? — Adic Analogues of Ramanujan Type Formulas for 1/?

Sarah Chisholm, Alyson Deines, Ling Long, Gabriele Nebe and Holly Swisher
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Sarah Chisholm: Department of Mathematics and Statistics, University of Calgary, Calgary AB, T2N 1N4, Canada
Alyson Deines: Department of Mathematics, University of Washington, Seattle, WA 98195, USA
Ling Long: Department of Mathematics, Cornell University, Ithaca, NY 14853, USA
Gabriele Nebe: Lehrstuhl D für Mathematik, RWTH Aachen University, 52056 Aachen, Germany
Holly Swisher: Department of Mathematics, Oregon State University, Corvallis, OR 97331, USA

Mathematics, 2013, vol. 1, issue 1, 1-22

Abstract: Following Ramanujan's work on modular equations and approximations of ? , there are formulas for 1 / ? of the form Following Ramanujan's work on modular equations and approximations of ? , there are formulas for 1 / ? of the form ? k = 0 ? ( 1 2 ) k ( 1 d ) k ( d - 1 d ) k k ! 3 ( a k + 1 ) ( ? d ) k = ? ? for d = 2 , 3 , 4 , 6 , where ? d are singular values that correspond to elliptic curves with complex multiplication, and a , ? are explicit algebraic numbers. In this paper we prove a p - 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 (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2013
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