Arithmetic of Ramanujan's continued Fraction

Bryden Cais

We identify Ramanujan's continued fraction R(q) with a generator for the function field of the genus 0 modular curve X(5) and use methods of Deuring and Shimura to establish arithmetic properties of R. In particular, we describe the class fields generated by singular values of R and show that these values are algebraic units. We establish "modular equations" of arbitrary degree and certain congruence properties mod p (for p not 5) that are analogous to the Kronecker congruences for the classical modular equations. Finally, we compute several modular equations and note the remarkably small size of their coefficients. This work is superceded and greatly generalized in Modular Curves and Ramanujan's Continued Fraction.