Mahler Measure of 3D Landau-Ginzburg Potentials

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.

36 pages, many tables, comments are welcome; v2 corrected a few typos, and incorporated non-squarefree cases in Conjecture 4.4

References in corpus (2)

• Mahler measures of elliptic modular surfaces
• Fano-Mathieu correspondence