Functional equations for Mahler measures of genus-one curves (original) (raw)

On the Mahler measure of a family of genus 2 curves

We establish a general identity between the Mahler measures m(Q k (x, y)) and m(P k (x, y)) of two polynomial families, where Q k (x, y) = 0 and P k (x, y) = 0 are generically hyperelliptic and elliptic curves, respectively. P k (x, y) = (x + 1)y 2 + (x 2 + kx + 1)y + (x 2 + x), k ∈ Z, which can be characterised by m(P k )/L ′ (E, 0) ∈ Q × , where E : P k (x, y) = 0 is (generically) an elliptic curve and L ′ (E, 0) is the derivative of its L-function L(E, s) at s = 0. Very few of these are proven so far: k = 0, 6 (conductor 36) by Rodriguez-Villegas [9], k = 1, 10, −5 (conductor 14) by Mellit [8], and k = 4, −2 (conductor 20) by Rogers and the second author . Note that these particular cases (as well as all other proven cases of elliptic type, for both CM and non-CM curves) accidentally fall under application of the Mellit-Brunault formula .

Mahler measure of some -variable polynomial families

Journal of Number Theory, 2006

The Mahler measures of some n-variable polynomial families are given in terms of special values of the Riemann zeta function and a Dirichlet L-series, generalizing the results of [13]. The technique introduced in this work also motivates certain identities among Bernoulli numbers and symmetric functions.

Mahler measure of some n-variable polynomial families

2004

The Mahler measures of some n-variable polynomial families are given in terms of special values of the Riemann zeta function and a Dirichlet L-series, generalizing the results of L. The technique introduced in this work also motivates certain identities among Bernoulli numbers and symmetric functions.

Hypergeometric functions and a family of algebraic curves

The Ramanujan Journal, 2012

Let λ ∈ Q \ {0, 1} and l ≥ 2, and denote by C l,λ the nonsingular projective algebraic curve over Q with affine equation given by y l = x(x − 1)(x − λ). In this paper we define Ω(C l,λ) analogous to the real periods of elliptic curves and find a relation with ordinary hypergeometric series. We also give a relation between the number of points on C l,λ over a finite field and Gaussian hypergeometric series. Finally we give an alternate proof of a result of [13].

An algebraic integration for Mahler measure

Duke Mathematical Journal, 2007

There are many examples of several-variable polynomials whose Mahler measure is expressed in terms of special values of polylogarithms. These examples are expected to be related to computations of regulators, as observed by Deninger [D], and later Rodriguez-Villegas [R-V], and Maillot [M]. While Rodriguez-Villegas made this relationship explicit for the two variable case, it is our goal to understand the three variable case and shed some light on the examples with more variables.

The Mahler measure of a Weierstrass form

International Journal of Number Theory, 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 measure of a Weierstrass form of conductor 17 given in terms of [Formula: see text]. Our proof involves a non-trivial identity between regulators which leads to the elliptic curve [Formula: see text]-function being expressed in terms of the regulator evaluated in a non-rational non-torsion point.

Regulator Proofs for Boyd's Identities on Genus 2 Curves

International Journal of Number Theory

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 differentiating the Mahler measures and using hypergeometric identities. Since our proofs involve the regulator, they yield light into the expected relation of each Mahler measure to special values of [Formula: see text]-functions of certain elliptic curves.

A new method for obtaining polylogarithmic Mahler measure formulas

Research in Number Theory, 2016

Given a formula for the Mahler measure of a rational function expressed in terms of polylogarithms, we describe a new method that allows us to construct a rational function with 2 more variables and whose Mahler measure is still expressed in terms of polylogarithms. We use this method to exhibit three new examples of Mahler measure and higher Mahler measure formulas. One of them involves a single term with ζ(5).