\(p\)-adic analogues of Ramanujan type formulas for \(1/\pi\)
type: Item
instance of: scholarly article
title: \(p\)-adic analogues of Ramanujan type formulas for \(1/\pi\)
zbMATH Open document ID: 1296.11058
author: Sarah Chisholm
author: Alyson Deines-Schartz
author: Ling Long
author: Gabriele Nebe
author: Holly Swisher
published in: Mathematics
publication date: 2014-08-27T00:00:00Z
review text: Summary: Following Ramanujan's work on modular equations and approximations of \(\pi\), there are formulas for \(1/\pi\) of the form \[ \sum\limits_{k=0}^\infty\frac{(\frac{1}{2})k(\frac{1}{d})k(\frac{d-1}{d})k}{k!^3}(ak+1)(\lambda_d)^k=\frac{\delta}{\pi} \] for \(d=2,3,4,6\), where \(\lambda d\) 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.
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 44A20
Mathematics Subject Classification ID: 11F33
zbMATH DE Number: 6334773
zbMATH Keywords: Ramanujan type supercongruences
zbMATH Keywords: Atkin and Swinnerton-Dyer congruences
zbMATH Keywords: hypergeometric series
zbMATH Keywords: elliptic curves
zbMATH Keywords: complex multiplication
zbMATH Keywords: periods
zbMATH Keywords: modular forms
zbMATH Keywords: Picard-Fuchs equation
MaRDI profile type: Publication
full work available at URL: math1010009
OpenAlex ID: W2111247341
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DOI: 10.3390/MATH1010009
description: scientific article; zbMATH DE number 6334773