Let p and ℓ be distinct primes, and ρ be an orthogonal or symplectic representation of the absolute Galois group of an ℓ-adic field over a finite field of characteristic p. We define and study a liftable deformation condition of lifts of ρ "ramified no worse than ρ", generalizing the minimally ramified deformation condition for GLn studied in [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.
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.
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