Tate module and bad reduction

TL;DR

This paper investigates the Galois action on the Tate module of a Jacobian of a curve over a local field, using the minimal regular model to relate it to the special fibre in cases of bad reduction.

Contribution

It provides a new description of the Galois action on the Tate module via the special fibre of the minimal regular model in semistable reduction cases.

Findings

Galois action can be described using the special fibre of the minimal regular model.

The approach applies to curves with bad reduction over local fields.

Provides tools for understanding Galois representations in arithmetic geometry.

Abstract

Let C/K be a curve over a local field. We study the natural semilinear action of Galois on the minimal regular model of C over a field F where it becomes semistable. This allows us to describe the Galois action on the l-adic Tate module of the Jacobian of C/K in terms of the special fibre of this model over F.