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, 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 $\exp (2\pi ic(A))$ 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 Néron model of $A$, which measures the behaviour of the Néron model under tame base change.

The proof of the monodromy conjecture is generalized to all tamely ramified abelian varieties, in arbitrary characteristic, using our results on the rationality of the Néron component series in arXiv:0910.1816. The results on motivic invariants of degenerating CY-varieties will appear in a separate paper

References in corpus (6)

• Desingularization of quasi-excellent schemes in characteristic zero
• A trace formula for rigid varieties, and motivic Weil generating series for formal schemes
• A trace formula for varieties over a discretely valued field
• Trace formula for component groups of Néron models
• The Néron component series of an abelian variety
• Galois actions on Neron models of Jacobians