Minimally Ramified Deformations when $\ell \neq p$

TL;DR

This paper introduces a new class of deformation conditions for Galois representations over $ ext{ell}$-adic fields, generalizing minimally ramified deformations for $GL_n$, by controlling unipotent element types during deformation.

Contribution

It defines and analyzes a liftable deformation condition for orthogonal or symplectic Galois representations, extending the minimally ramified framework to broader settings involving nilpotent orbits.

Findings

Established a deformation condition where unipotent types remain constant

Connected nilpotent orbit theory with Galois deformation problems

Generalized minimally ramified deformations beyond $GL_n$

Abstract

Let $p$ and $\ell$ be distinct primes, and $\rho$ be an orthogonal or symplectic representation of the absolute Galois group of an $\ell$-adic field over a finite field of characteristic $p$. We define and study a liftable deformation condition of lifts of $\rho$ "ramified no worse than $\rho$", generalizing the minimally ramified deformation condition for $GL_n$ studied in \cite{cht08}. The key insight is to restrict to deformations where an associated unipotent element does not change type when deforming. This requires an understanding of nilpotent orbits and centralizers of nilpotent elements in the relative situation, not just over fields.