Motivic zeta functions for degenerations of abelian varieties and Calabi-Yau varieties

TL;DR

This survey explores motivic zeta functions for abelian and Calabi-Yau varieties over discretely valued fields, linking them to singularity theory, monodromy eigenvalues, and arithmetic invariants, with explicit computations on models with normal crossings.

Contribution

It provides a comprehensive overview of motivic zeta functions for these varieties, relating them to singularity theory and arithmetic invariants, and discusses explicit computations.

Findings

Relation between motivic zeta functions and monodromy eigenvalues

Generalization of arithmetic invariants to Calabi-Yau varieties

Explicit computations on models with strict normal crossings

Abstract

This is a survey on motivic zeta functions associated to abelian varieties and Calabi-Yau varieties over a discretely valued field. We explain how they are related to Denef and Loeser's motivic zeta function associated to a complex hypersurface singularity and we investigate the relation between the poles of the zeta function and the eigenvalues of the monodromy action on the tame $\ell$-adic cohomology of the variety. The motivic zeta function allows to generalize many interesting arithmetic invariants from abelian varieties to Calabi-Yau varieties and to compute them explicitly on a model with strict normal crossings.