On some mod p representations of quaternion algebra over ℚ_p

2022/01/05 by Yongquan Hu, Haoran Wang, Hu, Yongquan +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2201.01464

openalex publication_date 2022/01/05 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28

Abstract

Let F be a totally real field in which p is unramified and B be a quaternion algebra which splits at at most one infinite place. Let r:Gal(F/F)→ GL₂(\mathbbF_p) be a modular Galois representation which satisfies the Taylor-Wiles hypotheses. Assume that for some fixed place v|p, B ramifies at v and F_v is isomorphic to ℚ_p and r is generic at v. We prove that the admissible smooth representations of the quaternion algebra over ℚ_p coming from mod p cohomology of Shimura varieties associated to B have Gelfand-Kirillov dimension 1. As an application we prove that the degree two Scholze's functor vanishes on supersingular representations of GL₂(ℚ_p). We also prove some finer structure theorem about the image of Scholze's functor in the reducible case.

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