# From L-series of elliptic curves to Mahler measures

@article{Rogers2012FromLO,
title={From L-series of elliptic curves to Mahler measures},
journal={Compositio Mathematica},
year={2012},
volume={148},
pages={385 - 414}
}
• Published 14 December 2010
• Mathematics
• Compositio Mathematica
Abstract 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.
Mahler measure and elliptic curve L-functions at s = 3
We study the Mahler measure of some three-variable polynomials that are conjectured to be related to L-functions of elliptic curves at s D 3 by Boyd. The connection with L-functions can be explained
The Beilinson conjectures for CM elliptic curves via hypergeometric functions
We consider certain CM elliptic curves which are related to Fermat curves, and express the values of L-functions at $$s=2$$s=2 in terms of special values of generalized hypergeometric functions. We
Further explorations of Boyd's conjectures and a conductor 21 elliptic curve
• Mathematics
J. Lond. Math. Soc.
• 2016
The modular parametrization of the elliptic curve $\tilde P(x,y)=0$, again of conductor 21, is used, due to Ramanujan and the Mellit--Brunault formula for the regulator of modular units.
The Mahler measure of a Calabi–Yau threefold and special L-values
• Mathematics
• 2013
The aim of this paper is to prove a Mahler measure formula of a four-variable Laurent polynomial whose zero locus defines a Calabi–Yau threefold. We show that its Mahler measure is a rational linear
Regulator proofs for Boyd’s identities on genus 2 curves
• Mathematics
International Journal of Number Theory
• 2019
We use the elliptic regulator to recover some identities between Mahler measures involving certain families of genus 2 curves that were conjectured by Boyd and proven by Bertin and Zudilin by
The Mahler measure of a Weierstrass form
• Mathematics
• 2017
We prove an identity between Mahler measures of polynomials that was originally conjectured by Boyd. The combination of this identity with a result of Zudilin leads to a formula involving a Mahler
The Mahler measure for arbitrary tori
• Mathematics
• 2017
We consider a variation of the Mahler measure where the defining integral is performed over a more general torus. We focus our investigation on two particular polynomials related to certain elliptic
On the Mahler Measure Of
We prove a conjectured formula relating the Mahler measure of the Laurent polynomial 1 + X + X−1 + Y + Y −1, to the L-series of a conductor 15 elliptic curve.

## References

SHOWING 1-10 OF 42 REFERENCES
Modular Mahler Measures I
We relate Boyd’s numerical examples, linking the Mahler measure m(P k ) of certain polynomials P k to special values of L-series of elliptic curves, to the Bloch-Beilinson conjectures. We study m(P k
Elliptic dilogarithms and parallel lines
• A. Mellit
• Mathematics
Journal of Number Theory
• 2019
Mahler measure and the WZ algorithm
• Mathematics
• 2010
We use the Wilf-Zeilberger method to prove identities between Mahler measures of polynomials. In particular, we oer a new proof of a formula due to Lal n, and we show how to translate the identity
Functional equations for Mahler measures of genus-one curves
• Mathematics
• 2006
In this paper we will establish functional equations for Mahler measures of families of genus-one two-variable polynomials. These families were previously studied by Beauville, and their Mahler
Modular Equations and Lattice Sums
• Mathematics
• 2013
We highlight modular equations due to Ramanujan and Somos and use them to prove new relations between lattice sums and hypergeometric functions. We also discuss progress towards solving Boyd’s Mahler
ETA-QUOTIENTS AND ELLIPTIC CURVES
In this paper we list all the weight 2 newforms f(τ) that are products and quotients of the Dedekind eta-function η(τ) := q ∞ Y n=1 (1− q), where q := e2πiτ . There are twelve such f(τ), and we give
Mahler's Measure and Special Values of L-functions
Some examples for which it appear that log M(P(x, y) = rL'(E, 0), where E is an elliptic curve and r is a rational number, often either an integer or the reciprocal of an integer.
Mahler Measure Variations, Eisenstein Series and Instanton Expansions
This paper points at an intriguing inverse function relation with on the one hand the coefficients of the Eisenstein series in Rodriguez Villegas’ paper on “Modular Mahler Measures” and on the
Fourier series of rational fractions of Jacobian elliptic functions
• Mathematics
• 1988
In this paper more than ninety of the Fourier series of rational fractions of Jacobian elliptic functions sn(u.k.), cn(u.k) and dn(u.k) are listed, which cannot be found in the handbook[1] and Ref.
Generalized Hypergeometric Functions
Introduction Multiplication by Xu (Gauss contiguity) Algebraic theory Variation of Wa with g Analytic theory Deformation theory Structure of Hg Linear differential equations over a ring Singularities