Motivic zeta functions of abelian varieties, and the monodromy conjecture

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 fu...

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Bibliographic Details
Published in:Advances in mathematics (New York. 1965) 2011-05, Vol.227 (1), p.610-653
Main Authors: Halle, Lars Halvard, Nicaise, Johannes
Format: Article
Language:English
Subjects:
Abelian varieties
Monodromy conjecture
Motivic zeta functions
Néron models
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Summary: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 Q ℓ in C , the value exp ( 2 π i c ( 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 Néron model of A, which measures the behavior of the Néron model under tame base change.
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2011.02.011