Motivic zeta functions of abelian varieties, and the monodromy conjecture

By L. H. Halle, Johannes Nicaise. 2009

48 citations recorded by the source corpus.

We prove for abelian varieties a global form of Denef and Loeserʼs motivic monodromy conjecture, in arbitrary characteristic. More precisely, we prove that for every tamely ramified abelian variety A over a complete discretely valued field with algebraically closed residue field, its motivic zeta function has a unique pole at Chaiʼs 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 Ql in C, the value exp(2πic(A)) is an l-adic tame monodromy eigenvalue of A. The main tool in the paper is Edixhovenʼs filtration on the special fiber of the Neron model of A, which measures the behavior of the Neron model under tame base change.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/218d50f3dd210e7901c7efc6c22755a801afc523