Galois actions on models of curves

TL;DR

This paper investigates how Galois group actions extend to regular models of algebraic curves over local fields, deriving formulas for traces and analyzing the structure of the Néron model's special fiber.

Contribution

It provides a formula for the Brauer trace of Galois actions on regular models and studies the filtration jumps of the Néron model of Jacobians, independent of residue characteristic.

Findings

Derived a formula for the Brauer trace of Galois actions.

Analyzed the jumps in the Néron model filtration based on fiber type.

Computed jumps for genus 1 and 2 curves.

Abstract

We study group actions on regular models of curves. If $X$ is a smooth curve defined over the fraction field $K$ of a complete d.v.r. $R$, every tamely ramified extension $K'/K$ with Galois group $G$ induces a $G$-action on $X_{K'}$. In this paper we study the extension of this $G$-action to certain regular models of $X_{K'}$. In particular, we obtain a formula for the Brauer trace of the endomorphism induced by a group element on the alternating sum of the cohomology groups of the structure sheaf of the special fiber of such a regular model. Inspired by this global study, we also consider similar questions for Galois actions on the exceptional locus of a tame cyclic quotient singularity. 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 $\Spec(R)$, 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 type for curves of genus 1 and 2.