that g and h split over F and hence their coefficients may be expressed as Z-polynomials in the roots of f (recall that g and h were chosen to be monic). But the roots of f in F lie in K, so the coefficients of g and h in F lie in K. Hence, g and h as elements in F [T] lie in (K ∩ F_{0})[T] = k[T], so our factorization of f takes place in k[T]. But f is irreducible in k[T], so our factorization is indeed trivial. ■

Thus, by renaming E_{n−1} as E and KE_{n−1} as K, we get to the special case n = 1, which is to say that F = E(a) with a^{p′} = b ∈ F^{×} for some prime p′ and K/E is an intermediate Galois extension of prime degree p. We claim that either [F : E] = p′ or else F = E(ζ) with ζ a primitive p′th root of unity (and the characteristic is distinct from p′). Consider the polynomial T^{p′} − b in E[T]. If this is irreducible, then clearly F is E-isomorphic to E[T]/(T^{p′} − b) and hence [F : E] = p′. On the other hand, if T^{p′} − b is reducible over E then b = c^{p′} for some c ∈ E^{×} (by [15, Thm. 9.1, Ch. VI]). In this case, ζ = a/c is a nontrivial p′th root of unity (since a ∉ E, as F = E(a) with [F : E] > 1). In particular, the characteristic is not equal to p′ and F = E(ζ) is generated by a primitive p′th root of unity.

If [F : E] = p′, then since p = [K : E] must divide [F : E], we get p′ = p and K = F, so in fact F/E is a finite Galois extension of degree p. However, this extension is generated by extracting a pth root a of an element b ∈ E^{×}, so the characteristic cannot be p. Since [F : E] > 1, the minimal polynomial of a over E is a factor of T^{p} − b ∈ E[T] with degree larger than 1 and this factor must split over the normal extension F. Any other root in F for this factor has to have the form aζ with ζ ≠ 1 a pth root of unity. Hence, we deduce that F contains a primitive pth root of unity when [F : E] = p′. But in the case [F : E] ≠ p′ we must have F = E(ζ) for a primitive p′th root of unity (with p′ distinct from the characteristic), and K/E is a degree-p subextension. Hence, [F : E] > 1, so p′ > 2. Thus, F contains a root of unity of odd prime order. ■

Appendix A. The Action of SL_{2}(Z/5Z) on j_{5}

Choose N ≥ 1 and let ζ ∈ C^{×} be a primitive Nth root of unity. Since S = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) and T = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) generate SL_{2}(Z) and hence generate the quotient SL_{2}(Z/NZ), we can determine the action of SL_{2}(Z/NZ) on C(X_{ζ}(N)) by determining the action of S and T on this function field. In genus-zero cases, it is enough to describe how S and T act on a rational parameter.

Example A.1. The rational parameter j_{5,ζ} on X_{ζ}(5) is transformed under the action of SL_{2}(Z/5Z) according to a representation ρ_{ζ}^{an} of SL_{2}(Z/5Z) in Aut(CP^{1}) ≃ PGL_{2}(C).

In order to better understand the representations ρ_{ζ}^{an} in Aut(X_{ζ}(N)), particularly their use in proving some classical identities with coefficients in Q (and not only in Q(ζ)), as we shall discuss in Appendix C for N = 5, it is convenient to develop a variant of ρ_{ζ}^{an} over Q that recovers the ζ-dependent construction ρ_{ζ}^{an} over C (or over any field of characteristic 0 that splits X^{N} − 1). The main point is to replace the constant group SL_{2}(Z/NZ) with a non-constant finite étale symplectic group over Spec Q

Fix N ≥ 1. There is a canonical non-degenerate symplectic pairing of étale sheaves on Spec Q

⟨·, ·⟩ : (μ_{N} × Z/NZ) × (μ_{N} × Z/NZ) → μ_{N}

given by ⟨(ζ, a), (ζ′, a′)⟩ = ζ^{a′}ζ′^{−a}. Let

G = Sp(μ_{N} × Z/NZ) = Aut(μ_{N} × Z/NZ, ⟨·, ·⟩)

be the associated symplectic group, so G is a finite étale Q-group. Over a splitting field K_{N}/Q of X^{N} − 1, a choice of primitive Nth root of unity ζ ∈ K_{N}^{×} identifies μ_{N} with Z/NZ and carries ⟨·, ·⟩ to the determinant form on (Z/NZ)^{2}, so this identifies G_{/K_{N}} with the constant group SL_{2}(Z/NZ). Of course, changing ζ changes this identification with SL_{2}(Z/NZ) (in a manner that we shall make explicit shortly).

Functorially, for a Q-algebra R we can uniquely describe any point γ ∈ G(R) as γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) where a, d ∈ (Z/NZ)^{×}(R) are R-automorphisms of μ_{N} and Z/NZ respectively, b ∈ μ_{N}(R), and c : μ_{N} → Z/NZ is an R-group map. In this description, the preservation of ⟨·, ·⟩ by γ says ad − c(b) = 1 and (following the conventions preceding Remark 2.5) there is a canonical left action

(A.1)  ρ : G × X_{μ}(N) → X_{μ}(N)

over Q given away from the cusps by

(γ, (E, ι)) ↦ (E, ι ∘ γ′)

with γ′ = ( \begin{matrix}d & b \\ c & a \end{matrix} ) for γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ). We can also describe ρ as a map of smooth Q-groups

ρ : G → Aut(X_{μ}(N)),

where Aut(Z) denotes the Aut-scheme of Grothendieck for any projective Q-scheme Z (this is a quasiprojective Q-group, and it is smooth by Cartier’s theorem). The scheme-theoretic image ρ(G) is a finite étale Q-subgroup of Aut(X_{μ}(N)), and since G → ρ(G) is a finite étale surjection we see that a field splitting G also splits ρ(G).

Upon extending scalars to a splitting field K_{N}/Q of X^{N} − 1 and using a primitive Nth root of unity ζ ∈ K_{N}^{×} to identify μ_{N} and Z/NZ, we may describe ρ_{/K_{N}} as a map of smooth K_{N}-groups

ρ_{ζ} : SL_{2}(Z/NZ) = G_{/K_{N}} → Aut(X_{ζ}(N))_{/K_{N}} ,

and since the source group is a constant group we can equivalently consider ρ_{ζ} as a map of ordinary groups (of K_{N}-points)

ρ_{ζ} : SL_{2}(Z/NZ) → Aut_{K_{N}} (X_{ζ}(N)).

Upon choosing an embedding ι : K_{N} ↪ C, this recovers ρ_{ι(ζ)}^{an} as in Example A.1.

If ζ′ = ζ^{e} for e ∈ (Z/NZ)^{×} then by using α_{ζ′,ζ} : X_{ζ}(N)_{/K_{N}} ≃ X_{ζ′}(N)_{/K_{N}} as in Remark 2.5 (this isomorphism respects the canonical identifications of source and target with X_{μ}(N)_{/K_{N}}) we see that ρ_{ζ} is carried to ρ_{ζ′} ∘ c_{e} where c_{e} denotes the conjugation action

γ ↦ ( \begin{matrix}1 & 0 \\ 0 & e \end{matrix} ) · γ · ( \begin{matrix}1 & 0 \\ 0 & e^{−1} \end{matrix} )

on SL_{2}(Z/NZ) (here we are using the intervention of γ ↦ γ′ in the definition of ρ and the fact that (γ_{1}γ_{2})′ = γ′_{2}γ′_{1}).

Under the natural action of Gal(K_{N}/Q) on

Aut_{K_{N}} (X_{μ}(N)_{/K_{N}}) = Aut(X_{μ}(N))(K_{N}),

any σ ∈ Gal(K_{N}/Q) gives rise to an isomorphism

[σ] : Aut_{K_{N}} (X_{ζ}(N)_{/K_{N}}) = Aut(X_{μ}(N))(K_{N}) ≃^{σ} Aut(X_{μ}(N))(K_{N}) = Aut_{K_{N}} (X_{σ(ζ)}(N)_{/K_{N}})

that is exactly intertwining with α_{σ(ζ),ζ}. Thus, by looking back at the original definition of the Q-group G we conclude that

(A.2)  [σ] ∘ ρ_{ζ} = ρ_{σ(ζ)} = ρ_{ζ} ∘ c_{e^{−1}}.

We conclude the first part of:

Theorem A.2. The subgroup ρ_{ζ}(SL_{2}(Z/NZ)) ⊆ Aut_{K_{N}} (X_{ζ}(N)) = Aut_{K_{N}} (X_{μ}(N)_{/K_{N}}) is independent of ζ and is Gal(K_{N}/Q)-stable.

When N = 5 and we identify X_{μ}(5) with P_{Q}^{1} by means of j_{5,μ} and thereby identify Aut(X_{μ}(5)) with PGL_{2/Q}, the resulting subgroups

ρ_{ζ}(SL_{2}(Z/5Z)) ⊆ PGL_{2}(Q(ζ_{5}))

are independent of the choice of primitive 5th root of unity ζ in Q(ζ_{5}) and this subgroup is Gal(Q(ζ_{5})/Q)- stable.

Proof. The only issue that requires clarification for the second part is that the classical action of σ ∈ Gal(Q(ζ_{5})/Q) on PGL_{2}(Q(ζ_{5})) agrees with the “base change” action denoted [σ] above that rested on the functorial description of PGL_{2} as the automorphism scheme of P_{Q}^{1} = X_{μ}(5) (where this final equality is defined by means of j_{5,μ} ∈ Q(X_{μ}(5))). This agreement comes down to the evident fact that the standard identification of GL_{n}(k)/k^{×} with Aut_{k}(P_{k}^{n}) for a field k carries the Aut(k)-action on matrices over to the base-change action on k-automorphisms of the k-scheme P_{k}^{n}. ■

For a primitive 5th root of unity ζ in C, we shall now describe the images of S = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) and T = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) in PGL_{2}(Q(ζ)) under ρ_{ζ}^{an} (cf. [7, vol. 2, p. 382]).

Theorem A.3. The action of SL_{2}(Z/5Z) on C(X_{ζ}(5)) satisfies

(A.3)  T j_{5,ζ} = ζ^{−1}j_{5,ζ}  Sj_{5,ζ} = \frac{−(ζ^{2} + ζ^{−2})j_{5,ζ} + 1}{j_{5,ζ} + (ζ^{2} + ζ^{−2})}.

Proof. Let K ⊆ C be the splitting field of X^{5} − 1. The moduli-theoretic approach over Spec K shows that T j_{5,ζ} and Sj_{5,ζ} must be degree-1 rational functions in j_{5,ζ} over K, and an application of (A.2) and the identity α_{ζ′,ζ}^{∗}(j_{5,ζ′}) = j_{5,ζ} ensure that these rational functions must compatible with change in ζ in the sense that if

ρ_{ζ}(g)(j_{5,ζ}) = \frac{a_{ζ}j_{5,ζ} + b_{ζ}}{c_{ζ}j_{5,ζ} + d_{ζ}}

for

M_{ζ} = (\begin{matrix}a_{ζ} & b_{ζ} \\ c_{ζ} & d_{ζ} \end{matrix}) ∈ PGL_{2}(K)

then for any σ ∈ Gal(K/Q) we have σ(M_{ζ}) = M_{σ(ζ)} in PGL_{2}(K). Thus, it is enough to prove the result for a single choice of ζ in C.

We shall now choose a connected component ℌ of C − R and let ζ = e^{2πi/5} with i = \sqrt{−1} ∈ ℌ; via π_{ζ} this lifts the SL_{2}(Z)-action on X_{ζ}(5) to the standard action of SL_{2}(Z) on ℌ and carries j_{5,ζ} back to j_{5}. Hence, T and S lift through π_{ζ} to the respective automorphisms [T] : τ ↦ τ + 1 and [S] : τ ↦ −1/τ on ℌ. The first relation in (A.3) is therefore obvious on ℌ via the q-expansion of j_{5} at ∞ (4.8).

For the second relation, observe that (4.3) and (4.5) give

([S]^{∗}j_{5})(τ) = j_{5}(−1/τ) = \frac{κ_{(0, \frac{2}{5} )}κ_{( \frac{1}{5} ,− \frac{2}{5} )}κ_{( \frac{2}{5} ,− \frac{2}{5} )}κ_{( \frac{2}{5} , \frac{2}{5} )}κ_{( \frac{1}{5} , \frac{2}{5} )}}{κ_{(0, \frac{1}{5} )}κ_{( \frac{1}{5} ,− \frac{1}{5} )}κ_{( \frac{2}{5} ,− \frac{1}{5} )}κ_{( \frac{2}{5} , \frac{1}{5} )}κ_{( \frac{1}{5} , \frac{1}{5} )}}

for τ ∈ ℌ. Now apply (4.6) to expand this as a q-series on ℌ. One finds

([S]^{∗}j_{5})(τ) = −(ζ^{2} + ζ^{−2}) + q_{τ}^{1/5}(3 + ζ + ζ^{−1}) + ⋯,

and since

j_{5}(τ) = q_{τ}^{−1/5} + q_{τ}^{4/5} − q_{τ}^{14/5} + ⋯,

it is not hard to deduce that

\frac{3 + ζ + ζ^{−1}}{Sj_{5,ζ} + (ζ^{2} + ζ^{−2})} − j_{5,ζ}

is a meromorphic function on X_{ζ}(5) with no poles. It is therefore a constant. Inspection of the q-series at ∞ shows that this constant is ζ^{2} + ζ^{−2}, and the proof of the theorem is complete upon simplification. ■

Remark A.4. It is now an easy matter to prove Ramanujan’s evaluation of F (i) for a primitive 4th root of unity i in C, without requiring Watson’s identity (1.3). Indeed, i is fixed by τ ↦ −1/τ, so we can use (A.3) with ζ = e^{±2πi/5} to obtain

j_{5}(i) = \frac{−(ζ^{2} + ζ^{−2})j_{5}(i) + 1}{j_{5}(i) + (ζ^{2} + ζ^{−2})};

either of the specific choices of ζ ensure that τ ↦ −1/τ on C − R lifts the S-action through π_{ζ}. For these choices of ζ, we have ζ^{2} + ζ^{−2} = −(1 + \sqrt{5})/2 with \sqrt{5} > 0. The visibly positive j_{5}(i) is therefore the unique positive root of

X^{2} − (1 + \sqrt{5})X − 1 = 0,

so F (i) = 1/j_{5}(i) is the unique positive root of Y^{2}+(1+\sqrt{5})Y −1 = 0. This yields (1.1). We can also similarly evaluate F (ζ_{3}) for a primitive 3rd root of unity ζ_{3} = (−1 ± \sqrt{−3})/2, as ζ_{3} is fixed by ST : τ ↦ −1/(τ + 1). The identity (1.2) may also be verified by these methods.

Appendix B. Values at cusps

To avoid confusion concerning how points in P^{1}(Q) are mapped into a modular curve, and to ensure that the moduli-theoretic action of SL_{2}(Z) on X_{ζ}(5) lifts (via π_{ζ}) to the standard linear-fractional action, fix a connected component ℌ in C − R and take ζ = e^{2πi/5} with i = \sqrt{−1} ∈ ℌ. We will identify ℌ with the quotient of C − R under negation, and so we will work with the modular curve X_{ζ}(5) as a quotient of ℌ. In particular, we consider c ∈ P^{1}(Q) as representing a cusp by means of punctured neighborhoods taken in the horocycle topology on ℌ. Using [19, Lemma 1.42], we get a set of representatives:

{0, 2/9, 1/4, 2/7, 1/3, 2/5, 1/2, 5/8, 2/3, 3/4, 1, ∞} .

The points 0, 5/8, ∞, and 2/5 are each Γ(5)-equivalent to their negatives, and hence at cusps represented by these points we can expect to get a formula for the value of j_{5} that is independent of the choice of ℌ; this cannot be expected (and indeed, does not happen) at the other cusps. There are two methods that we can use to evaluate j_{5} at the cusps. We can work algebraically via Theorem A.3 (this requires explicitly computing a word in S and T that carries ∞ to a chosen cusp) or we can work analytically with Klein forms. We shall present the analytic approach via Klein forms, and leave the computational details of the algebraic method to the interested reader.

Our list of transformation properties for Klein forms in §4 allows us to compute how the κ_{a}’s transform under the action of SL_{2}(Z). Thus, via (4.7) we can compute the q-expansion for j_{5} at any cusp and we can thereby compute the value of j_{5} at each cusp. For example, the cusp 2/9 is taken to ∞ by the matrix

α = (\begin{matrix}−4 & 1 \\ −9 & 2 \end{matrix}) .

We apply (4.5) to the product (4.7) and employ the relation (4.3) together with (4.2) to obtain

j_{5} ∘ α^{−1} = −\frac{κ_{(0, \frac{2}{5} )}κ_{( \frac{2}{5} ,0)}κ_{( \frac{1}{5} , \frac{1}{5} )}κ_{( \frac{1}{5} , \frac{2}{5} )}κ_{( \frac{2}{5} , \frac{1}{5} )}}{κ_{(0, \frac{1}{5} )}κ_{( \frac{1}{5} ,0)}κ_{( \frac{2}{5} , \frac{2}{5} )}κ_{( \frac{1}{5} ,− \frac{2}{5} )}κ_{( \frac{2}{5} ,− \frac{1}{5} )}} .

Using (4.6), we find that for τ ∈ C − R,

j_{5} ∘ α^{−1}(τ) = −(1 + ζ_{τ}^{−1}) + (1 + 3ζ_{τ} + ζ_{τ}^{2})q_{τ}^{1/5} + O(q_{τ}^{2/5})

with ζ_{τ} = e^{2πi_{τ} /5}. Let us now take τ ∈ ℌ, so ζ_{τ} = ζ = e^{2πi/5} ∈ ℌ. We conclude that the value of j_{5} at the cusp represented (via ℌ) by 2/9 is −(1 + ζ^{−1}). The computations for the other cusps are similar, and so we obtain the following table:

Table B.1

Cusp Representative  0  2/9  1/4  2/7  1/3  2/5
Value of j_{5}  −(ζ^{2} + ζ^{−2}) −(1 + ζ^{−1}) −(ζ^{−2} + ζ^{−1}) −(1 + ζ^{−2}) −(ζ^{2} + ζ^{−1}) 0
Cusp Representative  1/2  5/8  2/3  3/4  1  ∞
Value of j_{5}  −(ζ + ζ^{−2}) −(ζ + ζ^{−1}) −(1 + ζ^{2})  −(1 + ζ)  −(ζ + ζ^{2}) ∞

Remark B.2. Observe that the cusps 0, 5/8, ∞, and 2/5 represent Γ(5)-orbits that are invariant under negation, and the values of j_{5} at these cusps are independent of ℌ, or equivalently are invariant under replacing ζ = e^{2πi/5} with ζ^{−1} = e^{−2πi/5} (as we knew they had to be). In contrast, at a pair of cusp representatives such as 2/9 and 3/4, with 3/4 in the Γ(5)-orbit of −2/9, the values of j_{5} are related by replacing ζ with ζ^{−1}.

By Lemma 4.2, we can construct many meromorphic functions on X_{ζ}(5). Indeed, the construction of suitable finite subsets 𝒜 ⊆ (5^{−1}Z)^{2} and functions m : 𝒜 → Z as in Lemma 4.2 requires nothing more than linear algebra. Given any such meromorphic function

f = ∏_{a∈𝒜} κ_{a}^{m(a)}

on X_{ζ}(5), we can readily determine the order of f at each of the 12 cusps of X_{ζ}(5), and it is clear from (4.6) that any such f has divisor supported on the cusps of X_{ζ}(5).

Explicitly, the product formula (4.6) tells us f(τ) = q_{τ}^{d}(1 + ⋯), where

d = \frac{1}{2} ∑_{(a_{1},a_{2})∈𝒜} (a_{1}^{2} − a_{1})m(a_{1}, a_{2}).

Now let g_{c} ∈ SL_{2}(Z) take the cusp c to ∞, and write g_{c} = (\begin{matrix}α & β \\ γ & δ \end{matrix}). The order of f at c must be

\frac{1}{2} ∑_{(a_{1},a_{2})∈𝒜} ((a_{1}α + a_{2}γ mod 1)^{2} − (a_{1}α + a_{2}γ mod 1))m(a_{1}, a_{2}). (B.1)

In Table B.1, we have listed the value of j_{5} at each of the cusps of X_{ζ}(5), so for any cusp c we can explicitly write down the function

(B.2)  φ_{c} = \begin{cases}(j_{5} − j_{5}(c))^{−1} & if c ≠ ∞ \\ j_{5} & if c = ∞, \end{cases}

and since C(X_{ζ}(5)) = C(j_{5}) we see that φ_{c} has a simple pole at c and no other poles. Using these functions φ_{c}, the proof of any given modular identity is reduced to a finite q-series computation.

Example B.3. We illustrate the preceding considerations by proving the identity (6.14). By Theorem 6.9 we know that

f = j_{5}(τ/5)^{5} = (\frac{κ_{( \frac{2}{5} ,0)}}{κ_{( \frac{1}{5} ,0)}} )^{5}

is a rational function of j_{5}, though this also follows from Lemma 4.2 and Theorem 4.3. We find that f has simple poles at the cusps ∞, 1/4, 1/3, 1/2, 1, and simple zeroes at 2/9, 2/7, 2/5, 2/3, 3/4. Thus, by Table B.1,

\frac{f}{j_{5}} · \frac{(j_{5} + ζ^{−2} + ζ^{−1})(j_{5} + ζ^{2} + ζ^{−1})(j_{5} + ζ^{−2} + ζ)(j_{5} + ζ^{2} + ζ)}{(j_{5} + 1 + ζ^{−1})(j_{5} + 1 + ζ)(j_{5} + 1 + ζ^{−2})(j_{5} + 1 + ζ^{2})}

is constant. Evaluation at ∞ shows this constant is 1. Upon simplification we obtain

f = j_{5} · \frac{j_{5}^{4} + 3j_{5}^{3} + 4j_{5}^{2} + 2j_{5} + 1}{j_{5}^{4} − 2j_{5}^{3} + 4j_{5}^{2} − 3j_{5} + 1}.

Appendix C. Computing F_{n}(X, Y)

For any n ≥ 1 relatively prime to 5, in §6 we proved the existence and uniqueness of a primitive polynomial F_{n} ∈ Z[X, Y] such that F_{n} is absolutely irreducible over Q and F_{n} is an algebraic relation satisfied by the functions j_{5}(τ) and j_{5}(nτ). We now turn to the task of explicitly computing F_{n}(X, Y) for such n.

Klein [7, vol. 2, pp. 137–151] worked out F_{n}(X, Y) for n ≤ 11 and n = 13 with gcd(n, 5) = 1 using invariant theory and linear algebra on q-expansions. Some of these modular correspondences have been re-proved in recent years by other methods (see for example [23]). In this section, we aim to illustrate that modern computers allow us to apply Klein’s far superior techniques to efficiently compute F_{n} for many values of n. Throughout this section, we closely follow the methods of Klein [7, vol. 2] but we systematically work over Q whenever possible (as this seems the best perspective for explaining why many of the polynomialidentity formulas we shall establish only involve Q-coefficients).

We have seen in Appendix A that the action of the finite étale non-constant symplectic Q-group G = Sp(μ_{5} ×Z/5Z) on X_{μ}(5) together with the isomorphism j_{5,μ} : X_{ζ}(5) ⟶^{∼} P_{Q}^{1} gives a projective representation

ρ : G ⟶ PGL_{2/Q}

as Q-groups. We refer the reader to Appendix A for the description of ρ upon extending scalars to a field that splits X^{5} − 1. By inspecting the definition of ρ (and working on geometric points), we see that ker ρ is the order-2 center μ_{2} of G.

Theorem C.1. The projective representation ρ lifts uniquely to a representation

ρ̃: G → SL_{2/Q},

and this representation is faithful. This uniqueness persists upon any extension of the ground field.

Of course, upon extending scalars to a field that splits X^{5} − 1 and picking a primitive 5th root of unity ζ we can deduce from the theorem (and Theorem A.2) that the ordinary representation

ρ_{ζ} : SL_{2}(Z/5Z) → PGL_{2}(Q(ζ))

uniquely lifts to a representation

ρ̃_{ζ} : SL_{2}(Z/5Z) → SL_{2}(Q(ζ))

that is moreover faithful and has image that is both independent of ζ and Galois-stable.

Proof. We begin by giving a constructive proof of existence by means of modular forms, as this will be used later. The functorial left action of G on X_{μ}(5) lifts to an action on the universal generalized elliptic curve f : E → X_{μ}(5). Whereas the kernel of ρ is the center μ_{2} in G, clearly μ_{2} actions by negation on the universal elliptic curve.

The pushforward ω = ω_{E/X_{μ}(5)} of the relative dualizing sheaf is a line bundle on X_{μ}(5) and its formation commutes with arbitrary change of base on X_{μ}(5), so we get an action of G on each tensor power ω^{⊗k} covering the action on X_{μ}(5) for every k ≥ 0. We thereby get a representation of G on the Q-model

(C.1)  M_{k} = H^{0}(X_{μ}(5), ω^{⊗k})

for the space of weight-k modular forms of full level 5.

Let 𝒞 ↪ X_{μ}(5) be the reduced degree-12 divisor of cusps of X_{μ}(5). The Kodaira–Spencer isomorphism over Y_{μ}(5) induces an isomorphism of sheaves on X_{μ}(5)

(C.2)  ω^{⊗2} ≃ Ω_{X_{μ(5)}}^{1}(𝒞).

Since X_{μ}(5) ≃ P_{Q}^{1}, we know that Ω_{X_{μ(5)}/Q}^{1} has degree −2, and since 𝒞 has degree 12 we conclude from (C.1) and (C.2) that ω has degree 5 and that dim M_{k} = 5k + 1 for all k ≥ 0.

Lemma C.2. The subspace V ⊆ M_{5} of cusp forms of level 5 and weight 5 with q-expansion coefficients in Q at ∞ and vanishing to order at least 2 along 𝒞 is 2-dimensional, G-stable, and irreducible.

A Q-basis for V is

X_{a} = (κ_{( \frac{1+a}{5} ,0)}κ_{( \frac{1+a}{5} , \frac{1}{5} )}κ_{( \frac{1+a}{5} , \frac{2}{5} )}κ_{( \frac{1+a}{5} ,− \frac{2}{5} )}κ_{( \frac{1+a}{5} ,− \frac{1}{5} )})^{−1} , (C.3)

for a = 0, 1. Moreover, X_{a} has the q-product expansion

X_{a} = q^{(2+a)/5} ∏_{k=1}^{∞}(1 − q^{5k})(1 − q^{k})^{9} ∏_{\begin{matrix}k>0 \\ k≡±(2−a) mod 5 \end{matrix}}(1 − q^{k}) (C.4)

for a = 0, 1.

Proof. By our calculations above, for a G-stable effective divisor D on X_{μ}(5) we see that the G-stable subspace H^{0}(X_{μ}(5), ω^{⊗5}(−D)) in M_{5} has dimension 25 − deg(D) + 1. The j-map X_{μ}(5) → X(1) is Gequivariant with fibers of order 60 = | SL_{2}(Z/5Z)/{±1}| away from 0, 1728, ∞ ∈ X(1) and fibers of orders 30, 20, and 12 over 0, 1728, ∞ ∈ X(1) respectively, and the only pair of such orders that add up to 24 is 12+12. Thus, the only degree-24 effective G-invariant divisor on X_{μ}(5) is 2𝒞, and clearly V = H^{0}(X_{μ}(5), ω^{⊗5}(−2𝒞)) is a 2-dimensional G-stable subspace of M_{5}. It is irreducible because otherwise there would be a G-stable line and hence a copy of the trivial representation, yet the center μ_{2} acts on M_{5} through negation and hence there is no trivial subrepresentation.

Lemma 4.2 shows that X_{a} is a modular form of weight 5 on Γ(5) for a = 0, 1, and (4.5) may be employed to expand X_{0}, X_{1} about each cusp to infer that X_{0}, X_{1} have zeroes along 𝒞 with order at least 2. Using (4.6), one finds that X_{a} has the q-product (C.4) for a = 0, 1, and it is clear from these q-expansions that X_{0}, X_{1} are linearly independent. By the q-expansion principle, these elements of C ⊗_{Q} V lie in the Q-subspace V ⊆ M_{5}, and hence X_{0} and X_{1} are a basis of V. ■

Let ρ̃: G → GL(V) be the representation of G on V; since G has no nontrivial 1-dimensional characters, ρ̃ is a representation into SL(V). We have computed that the line bundle ℒ = ω^{⊗5}(−2𝒞) on X_{μ}(5) has degree 1, and since X_{μ}(5) is a projective line we therefore conclude that the basis X_{0} and X_{1} of V = H^{0}(X_{μ}(5), ℒ) generates ℒ. Thus, the data (ℒ; X_{0}, X_{1}) defines a morphism

X_{μ}(5) → P(V) = P_{Q}^{1}

over Q and this must be an isomorphism; in fact, a check of q-series shows X_{0}/X_{1} = j_{5,μ}, so this isomorphism is exactly the one that is defined by j_{5,μ}. ^{1} In this way we see that we have lifted the G-action on X_{μ}(5) to an action on its homogeneous coordinate ring Q[X_{0}, X_{1}] with respect to the rational parameter j_{5,μ}, and consequently we have lifted ρ to a representation into SL_{2}. This lift is faithful because ker ρ = μ_{2} acts by negation on ω over the trivial action on X_{μ}(5) and so also acts by negation on odd tensor-powers of ω.

For uniqueness of the lift it is enough to work over fields K that split X^{5} − 1 and to consider liftings of ρ_{ζ} for ζ ∈ K a primitive 5th root of unity. Let S and T be as in Appendix A.^{2} Recalling the presentation

SL_{2}(Z/5Z) = ⟨S, T | T^{5} = 1, S^{2} = (ST)^{3} = −1⟩,

lifting ρ_{ζ} is equivalent to giving matrices ρ̃_{ζ}(S) and ρ̃_{ζ}(T) satisfying the relations ρ̃_{ζ}(T)^{5} = 1 and ρ̃_{ζ}(S)^{2} = (ρ̃_{ζ}(S)ρ̃_{ζ}(T))^{3} = −1. In Theorem A.3 we explicitly computed ρ_{ζ}(S), ρ_{ζ}(T) ∈ PGL_{2}(K). As we require ρ̃_{ζ} to lift ρ_{ζ}, we see that our choices are

(C.5)  ρ̃_{ζ}(T) = ± (\begin{matrix}ζ^{2} & 0 \\ 0 & ζ^{3} \end{matrix})  ρ̃_{ζ}(S) = ±\frac{1}{\sqrt{5}} (\begin{matrix}ζ^{−1} − ζ & ζ^{−2} − ζ^{2} \\ ζ^{−2} − ζ^{2} & ζ − ζ^{−1} \end{matrix}) ,

where we take \sqrt{5} = ζ + ζ^{−1} − ζ^{2} − ζ^{−2}. A short computation with the relations reveals that we must select the positive sign in both cases, and that this works (thereby giving a second existence proof that descends to Q by uniqueness and Galois descent). ■

Put M_{n} = ( \begin{matrix}n & 0 \\ 0 & 1 \end{matrix} ) and note that since (n, 5) = 1, conjugation by M_{n} induces an automorphism of G depending only n mod 5. Define the representation

ρ̃_{n} : G ↪ SL_{2/Q}

on points by the functorial recipe

ρ̃_{n}(g) = ρ̃(M_{n}gM_{n}^{−1}),

so in particular we have ρ̃_{1} = ρ̃. For an extension K/Q splitting X^{5} − 1 and a primitive 5th root of unity ζ ∈ K^{×}, we write ρ̃_{ζ,n} to denote the representation of SL_{2}(Z/5Z) ≃ G(K) into SL_{2}(Q(ζ)) corresponding to ρ̃_{n/K}.

In §6 we defined the polynomial F_{n}(X, Y) as the dehomogenization (with respect to (∞, ∞)) of a certain absolutely irreducible bihomogenous polynomial F̃_{n}(X_{0}, X_{1}; Y_{0}, Y_{1}) whose zero-scheme Z_{5,n,Q} ⊆ P_{Q}^{1} × P_{Q}^{1} is the generic fiber of the scheme-theoretic image of the map

\overline{π}_{n} × \overline{π}′_{n} : X(Γ_{μ}(5), Γ_{0}(n)) → X_{μ}(5) ×_{Z[1/5]} X_{μ}(5)

1Throughout, Klein works with coordinates ζ_{1}, ζ_{2} satisfying ζ_{1}/ζ_{2} = 1/j_{5} = F and regards F as a rational parameter on X(5). This convention seems somewhat at odds with the case of X(1), where the classical rational parameter j has a simple pole at the cusp ∞. The function 1/j_{5} in fact has a simple zero at ∞.

2We follow the standard conventions; see for example [18, p. 77]. Klein [7] reverses the roles of S and T.

over Z[1/5]. As we noted in the proof of Lemma 6.2, the Kroneckerian model Z_{5,n} over Z[1/5] as in Definition 6.1 is the scheme-theoretic image of (\overline{π}_{n} × \overline{π}′_{n}) ∘ ϕ for any automorphism ϕ of X(Γ_{μ}(5), Γ_{0}(N)).

Lemma C.3. The action ρ × ρ_{n} of G on X_{μ}(5) ×_{Q} X_{μ}(5) restricts to an automorphism of Z_{5,n,Q}.

Proof. It suffices to check this over a field Q(ζ) generated by a primitive 5th root of unity ζ, over which we may identify X_{μ}(5) with X_{ζ}(5) and X(Γ_{μ}(5), Γ_{0}(n)) with X(Γ(5), Γ_{0}(n))_{Q(ζ)} For any γ ∈ SL_{2}(Z), let \overline{γ} denote the image of γ under the canonical map SL_{2}(Z) → SL_{2}(Z/5Z) = G(Q(ζ)). The diagram

(C.6)  \xymatrix{X(Γ(5), Γ_{0}(n))_{Q(ζ)} \ar[r]^{\overline{π}_{n}×\overline{π}′_{n}} \ar[d]_{γ} & X_{ζ}(5) × X_{ζ}(5) \ar[d]^{\overline{γ}×M_{n}\overline{γ}M_{n}^{−1}} \\ _{X(Γ(5), Γ_{0}(n))_{Q(ζ)}} \ar[r]_{\overline{π}_{n}×\overline{π}′_{n}} & X_{ζ}(5) × X_{ζ}(5)}

is readily seen to commute, where γ and \overline{γ} act (away from the cusps) as in (2.2). Thus, the action of G(Q(ζ)) through ρ × ρ_{n} is intertwined with a group action on X(Γ(5), Γ_{0}(n))_{Q(ζ)} and thus restricts to an action on the scheme-theoretic image Z_{5,n,Q}. ■

Let R ⊆ Q[X_{0}, X_{1}] ⊗_{Q} Q[Y_{0}, Y_{1}] be the homogeneous coordinate of X_{μ}(5) × X_{μ}(5); that is, it is the Q-subalgebra generated by all bihomogeneous polynomials of bidegree (v, v) for v ≥ 0. By the preceding lemma, the action of G on R through ρ̃ ⊗ ρ̃_{n} preserves the Q-line spanned by the irreducible generator F̃_{n}(X_{0}, X_{1}; Y_{0}, Y_{1}) of the ideal of Z_{5,n,Q}. Since G has no 1-dimensional characters, we conclude that F̃_{n} is G-invariant with respect to ρ̃⊗ ρ̃_{n}.

To ease notation, let G(n) denote G equipped with its action on R through ρ̃⊗ ρ̃_{n}. Following Klein, we now determine polynomial generators for R^{G(n)} for each n ∈ (Z/5Z)^{×}. Since F̃_{n} ∈ R^{G(n)}, in this way we will be able to compute F̃_{n} very efficiently.

Lemma C.4. There is a map R^{G(n)} → R^{G(−n)} given by

(C.7)  f(X_{0}, X_{1}; Y_{0}, Y_{1}) ↦ f(−X_{1}, X_{0}; Y_{0}, Y_{1}),

and this map is an isomorphism.

By this lemma, it suffices to determine generators for R^{G(n)} when n ≡ 1, 2 mod 5.

Proof. We may extend scalars to Q(ζ) with ζ a primitive 5th root of unity. Let δ_{n} be the automorphism of X_{μ}(5) defined by (6.2). Using the canonical isomorphism X_{ζ}(5)_{Q(ζ)} ≃ X_{μ}(5)_{Q(ζ)}, a short calculation shows

(δ_{2} × M_{n}δ_{2}^{−1}M_{n}^{−1}) ∘ (g × M_{n}gM_{n}^{−1}) ∘ (g × M_{n}δ_{2}M_{n}^{−1}) = g × M_{n}δ_{2}^{−1}gδ_{2}M_{n}^{−1} = g × M_{−n}gM_{−n}^{−1}

as automorphisms of X_{ζ}(5) × X_{ζ}(5) for any g ∈ SL_{2}(Z/5Z) = G(Q(ζ)), and hence

(ρ̃_{ζ} ⊗ ρ̃_{ζ,−n})(g) = (ρ̃_{ζ}(δ_{2})^{−1} ⊗ 1)(ρ̃_{ζ} ⊗ ρ̃_{ζ,n})(g)(ρ̃_{ζ}(δ_{2}) ⊗ 1). (C.8)

By Corollary 5.5 we have ρ̃_{ζ}(δ_{2}) = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) up to a sign ambiguity that is easily resolved by the proof of Theorem C.1, so ρ̃_{ζ}(δ_{2}) is self-inverse up to a sign and hence (C.8) provides the desired map. ■

The case n ≡ 1 mod 5 is simpler than the case n ≡ 2 mod 5 because the action ρ̃ × ρ̃_{n} is the diagonal action of ρ̃ = ρ̃_{1} for such n. Thus, we will first work out the case n ≡ 1 mod 5. Before we can determine the associated ring of invariants R^{G(1)} = R^{G}, we need some terminology and results from clasical invariant theory (cf. [12, §7]).

Let ρ : Γ → GL(V) be an algebraic representation of a finite étale K-group on a finite-dimensional vector space V over a field K of characteristic 0, and let K[V^{∗}] be the symmetric algebra of the dual space V^{∗}. For any positive integer r we let Γ act on K[V^{∗}^{⊕r}] ≃ K[V^{∗}]^{⊗r} via the action ρ^{∗}^{⊗r}, where ρ^{∗} denotes the dual right action of Γ on K[V^{∗}]. For any s ≥ r, the linear projection map pr : V^{⊕s} → V^{⊕r} onto the first r copies of V gives a Γ-equivariant inclusion

(C.9)  K[V^{∗}^{⊕r}] ↪ K[V^{∗}^{⊕s}],

and in this way we will identify K[V^{∗}^{⊕r}] as a subring of K[V^{∗}^{⊕s}].

Definition C.5. For each i, j with 1 ≤ i, j ≤ s, the polarization operators Δ_{ij} : K[V^{∗}^{⊕s}] → K[V^{∗}^{⊕s}] are the linear maps defined by

Δ_{ij}f(v_{1}, …, v_{s}) = \frac{f(v_{1}, …, v_{j} + tv_{i} …, v_{s}) − f(v_{1}, …, v_{s})}{t}|_{t=0} (C.10)

with v_{j} + tv_{i} in the jth component. The K-algebra generated by all Δ_{ij} with 1 ≤ i, j ≤ s will be denoted 𝒰(s).

The importance of the polarization algebra is that it sometimes allows us to construct K[V^{∗}^{⊕s}]^{Γ} from K[V^{∗}^{⊕r}]^{Γ} when s > r.

Definition C.6. Let dim V = r ≤ s, and for any ordered r-tuple 1 ≤ i_{1} ≤ … ≤ i_{r} ≤ s of integers let

[i_{1}, …, i_{r}] ∈ (V^{⊕s})^{∗} = V^{∗}^{⊕s}

be the composition V^{⊕s} → ∧^{r}V ⟶^{det} K, where V^{⊕s} → ∧^{r}V is given by

(v_{1}, …, v_{s}) ↦ v_{i_{1}} ∧ v_{i_{2}} ∧ … ∧ v_{i_{r}}.

Observe that [i_{1}, …, i_{r}] is SL(V) invariant.

Theorem C.7. Let r = dim V and suppose that Γ acts on V through SL(V). Let S ⊂ K[V^{∗}^{⊕(r−1)}]^{Γ} be a subset that generates K[V^{∗}^{⊕(r−1)}]^{Γ} as a K-algebra. Identify K[V^{∗}^{⊕(r−1)}]^{Γ} as a subring of K[V^{∗}^{⊕s}]^{Γ} as in (C.9).

For any s ≥ r, K[V^{∗}^{⊕s}]^{Γ} is generated as a K-algebra by the union of the set of all elements [i_{1}, …, i_{r}] ∈ K[V^{∗}^{s}] and the set 𝒰(s)(S) obtained by applying all polarization operators in 𝒰(s) to S.

Proof. It suffices to prove the result after extending the ground field K, and so we may suppose that the finite étale K-group Γ is an ordinary finite group. This case is [12, Cor. 1, §7.5]. ■

Returning to our original situation of interest, we wish to compute Q-algebra generators for the invariants R^{G(1)} ⊆ Q[X_{0}, X_{1}; Y_{0}, Y_{1}]^{G(1)} in the Q-subalgebra R generated by bihomogeneous polynomials of bidegree (v, v) for v ≥ 0. We first use Theorem C.7 with Γ = ρ̃(G) and s = r = 2 to compute Q-algebra generators of Q[X_{0}, X_{1}; Y_{0}, Y_{1}]^{G(1)} by computing Q-algebra generators of Q[X_{0}, X_{1}]^{G}.

We will prove the following classical theorem, formulated to work over Q and not just over C:

Theorem C.8. Let

Φ_{12} = X_{0}X_{1}(X_{0}^{10} + 11X_{0}^{5}X_{1}^{5} − X_{1}^{10})
Φ_{20} = −(X_{0}^{20} + X_{1}^{20}) + 228(X_{0}^{15}X_{1}^{5} − X_{1}^{15}X_{0}^{5}) − 494X_{0}^{10}X_{1}^{10}
Φ_{30} = (X_{0}^{30} + X_{1}^{30}) + 522(X_{1}^{25}X_{0}^{5} − X_{0}^{25}X_{1}^{5}) − 10005(X_{0}^{20}X_{1}^{10} + X_{1}^{20}X_{0}^{10}).

Then Q[X_{0}, X_{1}]^{G} is generated over Q by Φ_{12}, Φ_{20}, Φ_{30}, which satisfy the relation

1728Φ_{12}^{5} − Φ_{20}^{3} − Φ_{30}^{2} = 0. (C.11)

Moreover, if T is the graded polynomial ring Q[U_{12}, U_{20}, U_{30}] where U_{i} is given degree i then the kernel of the surjective graded Q-algebra homomorphism α : T ↠ Q[X_{0}, X_{1}]^{G} given by α : U_{i} ↦ Φ_{i} is a principal ideal, generated by 1728U_{12}^{5} − U_{20}^{3} − U_{30}^{2}.

Proof. Let f ∈ Q[X_{0}, X_{1}]^{G} be any nonzero homogeneous polynomial. The closed subscheme Z = V (f) ⊆ X_{μ}(5) is G-stable and is therefore topologically a finite union G-orbits. Conversely, any finite union of Gorbits defines a G-stable closed subscheme of X_{μ}(5) that is the zero-scheme of a nonzero radical homogeneous polynomial f that is unique up to unit-scaling and hence the line Q · f supports a 1-dimensional algebraic representation of G; the only such representation is the trivial one, so such f must be G-invariant.

Now the map j : X_{μ}(5) → P_{Q}^{1} is a degree-60 covering branched only over j = 0, j = ∞ and j = 1728, and it is generically a torsor for the generically-free action of the quotient ρ(G) = G/μ_{2} of G by its center. In particular, over Q(ζ_{5}) this becomes a Galois covering with Galois group SL_{2}(Z/5Z)/{±1} ≃ A_{5}. By identifying X_{μ}(5) with X_{ζ}(5) over Q(ζ) we see that j^{−1}(0) consists of 20 (geometrical) points with ramification index 3, j^{−1}(∞) consists of the 12 (geometrical) cusps with ramification index 5, and j^{−1}(1728) has 30 (geometrical) points with ramification index 2. For any field K of characteristic 0 and any t ∈ P^{1}(K), j^{−1}(t) is a G-orbit on X_{μ}(5)_{/K}. We write ϕ_{t} to denote the associated nonzero radical G-invariant homogeneous polynomial in K[X_{0}, X_{1}] (so ϕ_{t} is well-defined modulo K^{×}). (Klein [11, p. 50] calls the ϕ_{t} ground forms). We claim that the ring Q[X_{0}, X_{1}]^{G} is generated by ϕ_{0}, ϕ_{∞}, and ϕ_{1728}.

It suffices to work over K = \overline{Q}, and by working over this K we may identify every G-orbit in X_{μ}(5)(K) as j^{−1}(t) for a unique t ∈ P^{1}(K). If ϕ ∈ (K ⊗_{Q} R)^{G} is nonzero then its K-finite zero-scheme V (ϕ) on X_{μ}(5)_{/K} is a union of orbits and hence ϕ is a product of factors that are G-stable up to K^{×}-multiple and hence are G-stable (as G has no non-trivial 1-dimensional characters). Thus, we may suppose that the zero-scheme of ϕ is topologically a G-orbit, so V (ϕ) = j^{−1}(t) topologically for some t ∈ P^{1}(K). By G-invariance, it follows that ϕ has zeros of equal order at all points of V (ϕ) and hence ϕ is a scalar multiple of a power of the radical ϕ_{t} whose degree is equal to the size of j^{−1}(t). We therefore may restrict our attention to the ϕ_{t}’s.

Suppose that t ∉ {0, ∞, 1728}, pick v ∈ j^{−1}(t), and consider the polynomial

ϕ := ϕ_{∞}^{5}(v)ϕ_{0}^{3} − ϕ_{0}^{3}(v)ϕ_{∞}^{5}.

By construction, ϕ is a G-invariant homogeneous polynomial of degree 60 = |G|/2 vanishing at v, and hence it vanishes on the G-orbit j^{−1}(t) of v. This orbit has |G|/2 distinct geometrical points, so ϕ is a scalar multiple of ϕ_{t}. This shows that all ϕ_{t} lie in the subalgebra generated by ϕ_{0}, ϕ_{1728}, and ϕ_{∞}.

Since G is finite, Q[X_{0}, X_{1}] is a finite Q[X_{0}, X_{1}]^{G}-module, so both rings have Krull dimension 2. Moreover, Q[X_{0}, X_{1}]^{G} is a domain and hence ker α is a prime ideal which (by dimension considerations) must have height 1. But T is a polynomial ring, hence a UFD, so every height-1 prime is principal. Thus, up to Q^{×}- multiple there is a unique irreducible algebraic relation among U_{12}, U_{20}, and U_{30} over Q that is contained in ker α and it must generate ker α. The same holds when Q is replaced with any field of characteristic 0. The preceding considerations apply with t = 1728 and thereby provide an algebraic relation

(C.12)  ϕ_{1728}^{2} = ϕ_{∞}^{5}(v)ϕ_{0}^{3} − ϕ_{0}^{3}(v)ϕ_{∞}^{5}

over \overline{Q} for any v ∈ j^{−1}(1728); the coefficients ϕ_{∞}(v), ϕ_{0}(v) cannot both vanish (as ϕ_{1728} ≠ 0), so this is an irreducible algebraic relation over \overline{Q}. The monicity in ϕ_{1728}^{2} and the uniqueness of the irreducible relation up to \overline{Q}^{×}-multiple forces the relation (C.12) to have coefficients in Q.

It remains to compute ϕ_{0}, ϕ_{∞}, and ϕ_{1728} explicitly (up to Q^{×}-multiple) and to find the explicit coefficients in (C.12) upon making specific choices of ϕ_{0}, ϕ_{∞}, and ϕ_{1728}. In Table B.1, we computed the values of j_{5} on the geometrical cusps j^{−1}(∞). As G acts transitively on the geometrical cusps of X_{μ}(5) we can immediately write down the degree-12 G-invariant polynomial

Φ_{12} := X_{0}X_{1} ∏_{c∈j^{−1}(∞)−{∞,0}}(X_{0} − X_{1}j_{5}(c)) = X_{0}X_{1}(X_{0}^{10} + 11X_{0}^{5}X_{1}^{5} − X_{1}^{10}),

which by construction must be (a Q^{×}-multiple of) ϕ_{∞}. The coordinate-free theory of the Hessian ensures that the Hessian determinant

H(Φ_{12}) = |\begin{matrix}∂^{2}Φ_{12}/∂X_{0}^{2} & ∂^{2}Φ_{12}/∂X_{0}∂X_{1} \\ ∂^{2}Φ_{12}/∂X_{1}∂X_{0} & ∂^{2}Φ_{12}/∂X_{1}^{2} \end{matrix}|

is G-invariant (more specifically, by [11, pp. 56–62] this is a covariant of Φ_{12} of degree 2(12 − 2) = 20), so

Φ_{20} := \frac{1}{121}H(Φ_{12}) = −(X_{0}^{20} + X_{1}^{20}) + 228(X_{0}^{15}X_{1}^{5} − X_{1}^{15}X_{0}^{5}) − 494X_{0}^{10}X_{1}^{10}

is a degree-20 invariant polynomial. As there is a unique G-orbit of degree 20 on X_{μ}(5), namely j^{−1}(0), we conclude that Φ_{20} is a Q^{×}-multiple of ϕ_{0}. Similarly, the differential determinant

Φ_{30} := −\frac{1}{20} |\begin{matrix}∂Φ_{12}/∂X_{0} & ∂Φ_{12}/∂X_{1} \\ ∂Φ_{20}/∂X_{1} & ∂Φ_{20}/∂X_{1} \end{matrix}|

= (X_{0}^{30} + X_{1}^{30}) + 522(X_{1}^{25}X_{0}^{5} − X_{0}^{25}X_{1}^{5}) − 10005(X_{0}^{20}X_{1}^{10} + X_{1}^{20}X_{0}^{10})

of degree 3(12 − 2) = 30 must be a G-invariant polynomial, and must therefore coincide with a Q^{×}-multiple of ϕ_{1728} because there is a unique G-orbit of degree 30, and 30 cannot be written as a sum of positive integral multiples of 12 and 20.

We may now rephrase (C.12) as an algebraic relation

aΦ_{12}^{5} − bΦ_{20}^{3} − Φ_{30}^{2} = 0

for some unique a, b ∈ Q not both zero. Using (C.4) and the definitions of the Φ’s in terms of X_{0} and X_{1}, comparison of a few q-series coefficients allows us to solve for a and b to obtain the relation (C.11). ■

Before we proceed further with our computation of R^{G(n)}, let us use (C.4) to identify Φ_{12}, Φ_{20}, Φ_{30} as modular forms for SL_{2}(Z). By (C.4) we may identify the coordinates X_{0} and X_{1} with weight-5 modular forms for Γ(5), so Φ_{12}, Φ_{20}, and Φ_{30} are modular forms for Γ(5) with weights 60, 100, 150 respectively; the G(C)-invariance implies that these are in fact modular forms of full level 1. By comparing q-expansions, we thereby easily find

(C.13)  Φ_{12} = −Δ^{5} = −q^{5} + 120q^{6} − 7020q^{7} + ⋯
(C.14)  Φ_{20} = Δ^{8}E_{2} = q^{8} + 48q^{9} − 25776q^{10} + ⋯
(C.15)  Φ_{30} = Δ^{12}E_{3} = q^{12} − 792q^{13} + 169560q^{14} + ⋯

where

Δ = q ∏_{k=1}^{∞}(1 − q^{k})^{24},  E_{2} = 1 + 240 ∑_{k=1}^{∞} \frac{k^{3}q^{k}}{1 − q^{k}} ,  E_{3} = 1 − 504 ∑_{k=1}^{∞} \frac{k^{5}q^{k}}{1 − q^{k}} .

From these identifications, we immediately see that j = E_{2}^{3}/Δ = Φ_{20}^{3}/Φ_{12}^{5}. As we have the identification j_{5} = X_{0}/X_{1}, we use Theorem C.8 to find

j = \frac{1}{j_{5}^{5}} · \begin{matrix}(j_{5}^{20} − 228j_{5}^{15} + 494j_{5}^{10} + 228j_{5}^{5} + 1) & 3 \\ (j_{5}^{10} − 11j_{5}^{5} − 1) & 5 \end{matrix} . (C.16)

This “icosahedral equation” was of course well-known to Klein [11, pp. 60–66].

Returning now to the determination of R^{G(n)} for n ≡ 1 mod 5, we know from Theorem C.7 that Q[X_{0}, X_{1}; Y_{0}, Y_{1}]^{G(1)} is generated by the determinant [1, 2] as in Theorem C.7 and all polarizations of Φ_{12}, Φ_{20}, and Φ_{30}. Define

A_{1} := α_{1}[1, 2],

and for i = 6, 10, 15 put

(C.17)  A_{i} := α_{i}Δ_{12}^{i}Φ_{2i}

where α_{i} ∈ Q^{×} is chosen so as to make A_{i} primitive (with an arbitrary choice of sign) for i ∈ {1, 6, 10, 15}. After a short calculation with the explicit formula (C.10) (and an arbitrary choice of sign), we find

A_{1} = X_{0}Y_{1} − Y_{0}X_{1}
A_{6} = 42(X_{0}Y_{1} + Y_{0}X_{1})(X_{1}^{5}Y_{1}^{5} − X_{0}^{5}Y_{0}^{5}) + X_{0}^{6}Y_{1}^{6} + Y_{0}^{6}X_{1}^{6} + 36X_{0}Y_{0}X_{1}Y_{1}(X_{0}^{4}Y_{1}^{4} + Y_{0}^{4}X_{1}^{4})

+ 225X_{0}^{2}Y_{0}^{2}X_{1}^{2}Y_{1}^{2}(X_{0}^{2}Y_{1}^{2} + Y_{0}^{2}X_{1}^{2}) + 400X_{0}^{3}Y_{0}^{3}X_{1}^{3}Y_{1}^{3}

A_{10} = 374(X_{1}^{10}Y_{1}^{10} + X_{0}^{10}Y_{0}^{10}) − 66(X_{1}^{5}Y_{1}^{5} − X_{0}^{5}Y_{0}^{5})(21(X_{0}^{5}Y_{1}^{5} + Y_{0}^{5}X_{1}^{5}) + 175X_{0}Y_{0}X_{1}Y_{1}(X_{0}^{3}Y_{1}^{3} + Y_{0}^{3}X_{1}^{3})

+ 450X_{0}^{2}Y_{0}^{2}X_{1}^{2}Y_{1}^{2}(X_{0}Y_{1} + Y_{0}X_{1})) + (X_{0}^{10}Y_{1}^{10} + Y_{0}^{10}X_{1}^{10}) + 100X_{0}Y_{0}X_{1}Y_{1}(X_{0}^{8}Y_{1}^{8} + Y_{0}^{8}X_{1}^{8})

+ 2025X_{0}^{2}Y_{0}^{2}X_{1}^{2}Y_{1}^{2}(X_{0}^{6}Y_{1}^{6} + Y_{0}^{6}X_{1}^{6}) + 14400X_{0}^{3}Y_{0}^{3}X_{1}^{3}Y_{1}^{3}(X_{0}^{4}Y_{1}^{4} + Y_{0}^{4}X_{1}^{4})

+ 44100X_{0}^{4}Y_{0}^{4}X_{1}^{4}Y_{1}^{4}(X_{0}^{2}Y_{1}^{2} + Y_{0}^{2}X_{1}^{2}) + 63504X_{0}^{5}Y_{0}^{5}X_{1}^{5}Y_{1}^{5}

A_{15} = X_{0}^{15}Y_{0}^{15} + Y_{1}^{15}X_{1}^{15} + (X_{1}^{10}Y_{1}^{10} − Y_{0}^{10}X_{0}^{10})(11(X_{0}^{5}Y_{1}^{5} + Y_{0}^{5}X_{1}^{5}) + 75X_{0}Y_{0}X_{1}Y_{1}(X_{0}^{3}Y_{1}^{3} + Y_{0}^{3}X_{1}^{3})

+ 175X_{0}^{2}Y_{0}^{2}X_{1}^{2}Y_{1}^{2}(X_{0}Y_{1} + Y_{0}X_{1})) − (X_{0}^{5}Y_{0}^{5} + Y_{1}^{5}X_{1}^{5})(X_{0}^{10}Y_{1}^{10} + Y_{0}^{10}X_{1}^{10}

+ 25X_{0}Y_{0}X_{1}Y_{1}(X_{0}^{8}Y_{1}^{8} + Y_{0}^{8}X_{1}^{8}) + 225X_{0}^{2}Y_{0}^{2}X_{1}^{2}Y_{1}^{2}(X_{0}^{6}Y_{1}^{6} + Y_{0}^{6}X_{1}^{6}) + 975X_{0}^{3}Y_{0}^{3}X_{1}^{3}Y_{1}^{3}(X_{0}^{4}Y_{1}^{4} + Y_{0}^{4}X_{1}^{4})

+ 2275X_{0}^{4}Y_{0}^{4}X_{1}^{4}Y_{1}^{4}(X_{0}^{2}Y_{1}^{2} + Y_{0}^{2}X_{1}^{2}) + 3003X_{0}^{5}Y_{0}^{5}X_{1}^{5}Y_{1}^{5}).

Observe that each A_{i} is bihomogenous and G-invariant. Thus, the ring S := Q[A_{1}, A_{6}, A_{10}, A_{15}] is a subring of R^{G}. We claim that S = R^{G}. If not, we may choose (a possibly non-unique) f ∈ R^{G} − S of minimal positive bihomogeneous degree (d_{0}, d_{0}). The G(1)-equivariant linear map V → V^{⊕2} given by v ↦ (v, v) defines a Q-algebra homomorphism L : Q[V^{∗}^{⊕2}]^{G(1)} → Q[V^{∗}]^{G(1)} and we use Theorem C.8 to write L(f) = P (Φ_{12}, Φ_{20}, Φ_{30}) for some polynomial P with Q-coefficients. One checks easily from the definitions of the polarization operators (C.10) and the polynomials A_{i} for i ∈ {1, 6, 10, 15} (C.17) that

L(A_{i}) = \begin{cases}0 & if i = 1 \\ β_{i}Φ_{2i} & otherwise \end{cases} ,

for some β_{i} ∈ Q^{×}. Letting

g = f − P (A_{6}/β_{6}, A_{10}/β_{10}, A_{15}/β_{15}) ,

it follows that L(g) = 0 so the polynomial g vanishes on the zero-set of A_{1}. As A_{1} is absolutely irreducible, we conclude that A_{1}|g so we have

f = A_{1}f_{1} + P (A_{6}/β_{6}, A_{10}/β_{10}, A_{15}/β_{15}) ,

where f_{1} is evidently a bihomogeneous G(1)-invariant polynomial of bidegree (d_{0} − 1, d_{0} − 1), hence a polynomial in the A_{i} for i ∈ {1, 6, 10, 15} by our minimality assumption on d_{0}. It follows that f ∈ S afterall, and that the polynomials A_{i} for i ∈ {1, 6, 10, 15} generate R^{G(n)} as a Q-algebra for n ≡ 1 mod 5.

Observe that A_{15}, A_{10}, and A_{6} are uniquely determined as homogeneous generators of R^{G(1)} only modulo A_{1}. As Klein [11, p. 242] observes, A_{6}, A_{10}, and A_{15} do not have particularly nice geometric interpretations, and as a consequence, when n ≡ 1 mod 5 the formulae for F̃_{n} in terms of A_{1}, A_{6}, A_{10}, and A_{15} are somewhat messy and involve rather large coefficients. Following Klein, we choose more convenient generators by defining^{3}

(C.18)  \begin{aligned}W_{2} & = A_{1}^{2} \\ W_{6} & = \frac{−A_{6} + A_{1}^{6}}{42} \\ W_{10} & = \frac{A_{10} − A_{1}^{10} + 110A_{1}^{4}W_{6}}{374} \\ W_{15} & = A_{15}. \end{aligned}

It is immediate from these definitions that R^{G(1)} = Q[A_{1}, W_{6}, W_{10}, W_{15}]. By Lemma 6.2(2), we know that for n ≡ ±1 mod 5 the modular polynomial F̃_{n} satisfies F̃_{n}(X_{0}, X_{1}; Y_{0}, Y_{1}) = F̃_{n}(Y_{0}, Y_{1}; X_{0}, X_{1}). Observe that this transformation h(X_{0}, X_{1}; Y_{0}, Y_{1}) ↦ h(Y_{0}, Y_{1}; X_{0}, X_{1}) that preserves F̃_{n} also preserves W_{6}, W_{10}, W_{15} but sends A_{1} to −A_{1}. We conclude that F̃_{n} is a polynomial in W_{2} = A_{1}^{2}, W_{6}, W_{10}, W_{15}. Such an expression for F̃_{n} is not unique, as the relation (C.11) gives rise to a relation among the W_{i}:

(C.19) 16W_{15}^{2} = W_{2}^{5}W_{10}^{2} − 10W_{2}^{4}W_{6}^{2}W_{10} + 25W_{2}^{3}W_{6}^{4} − 40W_{2}^{2}W_{6}W_{10}^{2} + 360W_{2}W_{6}^{3}W_{10} − 864W_{6}^{5} + 16W_{10}^{3}.

However, since n ≡ 1 mod 5, so in particular n > 2, it follows that deg π_{n} = n ∏_{p|n}(1 + 1/p) is even and hence every homogeneous component of F̃_{n} is bihomogeneous with even bidegree. Since W_{2}, W_{6}, W_{10} are bihomogeneous of even bidegree and W_{15} is bihomogeneous of odd bidegree, F̃_{n} is in fact a polynomial in W_{2}, W_{6}, W_{10} and W_{15}^{2}. Thus, (C.19) shows that F̃_{n} is a polynomial in W_{2}, W_{6}, W_{10}. We claim that such a representation is unique. Indeed, the ring R is 3-dimensional (as it is the Segre product of Q[X_{0}, X_{1}] and Q[Y_{0}, Y_{1}]) and since G(1) is finite, R^{G(1)} also has dimension 3. As R^{G(1)} is generated as a Q-algebra by A_{1}, W_{6}, W_{10}, W_{15} and we have the relation (C.19), we conclude that A_{1}, W_{6}, W_{10} are algebraically independent, so W_{2} = A_{1}^{2}, W_{6}, W_{10} must be algebraically independent.

We next compute the ring of invariants R^{G(n)} when n ≡ 2 mod 5 and use it to analyze the element F̃_{n} for such n. As the action ρ̃⊗ ρ̃_{n} of G on R ⊆ Q[X_{0}, X_{1}, Y_{0}, Y_{1}] is not the same on each set of variables X_{0}, X_{1} and Y_{0}, Y_{1}, the polarization method used to treat the case of n ≡ 1 mod 5 is not available. Following Klein [7, pp. 139–141], we will explicitly construct generators for R^{G(n)}.

We have seen that the modular polynomials F̃_{n} are contained in the N_{0}-graded subring

R ⊆ Q[X_{0}, X_{1}] ⊗ Q[Y_{0}, Y_{1}]

whose vth graded piece R_{v} is the Q-vector space of bihomogeneous polynomials of bidegree (v, v). Moreover, it is clear that R is generated as a Q-algebra by R_{1}. Let V_{X} and V_{Y} be the representation spaces underlying ρ̃ and ρ̃_{n}, and identify Q[V_{X}^{∗} ⊗ V_{Y}^{∗}] with the polynomial ring Q[y_{1}, y_{2}, y_{3}, y_{4}] via

(C.20)  y_{1} ↦ X_{0}Y_{0}  y_{2} ↦ X_{1}Y_{0}  y_{3} ↦ −X_{0}Y_{1}  y_{4} ↦ X_{1}Y_{1}.

Observe that this gives a G-equivariant isomorphism Q[y_{1}, y_{2}, y_{3}, y_{4}]/(y_{1}y_{4} + y_{2}y_{3}) ≃ R as graded rings. Geometrically, this is simply the Segre embedding P^{1} × P^{1} → P^{3}.

Since μ_{2} ⊆ G acts trivially on each y_{i}, the representation G → SL(V_{X} ⊗ V_{Y}) factors through G/μ_{2}. We claim that the representation V_{X} ⊗ V_{Y} is geometrically irreducible. We may pass to an algebraically closed ground field K, so G/μ_{2} is identified with A_{5}. Since the irreducible representations of A_{5} have dimensions 1, 3, 3, 4, 5, geometrically V_{X} ⊗ V_{Y} is either irreducible or it decomposes as the direct sum of the trivial representation with one of the three-dimensional representations or as the direct sum of four copies of the trivial representation. As n ≡ 2 mod 5, a simple calculation shows that

3In Klein’s notation [11, p. 242], we have taken W_{6} = B/2, W_{10} = C, and W_{15} = D. We have decided to depart from Klein’s conventions in order to obtain more compact formulae and to ensure that our generators of R^{G(1)} are polynomials in X_{0} and X_{1} with integer coefficients.

M_{n}T M_{n}^{−1} = T^{2},

so using the explicit description (C.5) of ρ̃_{ζ} for a primitive 5th root of unity ζ ∈ K and using the definition of ρ̃_{ζ,n} we calculate that T ∈ SL_{2}(Z/5Z) carries y_{i} to ζ^{i}y_{i} and hence the character of V_{X} ⊗ V_{Y} takes the value ζ + ζ^{2} + ζ^{−2} + ζ^{−1} = −1 on the conjugacy class of the order-5 element T. The character table for A_{5} then rules out the possibility that V_{X} ⊗V_{Y} has the trivial representation as a direct summand, and this gives our claim. Thus, V_{X} ⊗ V_{Y} is a Q-model for the unique (absolutely) irreducible 4-dimensional representation of A_{5}; that is, just as G/μ_{2} is an étale twisted form of A_{5}, V_{X} ⊗ V_{Y} is an étale twisted form of the A_{5}-stable subspace of those vectors of trace 0 inside the standard 5-dimensional permutation representation of A_{5}.

Consider the transformations

(C.21)  i_{1} :h(X_{0}, X_{1}; Y_{0}, Y_{1}) ↦ h(−Y_{1}, Y_{0}; X_{0}, X_{1})
(C.22)  i_{2} :h(X_{0}, X_{1}; Y_{0}, Y_{1}) ↦ h(Y_{0}, Y_{1}; −X_{1}, X_{0})

on R.

Lemma C.9. For any n ≡ 2 mod 5 we have F̃_{n} ∘ i_{1} = F̃_{n} and F̃_{n} ∘ i_{2} = F̃_{n} if n > 2, while F̃_{2} ∘ i_{2} = −F̃_{2}.

The sign for i_{2} should be compared with the delicate symmetry for the dehomogenized polynomial F_{n} in Theorem 6.5.

Proof. We may and do extend scalars to a field K that splits X^{5} − 1 and we choose a primitive 5th root of unity ζ ∈ K, so we thereby identify G with SL_{2}(Z/5Z) and G/μ_{2} with A_{5}. As was explained above, we may (non-canonically) identify V_{X} ⊗ V_{Y} with the hyperplane ∑ z_{j} = 0 in the 5-dimensional standard representation W for A_{5} that we endow with coordinate functions z_{1}, …, z_{5}.

We let G act on K[W^{∗}] via the standard A_{5}-action that permutes the z_{j}’s, so we have a G-equivariant graded ring homomorphism

(C.23)  φ : K[y_{1}, y_{2}, y_{3}, y_{4}]/(y_{1}y_{4} + y_{2}y_{3}) ⟶ K[z_{1}, z_{2}, z_{3}, z_{4}, z_{5}]/(z_{1} + z_{2} + z_{3} + z_{4} + z_{5})

given explicitly as in [7, p. 140] by

(C.24)  φ(y_{j}) = \frac{1}{5} ∑_{k=1}^{5} ζ^{−kj}z_{k}.

As this is a graded homomorphism and the y_{i}’s satisfy a nontrivial G-invariant quadratic relation, the z_{i}’s must also satisfy a nontrivial G-invariant quadratic relation. Since the G-invariant quadratic polynomials in the z_{i}’s are precisely the K-span of s_{1}^{2} and s_{2} with

s_{j} := ∑_{k=1}^{5} z_{k}^{j}, (C.25)

we must have that s_{2} = 0. Letting B := K[z_{1}, z_{2}, z_{3}, z_{4}, z_{5}]/(s_{1}, s_{2}), it follows that (C.24) descends to a G-equivariant isomorphism

φ : K[y_{1}, y_{2}, y_{3}, y_{4}]/(y_{1}y_{4} + y_{2}y_{3}) ⟶^{∼} B (C.26)

and hence an isomorphism (K ⊗_{Q} R)^{G(n)} ≃ B^{G}.

The K-algebra of invariants B^{G} is generated by the images of generators of K[z_{1}, z_{2}, z_{3}, z_{4}, z_{5}]^{G}; that is, B^{G} is generated by the images of the power sums (C.25) for 1 ≤ j ≤ 5 and the discriminant Δ := ∏_{i<j}(z_{i} − z_{j}). The transformations i_{1} and i_{2} correspond to the automorphisms σ∘(1×[δ_{3}]) and σ∘([δ_{3}]×1) of X_{ζ}(5)×X_{ζ}(5) occurring in Lemma 6.2. Since

σ ∘ ([δ_{3}] × 1) = σ ∘ (1 × [δ_{3}]) ∘ δ_{2} × δ_{2},

it follows from Lemma 6.2(2) and the commutative diagram (C.6) with γ = δ_{2} that

F̃_{n} ∘ i_{j} = ±F̃_{n}

for j = 1, 2; the sign may depend on j, n. It remains to determine the sign.

Observe that i_{1}(y_{i}) = y_{3i}, and thus, as one computes, i_{1}(z_{i}) = z_{2i} for 1 ≤ i ≤ 5 (where the indices are taken modulo 5). This is an odd permutation of the z_{j}, so i_{1}(s) = s for any symmetric function s in z_{1}, …, z_{5}, while i_{1}(Δ) = −Δ.

Since Δ^{2} is a polynomial in the z_{j}, we may write the image of F̃_{n} in B as u+vΔ where u, v are symmetric in the z_{i}. Thus, the image of F̃_{n} ∘ i_{1} in B is then u − vΔ, so u + vΔ = ±(u − vΔ), whence either u = 0 or v = 0 (depending on whether the sign is −1 or +1 respectively). In the former case, F̃_{n} is reducible, which is a contradiction. Thus, v = 0 and the image of F̃_{n} = F̃_{n} ∘ i_{1} in B is a polynomial in the symmetric functions s_{3}, s_{4}, s_{5} of the z_{i}. For 3 ≤ j ≤ 5 we define W_{j} ∈ K[y_{1}, y_{2}, y_{3}, y_{4}] by

W_{j} := α_{j}φ^{−1}(s_{j}),

with α_{j} ∈ Q^{×} chosen (with arbitrary sign) so as to make W_{j} primitive. Since no nonzero polynomial relation holds between the s_{j}’s, we conclude that F̃_{n} is uniquely a Q-polynomial in W_{3}, W_{4}, W_{5}.

The W_{j}’s may be written as polynomials in the X_{i}, Y_{i} by using (C.20). Using the explicit description of the isomorphism φ as in (C.24), we compute

W_{3} = −X_{0}^{3}Y_{0}^{2}Y_{1} + X_{0}^{2}X_{1}Y_{1}^{3} + X_{0}X_{1}^{2}Y_{0}^{3} + X_{1}^{3}Y_{0}Y_{1}^{2}
W_{4} = −X_{0}^{4}Y_{0}Y_{1}^{3} + X_{0}^{3}X_{1}Y_{0}^{4} − 3X_{0}^{2}X_{1}^{2}Y_{0}^{2}Y_{1}^{2} − X_{0}X_{1}^{3}Y_{1}^{4} + X_{1}^{4}Y_{0}^{3}Y_{1}
W_{5} = X_{0}^{5}Y_{0}^{5} − X_{0}^{5}Y_{1}^{5} + 10X_{0}^{4}X_{1}Y_{0}^{3}Y_{1}^{2} + 10X_{0}^{3}X_{1}^{2}Y_{0}Y_{1}^{4} + 10X_{0}^{2}X_{1}^{3}Y_{0}^{4}Y_{1} − 10X_{0}X_{1}^{4}Y_{0}^{2}Y_{1}^{3} + X_{1}^{5}Y_{0}^{5} + X_{1}^{5}Y_{1}^{5}.

Note that i_{2}(W_{j}) = (−1)^{j}W_{j} for j = 3, 4, 5. As F̃_{n} is bihomogeneous of even bidegree for n > 2, every monomial term W_{3}^{a}W_{4}^{b}W_{5}^{c} must have 3a+4b+5c ≡ a+b ≡ 0 mod 2. This shows that F̃_{n} ∘i_{2} = F̃_{n} for n > 2, and by degree considerations we evidently must have F̃_{2} = W_{3} (up to Q^{×}-scaling), so F̃_{2} ∘ i_{2} = −F̃_{2}. ■

Remark C.10. Observe that the preceding argument would not have worked to compute the ring of invariants R^{G(1)}, as in this case the 4-dimensional representation V of G/μ_{2} that we obtain is reducible, being the sum of the trivial representation and a 3-dimensional representation, as one easily sees over an algebraically closed field by computing the character values.

Remark C.11. We have taken the power sums s_{3}, s_{4}, s_{5} ∈ B in the z_{i} as generators of the ring of invariants B^{G} and used the isomorphism (C.24) to define polynomials W_{3}, W_{4}, W_{5} in X_{0}, X_{1}, Y_{0}, Y_{1}. One might decide instead to use the elementary symmetric functions e_{3}, e_{4}, e_{5} in the z_{i} instead of the power sums s_{3}, s_{4}, s_{5} to generate the ring B^{G}. However, because of Newton’s formula

(−1)^{n+1}e_{n} − s_{n} = ∑_{j=1}^{n−1}(−1)^{j}e_{j}s_{n−j} = 0

and the fact that s_{1} = s_{2} = e_{1} = e_{2} = 0 in B, we see that s_{j} is a Q^{×}-multiple of e_{j} for 1 ≤ j ≤ 5, so we conclude that the formulae for W_{3}, W_{4}, W_{5} (up to sign) do not depend on whether one chooses the power sums or the elementary symmetric functions as generators for the algebra B^{G}.

Recall that we have an isomorphism R^{G(n)} ≃ R^{G(−n)} as in (C.7) given by

f(X_{0}, X_{1}; Y_{0}, Y_{1}) ↦ f(−X_{1}, X_{0}; Y_{0}, Y_{1}).

For i = 2, 3, 4, 5, 6, 10, 15 define

Z_{i}(X_{0}, X_{1}; Y_{0}, Y_{1}) = W_{i}(−X_{1}, X_{0}; Y_{0}, Y_{1});

the W_{i}’s were all defined in the proof of Lemma C.9 and the discussion following the proof of Theorem C.8. We have now proved the following theorem:

Theorem C.12. Let n > 1 be any integer with (n, 5) = 1 and let W_{i} and Z_{i} be as above. Then F̃_{n} is uniquely a Q-polynomial in:

(C.27)  \begin{cases}W_{2}, W_{6}, W_{10} & if n ≡ 1 mod 5 \\ W_{3}, W_{4}, W_{5} & if n ≡ 2 mod 5 \\ Z_{3}, Z_{4}, Z_{5} & if n ≡ 3 mod 5 \\ Z_{2}, Z_{6}, Z_{10} & if n ≡ 4 mod 5 \end{cases}

The point is that uniqueness is inherited under extension on the ground field and also descends through such extensions, so it suffices to work over an extension of Q. Upon introducing a primitive 5th root of unity, we can apply all preceding considerations (including the ones in the proof of Lemma C.9).

Computing F_{n} for any given n relatively prime to 5 is now a matter of simple linear algebra. We know that F̃_{n} can be written as a Q-linear combination of certain monomials in the W_{i}, Z_{j}. Knowledge of the common X- and Y -degrees of F_{n} and the W_{i}, Z_{j} enables us to predict exactly which monomials can occur. Using the identification (C.4) of X_{0}, X_{1} with weight-5 modular forms for Γ(5) and hence of Y_{0}, Y_{1} with weight-5 modular forms for Γ(5) ∩ Γ_{0}(n), we may compute the q-expansions of the modular forms W_{i}, Z_{j}. By computing enough terms, we are guaranteed to find a unique relation (up to Q^{×}-scaling) in the expected degrees, and this must be F̃_{n} (up to Q^{×}-scaling). One can even specify how many q-series terms are sufficient: indeed, with the identifications above, F̃_{n} is a modular form for SL_{2}(Z) of weight k := 5n ∏_{p|n}(1 + 1/p), so we need only check the q-expansion to order ⌊k/12⌋.

Let us illustrate this with an example.

Example C.13. Let n = 8. From Table C.1, we see that F̃_{8} is an element of Q[Z_{3}, Z_{4}, Z_{5}]. As F_{8} has common X- and Y -degree 2^{2} · 3 = 12, the only monomials that can occur are

Z_{3}^{4}, Z_{4}^{3}, Z_{3}Z_{4}Z_{5}.

We have

Z_{3}^{4}(X_{0}, X_{1}; Y_{0}, Y_{1}) = q^{44} − 112q^{45} + 6096q^{46} − 214592q^{47} + ⋯,
Z_{4}^{3}(X_{0}, X_{1}; Y_{0}, Y_{1}) = − q^{45} + 120q^{46} − 7020q^{47} + ⋯,
Z_{3}Z_{4}Z_{5}(X_{0}, X_{1}; Y_{0}, Y_{1}) = q^{44} − 114q^{45} + 6336q^{46} − 228632q^{47} + ⋯,

from which we determine that any linear relation among these monomials is contained in the kernel of the matrix

(\begin{matrix}1 & −112 \\ 0 & −1 \\ 1 & −114 \end{matrix}) .

But the kernel is already one-dimensional, being the Q-span of the row vector (1, 2, −1). Thus we conclude that F̃_{8} is a scalar multiple (which can easily be determined by examining the coefficient of the top X_{0}-degree term) of

Z_{3}^{4} + 2Z_{4}^{3} − Z_{3}Z_{4}Z_{5}.

We tabulate F̃_{n} (up to a Q^{×}-factor, so that equality below is to be interpreted as up to Q^{×}-scaling) for n ≤ 27 (Observe the errors in F̃_{13} and F̃_{8} in [7, vol. 2, p. 150]). We note that Klein writes F̃_{n} for n ≡ ±1 mod 5 in terms of the A_{i} and B_{i} := A_{i}(−X_{1}, X_{0}; Y_{0}, Y_{0}), and we use the W_{i} and Z_{i} because F̃_{n} is

MODULAR CURVES AND RAMANUJAN’S CONTINUED FRACTION 53

significantly simpler in these variables. We do not know why Klein did not use his variables A, B, C, D as in [11, p. 242] to write F̃_{n}, as the computations involved are much less arduous for even moderately large n when using the W_{i}’s and Z_{i}’s.

F_{2} = W_{3} F_{3} = Z_{4} F_{4} = Z_{6} F_{6} = W_{2}W_{10} − W_{6}^{2} F_{7} = W_{3}W_{5} − W_{4}^{2} F_{8} = −Z_{3}^{4} + Z_{3}Z_{4}Z_{5} − 2Z_{4}^{3} F_{9} = −4Z_{2}^{3}Z_{6} + Z_{2}Z_{10} + 3Z_{6}^{2} F_{11} = 4W_{2}^{6} − 4W_{2}^{3}W_{6} + W_{2}W_{10} − W_{6}^{2} F_{12} = 2W_{3}^{8} − W_{3}^{5}W_{4}W_{5} + W_{3}W_{4}^{4}W_{5} − W_{4}^{6} F_{13} = −Z_{3}^{3}Z_{5} − Z_{3}^{2}Z_{4}^{2} + Z_{4}Z_{5}^{2} F_{14} = −16Z_{2}^{9}Z_{6} + 4Z_{2}^{7}Z_{10} − 36Z_{2}^{6}Z_{6}^{2} + 4Z_{2}^{4}Z_{6}Z_{10} − 20Z_{2}^{3}Z_{6}^{3} + Z_{2}^{2}Z_{10}^{2} − 2Z_{2}Z_{6}^{2}Z_{10} + Z_{6}^{4} F_{16} = −8W_{2}^{9}W_{6} + 4W_{2}^{7}W_{10} + 52W_{2}^{6}W_{6}^{2} − 18W_{2}^{4}W_{6}W_{10} + 10W_{2}^{3}W_{6}^{3} + W_{2}^{2}W_{10}^{2} − W_{6}^{4} F_{17} = 8W_{3}^{6} − 10W_{3}^{3}W_{4}W_{5} + 9W_{3}^{2}W_{4}^{3} + W_{3}W_{5}^{3} − W_{4}^{2}W_{5}^{2}

54 BRYDEN CAIS AND BRIAN CONRAD

F_{18} = −8Z_{3}^{12} + 8Z_{3}^{9}Z_{4}Z_{5} − 12Z_{3}^{8}Z_{4}^{3} − Z_{3}^{7}Z_{5}^{3} + 4Z_{3}^{6}Z_{4}^{2}Z_{5}^{2} − 2Z_{3}^{5}Z_{4}^{4}Z_{5}

− 2Z_{3}^{4}Z_{4}^{6} + Z_{3}^{4}Z_{4}Z_{5}^{4} − 5Z_{3}^{3}Z_{4}^{3}Z_{5}^{3} + 10Z_{3}^{2}Z_{4}^{5}Z_{5}^{2} − 9Z_{3}Z_{4}^{7}Z_{5} + 3Z_{4}^{9}

F_{19} = −4Z_{2}^{7}Z_{6} + Z_{2}^{5}Z_{10} + 31Z_{2}^{4}Z_{6}^{2} − 24Z_{2}^{2}Z_{6}Z_{10} − 8Z_{2}Z_{6}^{3} + 4Z_{10}^{2} F_{21} = −16W_{2}^{7}W_{6}^{3} + W_{2}^{6}W_{10}^{2} + 2W_{2}^{5}W_{6}^{2}W_{10} + 285W_{2}^{4}W_{6}^{4}

− 16W_{2}^{3}W_{6}W_{10}^{2} − 72W_{2}^{2}W_{6}^{3}W_{10} + 72W_{2}W_{6}^{5} + 4W_{2}W_{10}^{3} − 4W_{6}^{2}W_{10}^{2}

F_{22} = 8W_{3}^{12} − 36W_{3}^{9}W_{4}W_{5} + 60W_{3}^{8}W_{4}^{3} + W_{3}^{7}W_{5}^{3} + 52W_{3}^{6}W_{4}^{2}W_{5}^{2} − 172W_{3}^{5}W_{4}^{4}W_{5} + 134W_{3}^{4}W_{4}^{6}

− 2W_{3}^{4}W_{4}W_{5}^{4} − 22W_{3}^{3}W_{4}^{3}W_{5}^{3} + 113W_{3}^{2}W_{4}^{5}W_{5}^{2} − 171W_{3}W_{4}^{7}W_{5} + W_{3}W_{4}^{2}W_{5}^{5} + 81W_{4}^{9} − W_{4}^{4}W_{5}^{4} F_{23} = −64Z_{3}^{8} + 120Z_{3}^{5}Z_{4}Z_{5} − 144Z_{3}^{4}Z_{4}^{3} − Z_{3}^{3}Z_{5}^{3} − 57Z_{3}^{2}Z_{4}^{2}Z_{5}^{2} + 135Z_{3}Z_{4}^{4}Z_{5} − 81Z_{4}^{6} + Z_{4}Z_{5}^{4} F_{24} = −64Z_{2}^{16}Z_{6}Z_{10} + 576Z_{2}^{15}Z_{6}^{3} + 16Z_{2}^{14}Z_{10}^{2} + 1888Z_{2}^{13}Z_{6}^{2}Z_{10} − 16752Z_{2}^{12}Z_{6}^{4} − 1264Z_{2}^{11}Z_{6}Z_{10}^{2} + 7840Z_{2}^{10}Z_{6}^{3}Z_{10} − 9648Z_{2}^{9}Z_{6}^{5} + 188Z_{2}^{9}Z_{10}^{3} − 1028Z_{2}^{8}Z_{6}^{2}Z_{10}^{2} + 2324Z_{2}^{7}Z_{6}^{4}Z_{10} − 1996Z_{2}^{6}Z_{6}^{6} + 12Z_{2}^{6}Z_{6}Z_{10}^{3}

− 52Z_{2}^{5}Z_{6}^{3}Z_{10}^{2} + 132Z_{2}^{4}Z_{6}^{5}Z_{10} + Z_{2}^{4}Z_{10}^{4} − 92Z_{2}^{3}Z_{6}^{7} − 4Z_{2}^{3}Z_{6}^{2}Z_{10}^{3} + 6Z_{2}^{2}Z_{6}^{4}Z_{10}^{2} − 4Z_{2}Z_{6}^{6}Z_{10} + Z_{6}^{8} F_{26} = 16W_{2}^{15}W_{6}^{2} − 8W_{2}^{13}W_{6}W_{10} + 136W_{2}^{12}W_{6}^{3} + W_{2}^{11}W_{10}^{2} − 134W_{2}^{10}W_{6}^{2}W_{10} + 805W_{2}^{9}W_{6}^{4} + 42W_{2}^{8}W_{6}W_{10}^{2}

− 588W_{2}^{7}W_{6}^{3}W_{10} + 1554W_{2}^{6}W_{6}^{5} − 4W_{2}^{6}W_{10}^{3} + 240W_{2}^{5}W_{6}^{2}W_{10}^{2} − 1008W_{2}^{4}W_{6}^{4}W_{10} + 1204W_{2}^{3}W_{6}^{6}

− 52W_{2}^{3}W_{6}W_{10}^{3} + 180W_{2}^{2}W_{6}^{3}W_{10}^{2} − 360W_{2}W_{6}^{5}W_{10} + 4W_{2}W_{10}^{4} + 216W_{6}^{7} − 4W_{6}^{2}W_{10}^{3}

F_{27} = 192W_{3}^{12} − 408W_{3}^{9}W_{4}W_{5} + 496W_{3}^{8}W_{4}^{3} + 3W_{3}^{7}W_{5}^{3} + 295W_{3}^{6}W_{4}^{2}W_{5}^{2} − 837W_{3}^{5}W_{4}^{4}W_{5} + 559W_{3}^{4}W_{4}^{6}

− 3W_{3}^{4}W_{4}W_{5}^{4} − 80W_{3}^{3}W_{4}^{3}W_{5}^{3} + 402W_{3}^{2}W_{4}^{5}W_{5}^{2} − 576W_{3}W_{4}^{7}W_{5} + W_{3}W_{4}^{2}W_{5}^{5} + 256W_{4}^{9} − W_{4}^{4}W_{5}^{4} Appendix D. Evaluation of Certain Arithmetic Sums

In this section, we prove two results that are used in §7. Lemma D.1. Let φ denote Euler’s function, and let m be a positive integer. Then

∑_{\begin{matrix}k=1 \\ k≡0 mod m \end{matrix}}^{N} \frac{φ(k)}{k^{2}} = \frac{6}{π^{2}} \frac{1}{[Γ(1) : Γ_{0}(m)]} log N + O(1),

where the O-constant is absolute: it is independent of m and N.

Proof. We will study sums of the form ∑_{k>0,k≡a mod m} φ(dk)χ(k)/k^{s} as s → 1^{+}, where χ is a Dirichlet character modulo m and d = gcd(a, m) ≥ 1. We may write a = da′ and m = dm′, so

∑_{\begin{matrix}k>0 \\ k≡a mod m \end{matrix}} \frac{φ(k)}{k^{s}} = ∑_{\begin{matrix}k>0 \\ k≡a′ mod m′ \end{matrix}} \frac{φ(dk)}{(dk)^{s}}

= \frac{1}{φ(m′)d^{s}} ∑_{χ mod m′} \overline{χ}(a′) ∑_{k>0} \frac{φ(dk)χ(k)}{k^{s}}. (D.1)

We are therefore reduced to understanding sums of the form ∑_{k>0} φ(dk)χ(k)/k^{s}. Since φ(nd)/φ(d) is a multiplicative function of n, we have an Euler product:

∑_{k>0} \frac{φ(dk)χ(k)}{k^{s}} = φ(d) ∑_{k>0} \frac{φ(dk)}{φ(d)} \frac{χ(k)}{k^{s}}

= φ(d) ∏_{p} (∑_{r≥0} \frac{φ(dp^{r})}{φ(d)} \frac{χ(p^{r})}{p^{rs}} ) .

We would like to simplify the inner sum occuring in the product above; in order to do this, we must distinguish two cases.

If p ∤ d then we have

∑_{r≥0} \frac{φ(dp^{r})}{φ(d)} \frac{χ(p^{r})}{p^{rs}} = ∑_{r≥0} \frac{φ(p^{r})χ(p^{r})}{p^{rs}}

= 1 + ∑_{r>0} \frac{p − 1}{p} · \frac{p^{r}χ(p^{r})}{p^{rs}}

= 1 + \frac{p − 1}{p} \frac{pχ(p)/p^{s}}{1 − pχ(p)/p^{s}}

= 1 + (p − 1)\frac{χ(p)}{p^{s} − pχ(p)}

= \frac{p^{s} − χ(p)}{p^{s} − pχ(p)}

= \frac{1 − χ(p)/p^{s}}{1 − χ(p)/p^{s−1}} .

On the other hand, if p|d then let us write d = p^{e}d′ with p ∤ d′ and e > 0. Then since

\frac{φ(dp^{r})}{φ(d)} = \frac{φ(d′)φ(p^{r+e})}{φ(d′)φ(p^{e})} = p^{r},

we have

∑_{r≥0} \frac{φ(dp^{r})}{φ(d)} \frac{χ(p^{r})}{p^{rs}} = ∑_{r≥0} \frac{p^{r}χ(p^{r})}{p^{rs}}

= ∑_{r≥0} \frac{χ(p)^{r}}{p^{r(s−1)}}

= \frac{1}{1 − χ(p)/p^{s−1}} .

Therefore,

∑_{k>0} \frac{φ(dk)χ(k)}{k^{s}} = φ(d) ∏_{p|d} \frac{1}{1 − χ(p)/p^{s−1}} ∏_{p|̸d} \frac{1 − χ(p)/p^{s}}{1 − χ(p)/p^{s−1}}

= φ(d)\frac{L(s − 1, χ)}{L(s, χ)} \frac{1}{∏_{p|d} (1 − χ(p)/p^{s})}.

Together with (D.1), this gives

∑_{\begin{matrix}k>0 \\ k≡a mod m \end{matrix}} \frac{φ(k)}{k^{s}} = \frac{φ(d)}{d^{s}φ(m′)} ∑_{χ mod m′} \overline{χ}(a′)\frac{L(s − 1, χ)}{L(s, χ)} ∏_{p|d} \frac{1}{1 − χ(p)/p^{s}} .

Observe that for nontrivial χ mod m′, the term L(s − 1, χ)/L(s, χ) is holomorphic for Re(s) ≥ 2, while L(s − 1, χ_{triv})/L(s, χ_{triv}) has a simple pole at s = 2. Using the fact that

L(s, χ_{triv}) = ζ(s) ∏_{p|m′} (1 − 1/p^{s})

and the expansion ζ(s) = 1/(s − 1) + h(s) where h(s) is holomorphic, we see that

∑_{\begin{matrix}k>0 \\ k≡a mod m \end{matrix}} φ(k)/k^{s}

has a simple pole at s = 2 with residue

(D.2)  \frac{φ(d)}{φ(m′)d^{2}} \frac{∏_{p|m′} (1 − 1/p)}{∏_{p|m′} (1 − 1/p^{2}) · ζ(2)} ∏_{\begin{matrix}p|d \\ p∤m′ \end{matrix}} \frac{1}{1 − 1/p^{2}} = \frac{6}{π^{2}d^{2}} \frac{φ(d)}{φ(m′)} ∏_{p|m′} (1 − \frac{1}{p}) ∏_{p|dm′} \frac{1}{1 − 1/p^{2}} .

We now appeal to the following well-known theorem (make the change of variable s′ = s/2):

Theorem D.2. Let f(s) = ∑_{k>0} a_{k}/k^{s} converge absolutely for Re(s) > 2 with analytic continuation to Re(s) = 2 except for a simple pole at s = 2 with residue R. Then

∑_{k≤x} \frac{a_{k}}{k^{2}} = R log x + O(1).

Theorem D.1 now follows on setting a = m in (D.2) and using the well-known identity [Γ(1) : Γ_{0}(m)] = m ∏_{p|m} (1 + 1/p). ■

Lemma D.3. Let a and k be integers with k > 0, and let p be a prime that divides k. Assume p ∤ a, and let ζ be a primitive kth root of unity in C. Define

c_{k}(m) = ∑_{\begin{matrix}h∈(Z/kZ)^{×} \\ h≡a mod p \end{matrix}} ζ^{hm}.

Then

c_{k}(m) = \begin{cases}(μ(k/(m, k))φ(k))/((p − 1)φ(k/(m, k))) & if ord_{p}(k) ≤ ord_{p}(m), \\ 0 & otherwise. \end{cases}

Proof. Since ζ^{k/p} is a primitive pth root of unity, we may write

c_{k}(m) = \frac{1}{p} ∑_{ζ_{0}^{p}=1} ∑_{h∈(Z/kZ)^{×}} ζ_{0}^{−(a−h)}ζ^{mh}

= \frac{1}{p} ∑_{j∈Z/pZ} ζ^{−kja/p} ∑_{h∈(Z/kZ)^{×}} ζ^{(m+jk/p)h}.

We can remove the a in the first exponent since p ∤ a. Combining this with the standard evaluation

∑_{h∈(Z/kZ)^{×}} ζ^{ℓh} = μ(k/(k, ℓ)) · \frac{φ(k)}{φ(k/(k, ℓ))}

for ℓ ∈ Z, we get

c_{k}(m) = \frac{1}{p} ∑_{j∈Z/pZ} ζ^{−kj/p}μ (\frac{k}{(k, m + jk/p)}) \frac{φ(k)}{φ(k/(k, m + jk/p))}.

Evidently,

(k, m + jk/p) = \begin{cases}(m, k/p) & if j ≢ 0 mod p \\ (m, k) & otherwise \end{cases} .

Therefore,

c_{k}(m) = \frac{1}{p} (−μ (\frac{k}{(m, k/p)}) \frac{φ(k)}{φ(k/(m, k/p))} + μ (\frac{k}{(m, k)}) \frac{φ(k)}{φ(k/(m, k))}) .

If ord_{p}(m) < ord_{p}(k) then (m, k) = (m, k/p) so c_{k}(m) = 0. On the other hand, if ord_{p}(m) ≥ ord_{p}(k) then (m, k/p) = (m, k)/p and ord_{p}(k/(m, k)) = 0, so we have

c_{k}(m) = \frac{1}{p} (−μ (\frac{pk}{(m, k)}) \frac{φ(k)}{φ(pk/(m, k))} + μ (\frac{k}{(m, k)}) \frac{φ(k)}{φ(k/(m, k))})

= \frac{1}{p}μ (\frac{k}{(m, k)}) (\frac{φ(k)}{φ(p)φ(k/(m, k))} + \frac{φ(k)}{φ(k/(m, k))})

= \frac{1}{φ(p)}μ (\frac{k}{(m, k)}) \frac{φ(k)}{φ(k/(m, k))}

This completes the proof. ■

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Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA

   E-mail address: bcais@umich.edu

Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA

   E-mail address: bdconrad@umich.edu