On the existence of analytic families of G-stable lattices and their reductions

2023/12/31 by Torti, Emiliano
#11F03 #11F80 #11F85 #14G20 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2401.00462

Abstract

In this article, we prove the existence of rigid analytic families of G-stable lattices with locally constant reductions inside families of representations of a topologically compact group G, extending a result of Hellman obtained in the semi-simple residual case. Implementing this generalization in the context of Galois representations, we prove a local constancy result for reductions modulo prime powers of trianguline representations of generic dimension d. Moreover, we present two explicit applications. First, in dimension two, we extend to a prime power setting and to the whole rigid projective line a recent result of Bergdall, Levin and Liu concerning reductions of semi-stable representations of Gal(ℚ_p / ℚ_p) with fixed Hodge-Tate weights and large L-invariant. Second, in dimension d, let V_n be a sequence of crystalline representations converging in a certain geometric sense to a crystalline representation V. We show that for any refined version (V, σ) of V (or equivalently for any chosen triangulation of its attached (φ, Γ)-module D_rig (V) over the Robba ring), there exists a sequence of refinement σ_n of each of the V_n such that the limit as refined representations (V_n , σ_n ) converges to the (V, σ). This result does not hold under the weaker assumption that V_n converges only uniformly p-adically to V (in the sense of Chenevier, Khare and Larsen).

Related

• Semi-stable representations as limits of crystalline representations
• Local constancy for reductions of two-dimensional crystalline representations
• Reduction modulo p of certain semi-stable representations
• A note on reductions of 2-dimensional crystalline Galois representations
• Reductions of Some Crystalline Representations in the Unramified Setting