Unirational algebraic groups and tame ramification

TL;DR

This paper studies the behavior of Néron models of certain algebraic groups over local fields, showing rationality of jumps and motivic zeta functions for unirational groups, with implications for algebraic and arithmetic geometry.

Contribution

It proves the rationality of jumps and motivic zeta functions for unirational algebraic groups, addressing questions posed by Halle-Nicaise and Edixhoven.

Findings

Jumps of unirational groups are rational numbers.

Motivic zeta functions of these groups are rational functions.

Results extend to Abelian varieties with specific reduction types.

Abstract

Let $\mathcal{O}_K$ be a complete discrete valuation ring with field of fractions $K$ and algebraically closed residue field $k.$ Let $G$ be a smooth connected commutative algebraic group over $K$ which does not contain a copy of $\mathbf{G}_{\mathrm{a}}.$ For each $d$ prime to $p:=\mathrm{char}\, k,$ let $K(d)$ be the unique extension of $K$ of degree $d.$ We investigate how the N\'eron lft-model of $G$ behaves under base change to the ring of integers $\mathcal{O}_{K(d)}.$ Information about this behaviour is encoded in the "jumps" of Edixhoven's filtration on the special fibre of the N\'eron lft-model of $G,$ as well as in Halle-Nicaise's motivic zeta function of $G.$ If $G$ is unirational (e. g. an algebraic torus), we show that the jumps of $G$ are rational numbers and that the motivic zeta function of $G$ is a rational function. We also deduce analogous results for Abelian varieties with potentially totally multiplicative reduction. This answers a question of Halle-Nicaise and partially one of Edixhoven. Along the way, we answer a question of Oesterl\'e about the structure of unipotent algebraic groups over function fields in positive characteristic. Under stronger conditions on $G,$ we obtain rationality of jumps even for separably closed but imperfect $k.$