Asymptotic Mahler measure of Gaussian periodsMPIM

by Gunther Cornelissen (Universiteit Utrecht/MPIM)

Thursday Jun 12, 2025, 3:00 PM → 4:00 PM Europe/Berlin

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

MPI-Oberseminar

I will describe a construction of a family of algebraic integers (Gaussian periods) with cyclic Galois group and small Mahler measure/height. Studying their asymptotics as a function of the degree (essentially a problem of Quasi-Monte-Carlo integration) leads to a family of Calabi-Yau varieties constructed from reflexive polytopes, whose Mahler measure in turn can be studied via probability theory (similar to the computation of density of states in graphite) as their dimension grows. Finally, we run into a problem related to the smallest prime in an arithmetic progression. Computations indicate interesting growth behaviour (better than the currently proven non-zero constant) of the smallest height of a non-cyclotomic integer with cyclic Galois group, as a function of its degree.

All notions will be defined and a lot of examples will be given. (Joint work with David Hokken and Berend Ringeling.)