Deformation of $\ell$-adic sheaves with Undeformed Local Monodromy

TL;DR

This paper investigates how $ ext{ell}$-adic Galois representations of a function field can be deformed while preserving their local monodromy data, providing insights into the deformation theory of such representations.

Contribution

It introduces a framework for deforming $ ext{ell}$-adic sheaves with fixed local monodromy on algebraic curves, advancing understanding of their deformation spaces.

Findings

Deformation spaces are characterized under fixed local monodromy conditions.

Conditions for unobstructed deformations are identified.

New techniques for controlling local monodromy in deformations are developed.

Abstract

Let $X$ be a smooth connected algebraic curve over an algebraically closed field $k$. We study the deformation of $\ell$-adic Galois representations of the function field of $X$ while keeping the local Galois representations at all places undeformed.