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2005: Geometric non-vanishing
Inventiones Mathematicae 159 (2005), 133-186.
SpringerLink version, ArXiV version, local version
We prove a very general non-vanishing result for twists of L-functions associated to Galois representations over function fields. The word “geometric” in the title refers to the fact that algebro-geometric techniques play the main role, as opposed to the automorphic techniques usually involved in non-vanishing results. The motivation for considering the question came from applications to the Birch and Swinnerton-Dyer conjecture; in particular, this article reduces the BSD conjecture for elliptic curves over function fields in the case where the L-series vanishes to order at most 1 to a relatively mild version of the Gross-Zagier formula.
2004: Elliptic curves and analogies between number fields and
function fields
in “Heegner points and Rankin
L-series,” MSRI Publications 49, 285-315.
CUP version at MSRI, ArXiV version, local version
This article is the conjunction
of a survey article on one topic and a research article on another
loosely related topic. The survey part gives a rough ouline of how to
prove the Birch and Swinnerton-Dyer conjecture for elliptic curves
over function fields when the L-function vanishes to order at most 1
using a Gross-Zagier formula and the non-vanishing results of 2005.
The research part explains some questions related to ranks of elliptic
curves in various towers of fields which were motivated by the last
part of 2002 and well-known analogies. There has since been
interesting work by Silverman, Ellenberg, and others related to the
case of towers of function fields over number fields. (Update of
November 8, 2007: Martin Brown informs me that his Springer Lecture
Notes volume 1849, in particular the errata on pages 433-434, fixes
the problems mentioned in Section 3.4 of my article. I have not
checked this claim.)
2002: Elliptic curves with large rank over function fields
Annals of Mathematics (2) 155 (2002), 295-315
Project Euclid version, ArXiV version, local version
We show the existence of non-isotrivial elliptic curves of arbitrarily large rank over any function field over a finite field. Except for the “non-isotrivial” this was shown long before by Tate and Shafarevitch, but the non-isotrivial examples are considered (by some) to be much more compelling evidence for the analogous question over number fields. The curves in question show that bounds for ranks in terms of conductors due to Brumer following Mestre are sharp.
1996b: Slopes of modular forms and congruences
Annales de l'Institut Fourier 46 (1996), 1-32
Corrigendum, same volume, page 1519
Numdam version, local version, local version of corrigendum,
This paper exploits the results of 1996a to give congruences between forms of level pN and weight > 2 and forms of level 2, including a curious relationship between the leading terms of the p-adic expansions of the Up eigenvalues of the forms which brings in the slope (p-adic valuation) of the higher weight form. This leads to a complete description of the local mod p Galois representation attached to certain non-ordinary forms of higher weight. Barry Mazur suggested, and the author and Breuil confirmed in weight 3 (unpublished), that there is a purely local, Galois representation-theoretic explanation for the latter result.
1996a: On the Fourier coefficients of modular forms II
Mathematische Annalen 304 (1996), 363-422
SpringerLink version, local version
The aim of this paper is to improve the results of 1995a by computing the highest polygon with integer slopes which lies on or below the Newton polygon of the Up operator. The results ultimately boil down to fine information on certain crystalline cohomology groups obtained via cohomology of exact and logarithmic differentials.
1995b: A construction of local points on elliptic curves over modular curves
International Mathematics Research Notices 1995 (1995), 349-363
Hindawi version, local version
For the universal elliptic curve E over the function field K of a modular curve over a finite field, this paper constructs a Zp-submodule of the group of local points E(Kv), for a suitable place v, which has rank equal to the order of vanishing of the L-function at s=1 and which contains a finite index subgroup of the global points E(K). It is tantalizing problem to characterize the global points among these explicitly constructed local points, since this would give a completely new construction of global points on elliptic curves over function fields. Many of the ideas of this paper can be transferred to a much more general, non-modular, situation (to be written). The paper also shows that the Birch and Swinnerton-Dyer conjecture implies the semi-simplicity of the action of certain Hecke operators on modular forms of weight 3. Coleman and Edixhoven (Math. Ann. 310) later generalized this, replacing BSD with the Tate conjecture, to higher weights.
1995a: On the Fourier coefficients of modular forms
Annales Scientifiques de l'École Normale Supérieure 28 (1995), 129-160
local version
This paper studies the p-adic valuations of the eigenvalues of the Hecke operator Up on modular forms of level divisible by p and weight between 2 and p+1. The results are stated in terms of Newton and Hodge polygons and they say roughly that the eigenvalues of Up are more divisible by p than one might a priori expect. The results are compatible with, but nowhere near strong enough to prove, conjectures of Gouvea and Mazur on p-adic families of modular forms. Conceptually, the proof is a simple consequence of Scholl's ideas on motives for modular forms and a motivic variant of the Katz conjecture on Newton and Hodge polygons associated to crystalline cohomology, but there are messy technical difficulties related to the cusps which are resolved using log schemes.
1994: Slopes of modular forms
in “Arithmetic Geometry,” Contemporary Mathematics 174 (1994), 167-183
local version
Although it appeared earlier, this paper is a continuation of 1995a and 1996a. It extends the results of 1995a to all weights and explains what is needed to do the same for 1996a. This would be enough to prove conjectures of Gouvea and Mazur on the number of eigenforms whose Up eigenvalue has a given p-adic valuation.
1993: Curves of genus ten on K3 surfaces (with Fernando Cukierman)
Compositio Mathematica 89 (1993), 81-90
local version
J. Wahl proved that if a curve C admits an embedding into a K3 surface, then its Gaussian map (taking pairs of holomorphic 1-forms to sections of the tri-canonical bundle) fails to be surjective. In this paper, we prove the converse for curves of genus 10 (the first interesting case). A key point is a partial computation of the classes in the Picard group of the moduli space of the divisor of curves where the Gaussian map is not surjective and the divisor of curves embeddable in a K3. A more complete calculation of the latter class was recently carried out by Farkas and Popa (J. Alg. Geom. 14), leading to a counterexample to the Harris-Morrison slope conjecture.
1991: p-descent in characteristic p
Duke Mathematical Journal 62 (1991), 237-265
scanned pdf at Project Euclid, local version
The first part of the paper is devoted to computing the Selmer group for the multiplication-by-p isogeny on an elliptic curve over a global field of characteristic p in terms of the arithmetic (differentials and the Cariter operator, p-torsion in the Jacobian) of the base curve. Ultimately a rather detailed knowledge of the group scheme Ker(p) is required. In the second part of the paper, these tools are applied to the universal elliptic curve over an Igusa curve studied in 1990a. Elements of the Selmer group predicted by the L-function computation of 1990a and the BSD conjecture are constructed using modular forms modulo p.
1990b: L-functions of universal elliptic curves over Igusa curves
American Journal of Mathematics 112 (1990), 687-712
scanned pdf at JSTOR, local version
This paper gives a computation of the Hasse-Weil L-function of powers of the universal elliptic curve over the function field of an Igusa curve in terms of modular forms. In 1990a this type of result was proven using “point counting” and the trace formula whereas this paper uses étale cohomology and results of Katz and Mazur to obtain a much more general statement. A definition of the Hecke operator Up for primes p dividing the level is given which has seen other uses (e.g., Faltings-Jordan, Israel J. Math. 90).
1990a: On universal elliptic curves over Igusa curves
Inventiones Mathematicae 99 (1990), 377-391
scanned pdf at Goettingen, local version
Igusa curves appear naturally as components of the reduction of modular curves at primes dividing the level. This paper studies the arithmetic of the universal elliptic curve over an Igusa curve, viewed as an elliptic curve over the function field of the base curve. In particular, its Hasse-Weil L-function is computed in terms of modular forms and a systematic source of zeroes of the L-function is exhibited. Other arithmetic invariants of the universal curve are computed and in some cases the Birch and Swinnerton-Dyer conjecture is verified. To date, there is no known construction of the systematic supply of rational points predicted by the L-function computation and the BSD conjecture.
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