Ramanujan-type identities for 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 periods of elliptic curves over Q with complex multiplication by .