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

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

2010-12-22

arxiv.org/abs/1012.4969v1

Mathematics

Algebraic Geometry

Scientific paper

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.

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