Mahler Measures and Dirichlet L-Functions: New Results on Chinburg’s ConjecturesMPIM

by Mahya Mehrabdollahi (MPIM)

Wednesday Apr 29, 2026, 2:30 PM → 3:30 PM Europe/Berlin

MPIM, Vivatsgasse, 7 - Lecture Hall (Max Planck Institute for Mathematics)

Number theory lunch seminar

This talk concerns Chinburg’s conjectures, which propose a connection
between two a priori different objects: Mahler measures and certain special
values of Dirichlet L-functions associated with odd quadratic characters.

The Mahler measure of a polynomial is the arithmetic mean of log|P| over
the unit torus. Chinburg’s conjecture (1984) states that, for each odd
quadratic Dirichlet character, there exists an integral bivariate rational
function (or, in its strongest form, an integral polynomial) whose Mahler
measure is equal to a rational multiple of the derivative at −1 of the
corresponding L-function. This relationship is currently known only in a
limited number of cases (18 values of the conductor of the Dirichlet
character).

I will present recent results obtained in collaboration with David Hokken
and Berend Ringeling, in which we construct new examples for previously
unknown conductors, thereby doubling the number of verified cases. Finally,
we establish a special case of the conjecture when coefficients in a
cyclotomic extension are allowed.