# From L-series of elliptic curves to Mahler measures

@article{Rogers2012FromLO, title={From L-series of elliptic curves to Mahler measures}, author={Mathew Rogers and Wadim Zudilin}, journal={Compositio Mathematica}, year={2012}, volume={148}, pages={385 - 414} }

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.

## 55 Citations

Mahler measure and elliptic curve L-functions at s = 3

- Mathematics
- 2015

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

- Mathematics
- 2016

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

- MathematicsJ. 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…

Identities of Mahler measures of certain polynomials defining curves of arbitrary genus

- Mathematics
- 2021

Regulator proofs for Boyd’s identities on genus 2 curves

- MathematicsInternational 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

- Mathematics
- 2011

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.

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