Open AccessThis article is

$\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.

- freely available
- re-usable

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

Download PDF [301 KB, uploaded 13 March 2013]

# Abstract

Following Ramanujan's work on modular equations and approximations of*Keywords:*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) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.*

# Share & Cite This Article

**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.

# Related Articles

# Article Metrics

# Comments

# Cited By

[Return to top]*Mathematics*EISSN 2227-7390 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert