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

doi:10.3390/math1010009

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Chisholm, S
Deines, A
Long, L
Nebe, G
Swisher, H

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Deines, A
Long, L
Nebe, G
Swisher, H

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Deines, A
Long, L
Nebe, G
Swisher, H

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Ramanujan type supercongruences

Atkin and Swinnerton-Dyer congruences

hypergeometric series

elliptic curves

complex multiplication

periods

modular forms

Picard–Fuchs equation

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Alyson Deines

Ling Long

Gabriele Nebe

Holly Swisher

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

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

Sarah Chisholm¹

, Alyson Deines²

, Ling Long^3,4

, Gabriele Nebe⁵

and Holly Swisher^6,*

Department of Mathematics and Statistics, University of Calgary, Calgary AB, T2N 1N4, Canada

Department of Mathematics, University of Washington, Seattle, WA 98195, USA

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

View Full-Text | Download PDF [301 KB, uploaded 13 March 2013]

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

\sum_{k = 0}^{\infty} \frac{{(\frac{1}{2})}_{k} {(\frac{1}{d})}_{k} {(\frac{d - 1}{d})}_{k}}{k!^{3}} (a k + 1) {(λ_{d})}^{k} = \frac{δ}{π}

for

d = 2, 3, 4, 6,

where

ł_{d}

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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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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