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*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; in revised form: 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 $\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 $\sum _{k=0}^{\infty}\frac{{\left(\frac{1}{2}\right)}_{k}{\left(\frac{1}{d}\right)}_{k}{\left(\frac{d-1}{d}\right)}_{k}}{k{!}^{3}}(ak+1){\left({\lambda}_{d}\right)}^{k}=\frac{\delta}{\pi}$ 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 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.

**AMA Style**

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.

**Chicago/Turabian Style**

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.

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