Motivic zeta functions of abelian varieties, and the monodromy conjecture

TL;DR

This paper proves a global motivic monodromy conjecture for abelian varieties, linking the poles of their motivic zeta functions to monodromy eigenvalues and the structure of their Néron models.

Contribution

It establishes the conjecture for tamely ramified abelian varieties in arbitrary characteristic, connecting motivic zeta function poles to monodromy and Néron model filtrations.

Findings

Motivic zeta function has a unique pole at the base change conductor c(A).

Pole order equals one plus the potential toric rank of A.

Exponential of 2πi times c(A) is an ℓ-adic tame monodromy eigenvalue.

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 $\exp(2\pi i c(A))$ 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.