Quantitative level lowering for 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".

Minor changes and corrections based on a referee's report. To appear in the Journal of the London Mathematical Society

References in corpus (3)

• Surjectivity Criteria for p-adic Representations, Part I
• Minimally Ramified Deformations when $\ell \ne p$
• Producing Geometric Deformations of Orthogonal and Symplectic Galois Representations