From $L$-series of elliptic curves to Mahler measures

TL;DR

This paper establishes new links between Mahler measures and L-values of elliptic curves, providing explicit formulas, hypergeometric expressions, and functional equations for specific conductors and polynomial families.

Contribution

It proves conjectural relations between Mahler measures and L-values for certain elliptic curves and introduces new hypergeometric formulas and a functional equation for Mahler measures.

Findings

Proved relations between Mahler measures and L-values for conductors 20 and 24.

Derived hypergeometric expressions for L-values of CM elliptic curves with conductors 27 and 36.

Established a new functional equation for Mahler measures of a specific polynomial family.

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 CM elliptic curves of conductors 27 and 36. Furthermore, we prove a new functional equation for the Mahler measure of the polynomial family $(1+X)(1+Y)(X+Y)-\alpha XY$, $\alpha\in\mathbb R$.