Producing geometric deformations of orthogonal and symplectic 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.

Deformation of rigid conjugate self-dual Galois representations

“…At the end, we remark that in the self-dual (not conjugate self-dual) case, the Fontaine-Laffaille deformations have been studied in [Boo19a].…”

Quantitative level lowering for Galois representations

“…Remark It is not known exactly when this assumption is satisfied, especially at primes dividing . For it is known to hold for classical groups, albeit after increasing , and for in the Fontaine–Laffaille case (see [3, 4, 6]).…”

Minimally ramified deformations when

“…This work forms part of my thesis [Boo16], and I am extremely grateful for the generosity and support of my advisor Brian Conrad, and for his extensive and helpful comments on drafts of my thesis. The thesis was originally submitted as a single paper before being split into this paper and [Boo19]: I thank the referee for a careful reading.…”

“…This can equivalently be expressed as exhibiting a formally smooth quotient of the universal lifting ring . In this paper, we study only the local theory: the applications to producing geometric lifts are discussed in [Boo19]. In the remainder of the introduction, we will sketch how to correctly generalize the minimally ramified deformation condition introduced for and analyze it.…”