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ρ — Adic Analogues of Ramanujan Type Formulas for 1/π
Department of Mathematics and Statistics, University of Calgary, Calgary AB, T2N 1N4, Canada
Department of Mathematics, University of Washington, Seattle, WA 98195, USA
Department of Mathematics, Cornell University, Ithaca, NY 14853, USA
Department of Mathematics, Iowa State University, Ames, IA 50011, USA
Lehrstuhl D für Mathematik, RWTH Aachen University, 52056 Aachen, Germany
Department of Mathematics, Oregon State University, Corvallis, OR 97331, USA
* Author to whom correspondence should be addressed.
Received: 18 February 2013; in revised form: 26 February 2013 / Accepted: 1 March 2013 / Published: 13 March 2013
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
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MDPI and ACS Style
Chisholm, S.; Deines, A.; Long, L.; Nebe, G.; Swisher, H. ρ — Adic Analogues of Ramanujan Type Formulas for 1/π. Mathematics 2013, 1, 9-30.
Chisholm S, Deines A, Long L, Nebe G, Swisher H. ρ — Adic Analogues of Ramanujan Type Formulas for 1/π. Mathematics. 2013; 1(1):9-30.
Chisholm, Sarah; Deines, Alyson; Long, Ling; Nebe, Gabriele; Swisher, Holly. 2013. "ρ — Adic Analogues of Ramanujan Type Formulas for 1/π." Mathematics 1, no. 1: 9-30.