ρ — Adic Analogues of Ramanujan Type Formulas for 1/π

Abstract

Following Ramanujan’s work on modular equations and approximations of π, there are formulas for 1/π of the form [PLEASE CHECK FORMULA IN THE PDF] for d = 2, 3, 4, 6, where λd are singular values that correspond to elliptic curves with complex multiplication, and α, δ are explicit algebraic numbers. In this paper we prove a ρ-adic version of this formula in terms of the so-called Ramanujan type congruence. In addition, we obtain a new supercongruence result for elliptic curves with complex multiplication.

Keywords

• Ramanujan type supercongruences
• Atkin and Swinnerton-Dyer congruences
• hypergeometric series
• elliptic curves
• complex multiplication
• periods
• modular forms
• Picard–Fuchs equation