Galois actions on Neron models of Jacobians

TL;DR

This paper investigates how Galois group actions extend to regular models of curves over discrete valuation rings, analyzing their impact on cohomology and the structure of Néron models, with explicit results for genus 1 and 2 curves.

Contribution

It provides a formula for the Brauer trace of Galois actions on cohomology and characterizes the jumps in the Néron model filtration based on fiber type, independent of residue characteristic.

Findings

Derived a formula for the Brauer trace of Galois actions.

Showed jumps in the Néron model filtration depend only on fiber type.

Computed jumps explicitly for genus 1 and 2 curves.

Abstract

Let $X$ be a smooth curve defined over the fraction field $K$ of a complete d.v.r. $R$, and let $K'/K$ be a tame extension. We study extensions of the $G = \Gal(K'/K)$-action on $ X_{K'} $ to certain regular models of $X_{K'}$ over $R'$, the integral closure of $R$ in $K'$. In particular, we consider the induced action on the cohomology groups of the structure sheaf of the special fiber of such a regular model, and obtain a formula for the Brauer trace of the endomorphism induced by a group element on the alternating sum of the cohomology groups. We apply these results to study a natural filtration of the special fiber of the N\'eron model of the Jacobian of $X$ by closed, unipotent subgroup schemes. We show that the jumps in this filtration only depend on the fiber type of the special fiber of the minimal regular model with strict normal crossings for $X$ over $R$, and in particular are independent of the residue characteristic. Furthermore, we obtain information about where these jumps occur. We also compute the jumps for each of the finitely many possible fiber types for curves of genus 1 and 2.