The Mahler measure for arbitrary tori

TL;DR

This paper explores a generalized Mahler measure over arbitrary tori, deriving new formulas connecting it to the derivatives of L-functions of elliptic curves, expanding understanding of Mahler measures in algebraic geometry.

Contribution

It introduces a variation of the Mahler measure over general tori and establishes formulas linking it to L'(E,0) for specific elliptic curves, a novel connection in the field.

Findings

Derived new formulas for the Mahler measure variation in terms of L'(E,0).

Connected Mahler measure variations to elliptic curve L-functions.

Extended the understanding of Mahler measures beyond classical settings.

Abstract

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 curve $E$ and we establish new formulas for this variation of the Mahler measure in terms of $L'(E,0)$.