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From L-series of elliptic curves to Mahler measures

Part of: Other special functions Algebraic number theory: global fields $K$-theory in number theory Curves Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  23 January 2012

Mathew Rogers  and
Wadim Zudilin
Mathew Rogers
Affiliation:
Department of Mathematics and Statistics, Université de Montréal, CP 6128 succ. Centre-ville, Montréal Québec H3C 3J7, Canada (email: mathewrogers@gmail.com)
Wadim Zudilin
Affiliation:
School of Mathematical and Physical Sciences, The University of Newcastle, Callaghan NSW 2308, Australia (email: wadim.zudilin@newcastle.edu.au)
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Abstract

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We prove the conjectural relations between Mahler measures and L-values of elliptic curves of conductors 20 and 24. We also present new hypergeometric expressions for L-values of elliptic curves of conductors 27 and 36. Furthermore, we prove a new functional equation for the Mahler measure of the polynomial family (1+X) (1+Y )(X+Y )−αXY, α∈ℝ.

Keywords

Mahler measureL-value of elliptic curvehypergeometric serieslattice sumelliptic dilogarithm

MSC classification

Secondary: 11R06: PV-numbers and generalizations; other special algebraic numbers; Mahler measure 33C20: Generalized hypergeometric series, $_pF_q$ 11F03: Modular and automorphic functions 14H52: Elliptic curves 19F27: Étale cohomology, higher regulators, zeta and $L$-functions 33C75: Elliptic integrals as hypergeometric functions 33E05: Elliptic functions and integrals
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 2 , March 2012 , pp. 385 - 414
Copyright
Copyright © Foundation Compositio Mathematica 2012

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