Hodge filtration and crystalline representations of GL_n

2025/12/13 by Christophe Breuil, Breuil, Christophe, Yiwen Ding +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research

paper · pdf · doi:10.48550/arxiv.2512.12153

Abstract

Let p be a prime number, n an integer ≥ 2 and ρ an n-dimensional automorphic p-adic Galois representation (for a compact unitary group) such that r:=ρ\vert_Gal(ℚ_p/ℚ_p) is crystalline. Under a mild assumption on the Frobenius eigenvalues of D:=D_cris(r) and under the usual Taylor-Wiles conditions, we show that the locally analytic representation of GL_n(ℚ_p) associated to ρ in the corresponding Hecke eigenspace of the completed H⁰ contains an explicit finite length subrepresentation which determines and only depends on r. This generalizes previous results of the second author which assumed that the Hodge filtration on D was as generic as possible. Our approach provides a much more explicit link to this Hodge filtration (in all cases), which allows to study the internal structure of this finite length locally analytic subrepresentation.

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