Motivic zeta functions of abelian varieties, and the monodromy conjecture

Details Motivic zeta functions of abelian varieties, and the monodromy conjecture Motivic zeta functions of abelian varieties, and the monodromy conjecture

2009-02-21

arxiv.org/abs/0902.3755v4

Mathematics

Algebraic Geometry

The proof of the monodromy conjecture is generalized to all tamely ramified abelian varieties, in arbitrary characteristic, us

Scientific paper

We prove for abelian varieties a global form of Denef and Loeser's motivic monodromy conjecture, in arbitrary characteristic. More precisely, we prove that for every tamely ramified abelian variety $A$ over a complete discretely valued field, its motivic zeta function has a unique pole at Chai's base change conductor $c(A)$ of $A$, and that the order of this pole equals one plus the potential toric rank of $A$. Moreover, we show that for every embedding of $\Q_\ell$ in $\C$, the value $\exp(2\pi i c(A))$ is an $\ell$-adic tame monodromy eigenvalue of $A$. The main tool in the paper is Edixhoven's filtration on the special fiber of the N\'eron model of $A$, which measures the behaviour of the N\'eron model under tame base change.

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