Producing Geometric Deformations of Orthogonal and Symplectic Galois Representations

TL;DR

This paper extends the theory of geometric lifts of Galois representations from two dimensions to higher-dimensional orthogonal and symplectic cases, using Fontaine-Laffaille theory to analyze local deformation conditions.

Contribution

It generalizes Ramakrishna's results on geometric lifts to higher dimensions, introducing new local deformation conditions at p based on Fontaine-Laffaille theory.

Findings

Established existence of geometric lifts for higher-dimensional orthogonal and symplectic Galois representations.

Generalized local deformation conditions at p using Fontaine-Laffaille theory.

Extended Ramakrishna's methods to broader classes of Galois representations.

Abstract

For a representation of the absolute Galois group of the rationals over a finite field of characteristic $p$, we study the existence of a lift to characteristic zero that is geometric in the sense of the Fontaine-Mazur conjecture. For two-dimensional representations, Ramakrishna proved that under technical assumptions odd representations admit geometric lifts. We generalize this to higher dimensional orthogonal and symplectic representations. A key step is generalizing and studying a local deformation condition at $p$ arising from Fontaine-Laffaille theory.