The Mahler measure of exact polynomials and special $L$-values of $K3$ surfaces

TL;DR

This paper links the Mahler measure of exact polynomials to special values of $L$-functions of $K3$ surfaces and modular forms, extending previous results to higher variables and specific cases.

Contribution

It generalizes the relationship between Mahler measures and $L$-values to four-variable polynomials and under Beilinson's conjecture, relates a specific polynomial's Mahler measure to zeta and $L$-functions.

Findings

Mahler measure expressed via Deligne-Beilinson cohomology

Extension of the Mahler measure-$L$-value relationship to four variables

Conditional expression of a polynomial's Mahler measure in terms of zeta and $L$-functions

Abstract

We express the Mahler measure of an exact polynomial in arbitrarily many variables in terms of Deligne-Beilinson cohomology. We then focus on the relationship between the Mahler measure of four-variable exact polynomials and the special value of the $L$-function of $K3$ surfaces at $s = 4$. This result extends the three-variable case studied in \cite{Tri23}. Finally, we prove, under Beilinson's conjecture, that the Mahler measure of the polynomial $(x+1)(y+1)(z+1) + t$ is expressed in terms of the Riemann zeta function and the $L$-function of the modular form of weight 3 and level 7.