The N\'eron component series of an abelian variety

TL;DR

The paper introduces the Néron component series for abelian varieties over complete discretely valued fields, analyzing its properties under tame ramification and its role in the motivic monodromy conjecture.

Contribution

It defines the Néron component series, proves its rationality under tame ramification, and links its pole order to the potential toric rank, advancing understanding of the motivic monodromy conjecture.

Findings

Néron component series is rational for tamely ramified abelian varieties.

The pole at T=1 relates to the potential toric rank.

Results extend to elliptic curves and multiplicative reduction cases.

Abstract

We introduce the N\'eron component series of an abelian variety $A$ over a complete discretely valued field. This is a power series in $\Z[[T]]$, which measures the behaviour of the number of components of the N\'eron model of $A$ under tame ramification of the base field. If $A$ is tamely ramified, then we prove that the N\'eron component series is rational. It has a pole at T=1, whose order equals one plus the potential toric rank of $A$. This result is a crucial ingredient of our proof of the motivic monodromy conjecture for abelian varieties. We expect that it extends to the wildly ramified case; we prove this if $A$ is an elliptic curve, and if $A$ has potential purely multiplicative reduction.