Mahler Measure of 3D Landau-Ginzburg Potentials

TL;DR

This paper computes Mahler measures of 23 Laurent polynomial families linked to Fano 3-folds, expressing them via Eisenstein-Kronecker series and relating them to special values of L-functions, revealing new relations among these measures.

Contribution

It provides explicit formulas for Mahler measures of Landau-Ginzburg potentials on Fano 3-folds and uncovers novel relations among these measures.

Findings

Mahler measures expressed in terms of Eisenstein-Kronecker series

Connection between Mahler measures and L-values of weight-3 newforms

Discovery of 10 exotic relations among Mahler measures

Abstract

We express the Mahler measures of $23$ families of Laurent polynomials in terms of Eisenstein-Kronecker series. These Laurent polynomials arise as Landau-Ginzburg potentials on Fano $3$-folds, $16$ of which define $K3$ hypersurfaces of generic Picard rank $19$, and the rest are of generic Picard rank $< 19$. We relate the Mahler measure at each rational singular moduli to the value at $3$ of the $L$-function of some weight-$3$ newform. Moreover, we find $10$ exotic relations among the Mahler measures of these families.