N\'eron models and base change

TL;DR

This paper investigates how Néron models of semi-abelian varieties, especially Jacobians, change under finite field extensions, focusing on component groups, conductors, and motivic zeta functions.

Contribution

It provides new insights into the behavior of Néron models under base change, including rationality results and relations between various conductors and filtrations.

Findings

Rationality results for generating series of component groups

Relations between Chai's base change conductor, Edixhoven's filtration, and the Artin conductor

Applications to motivic zeta functions of semi-abelian varieties

Abstract

We study various aspects of the behaviour of N\'eron models of semi-abelian varieties under finite extensions of the base field, with a special emphasis on wildly ramified Jacobians. In Part 1, we analyze the behaviour of the component groups of the N\'eron models, and we prove rationality results for a certain generating series encoding their orders. In Part 2, we discuss Chai's base change conductor and Edixhoven's filtration, and their relation to the Artin conductor. All of these results are applied in Part 3 to the study of motivic zeta functions of semi-abelian varieties. Part 4 contains some intriguing open problems and directions for further research. The main tools in this work are non-archimedean uniformization and a detailed analysis of the behaviour of regular models of curves under base change.