Modular Curves and Ramanujan's Continued Fraction

Bryden Cais and Brian Conrad

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 real 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.