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Tate module and bad reduction

Tim Dokchitser, Vladimir Dokchitser, Adam Morgan

TL;DR

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 where it becomes semistable is described.

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.

Tate module and bad reduction

TL;DR

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 where it becomes semistable is described.

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.

Paper Structure

This paper contains 6 sections, 9 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Semilinear actions
  3. Semilinear morphisms
  4. Semilinear actions
  5. Geometric action over local fields
  6. Curves and Jacobians

Key Result

Theorem 1.5

The filtration eq1 of $T_l(A)$ is independent of the choice of $F/K$ and is $G_K$-stable. Moreover, $G_K$ acts semilinearlysee Definition def:semilinearaction on $\mathcal{C}/\mathcal{O}_F$, inducing actions on $\mathcal{C}_{k_F}$, $\Upsilon$, $\mathop{\mathrm{Pic}}\nolimits^0 \mathcal{C}_{\bar{k}_F The action of $\sigma \in G_K$ on $\mathcal{C}_{k_F}$ is uniquely determined by its action on non-s

Theorems & Definitions (36)

  • Theorem 1.5
  • Corollary 1.6
  • Remark 1.7
  • Remark 1.8
  • Example 1.9
  • Example 1.11
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Remark 2.5
  • ...and 26 more