Motivic zeta functions of abelian varieties, and the monodromy conjecture

Abstract

We prove for abelian varieties a global form of Denef and Loeser's motivic monodromy conjecture. More precisely, we prove that for any abelian $\C((t))$-variety $A$, 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 any embedding of $\Q_\ell$ in $\C$, the value $\exp(2\pi i c(A))$ is an $\ell$-adic monodromy eigenvalue of $A$. In mixed and positive characteristic we obtain partial results under the assumption that $A$ is tamely ramified. In particular, we show that the above properties still hold for tamely ramified Jacobians. 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. We also extend certain arithmetic invariants of abelian varieties to Calabi-Yau varieties, using motivic integration.