Mahler measure and L-functions of motives

Time: 9:30 -- 11: 00, April 2, 2025

Venue: Room 612, A6, Institute of Mathematics-VAST

Abstract: The Mahler measure of polynomials was introduced by Mahler in 1962 as a tool to study the transcendental number theory. Over time, numerous connections have been discovered between Mahler measures and objects in number theory, such as the special values of L-functions and the Riemann zeta function.

In the first part of the talk, I will first give an introduction to the Mahler measure of polynomials and highlight a fundamental result by Deninger from 1997. I will then express the Mahler measure of an exact polynomial in arbitrarily many variables in terms of Deligne-Beilinson cohomology. In the second part, I will reveal the relationship between the Mahler measure of three- and four-variable exact polynomials and Beilinson regulators. More precisely, I will construct motivic cohomology classes such that their Beilinson regulators compute the Mahler measure. This, therefore, establishes a connection between Mahler measures and the special values of L- functions of motives via Beilinson’s conjectures. Finally, I will present some examples to which my theorems apply. If time permits, I will also briefly discuss some open questions in this research direction.