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

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Author(s): Sarah Chisholm | Alyson Deines | Ling Long | Gabriele Nebe | Holly Swisher

Journal: Mathematics
ISSN 2227-7390

Volume: 1;
Issue: 1;
Start page: 9;
Date: 2013;
Original page

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

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.

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