Ramanujan-type identities for Shimura curves

TL;DR

This paper extends Ramanujan-type identities for 1/π to Shimura curves, connecting hypergeometric functions, Gamma values, and elliptic curve periods with complex multiplication.

Contribution

It develops new Ramanujan-type formulas for Shimura curves involving hypergeometric functions and Gamma products, expanding the scope beyond classical modular curves.

Findings

Derived explicit formulas involving hypergeometric functions and Gamma values.

Connected Gamma products to periods of elliptic curves with complex multiplication.

Extended Ramanujan identities to the setting of Shimura curves.

Abstract

In 1914, Ramanujan gave a list of 17 identities expressing $1/\pi$ as linear combinations of values of hypergeometric functions at certain rational numbers. Since then, identities of similar nature have been discovered by many authors. Nowadays, one of the standard approaches to this kind of identities uses the theory of modular curves. In this paper, we will consider the case of Shimura curves and obtain Ramanujan-type formulas involving special values of hypergeometric functions and products of Gamma values. The product of Gamma values are related to periods of elliptic curves with complex multiplication by Q(\sqrt{-3}) and Q(\sqrt{-4}).