Ramanujan-type identities for Shimura curves

Abstract

ABSTRACT. In 1914, Ramanujan gave a list of 17 identities expressing 1/π 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. These products of Gamma values are related to periods of elliptic curves with complex multiplication by Q( √ −3) and Q( √ −4).

Special values of hypergeometric functions and periods of CM elliptic curves

“…In Section 6, we will work out an example to illustrate a general technique to determine special values of the hypergeometric functions when there are more than one CM-points of discriminant d. be the hypergeometric functions in Theorem 2. The Ramanujan-type identities obtained in [35] can be written as…”

“…The first part is the content of Lemmas 3 and 4 of [35]. For the second part, the proof of Lemma 14 of [36] shows that …”

“…, 5, form a complete set of coset representatives of Γ\Γι(α)Γ. In Section 4 of [35], by using results from [36], we find that (It is possible to determine the precise value, not just the absolute value. The two zeros of the factor of degree 2 are F (τ 1 )/F (τ 1 ) and the value of (F 8 ι(α ))/F at τ 1 , where α = (3 − 9I + J − 3IJ)/2.…”

“…Then by utilizing the Jacquet-Langlands correspondence and explicit covers between Shimura curves, we devised a method to compute Hecke operators with respect to the explicitly given basis of modular forms. As applications of this computation of Hecke operators, we computed modular equations for Shimura curves, which can be regarded as equations for Shimura curves associated to Eichler orders of higher levels, in [34] and obtained Ramanujan-type identities for Shimura curves in [35]. In addition, since some Schwarzian differential equations are essentially hypergeometric differential equations, this realization of modular forms yields many beautiful identities among hypergeometric functions.…”

“…We first recall a technical lemma from [35]. Recall that ψ F f (τ ) and ψ Fg (τ ) are the Borcherds forms defined in (8) and (9), respectively.…”