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Mathematics 2013, 1(1), 9-30; doi:10.3390/math1010009
Article

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

1
,
2
,
3,4
,
5
 and
6,*
1 Department of Mathematics and Statistics, University of Calgary, Calgary AB, T2N 1N4, Canada 2 Department of Mathematics, University of Washington, Seattle, WA 98195, USA 3 Department of Mathematics, Cornell University, Ithaca, NY 14853, USA 4 Department of Mathematics, Iowa State University, Ames, IA 50011, USA 5 Lehrstuhl D für Mathematik, RWTH Aachen University, 52056 Aachen, Germany 6 Department of Mathematics, Oregon State University, Corvallis, OR 97331, USA
* Author to whom correspondence should be addressed.
Received: 18 February 2013 / Revised: 26 February 2013 / Accepted: 1 March 2013 / Published: 13 March 2013
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Abstract

Following Ramanujan's work on modular equations and approximations of π, there are formulas for 1/π of the form Following Ramanujan's work on modular equations and approximations of π, there are formulas for 1/π of the form k = 0 ( 1 2 ) k ( 1 d ) k ( d - 1 d ) k k ! 3 ( a k + 1 ) ( λ d ) k = δ π for d=2,3,4,6, where łd are singular values that correspond to elliptic curves with complex multiplication, and a,δ 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 Ramanujan type supercongruences; Atkin and Swinnerton-Dyer congruences; hypergeometric series; elliptic curves; complex multiplication; periods; modular forms; Picard–Fuchs equation
This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).
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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.

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