Quantitative level lowering for Galois representations

TL;DR

This paper develops explicit Galois cohomology techniques to construct optimal level-lowering congruences for p-adic Galois representations, controlling ramification and independence properties.

Contribution

It introduces a method to produce finite sets of lifts of residual Galois representations with precise ramification and congruence properties, extending level-lowering techniques.

Findings

Constructs lifts ramified at specified primes

Ensures congruences mod p^d between lifts

Produces independent Galois representations

Abstract

We use Galois cohomology methods to produce optimal mod $p^d$ level lowering congruences to a $p$-adic Galois representation that we construct as a well chosen lift of a given residual mod $p$ representation. Using our explicit Galois cohomology methods, we construct for a reductive group $G$ and a given residual representation $\bar{\rho}: \Gamma_F \to G(k)$, ramified at a finite set of primes $S$, in favorable conditions that we identify, a finite set of lifts $\rho$, $\{\rho^q\}$ of $\bar{\rho}$ to $G(W(k))$ with the following properties: $\rho: \Gamma_F \to G(W(k))$ is ramified precisely at $S \cup Q$, with $Q$ a finite set of primes disjoint from $S$. For $q \in Q$, $\rho^q:G_F \to G(W(k))$ is unramified outside $S \cup Q \backslash \{q\}$ and $\rho$ and $\rho^q$ are congruent mod $p^d$ if $\rho$ mod $p^d$ is unramified at $q$. Furthermore, the Galois representations $\{\rho^q\}$ are "independent".