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$\pi $ , there are formulas for $1/\pi $ of the form Following Ramanujan's work on modular equations and approximations of $\pi $ , there are formulas for $1/\pi $ of the form $\sum _{k=0}^{\infty}\frac{{\left(\frac{1}{2}\right)}_{k}{\left(\frac{1}{d}\right)}_{k}{\left(\frac{d1}{d}\right)}_{k}}{k{!}^{3}}(ak+1){\left({\lambda}_{d}\right)}^{k}=\frac{\delta}{\pi}$ for $d=2,3,4,6,$ where ${\u0142}_{d}$ are singular values that correspond to elliptic curves with complex multiplication, and $a,\delta $ are explicit algebraic numbers. In this paper we prove a $p$ adic version of this formula in terms of the socalled Ramanujan type congruence. In addition, we obtain a new supercongruence result for elliptic curves with complex multiplication.
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Mathematics 2013, 1(1), 930; doi:10.3390/math1010009
Article
ρ — Adic Analogues of Ramanujan Type Formulas for 1/π
^{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 ofKeywords:
Ramanujan type supercongruences; Atkin and SwinnertonDyer congruences; hypergeometric series; elliptic curves; complex multiplication; periods; modular forms; Picard–Fuchs equation
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