Cohomology of groups acting on vector spaces over finite fields

2020/06/30 by Davide Lombardo, Lombardo, Davide, Laura Paladino +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2006.16857

openalex publication_date 2020/06/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let F_q be the finite field with q=p^m elements and G be a subgroup of \rmGL_n(F_q). A famous theorem of Nori published in 1987 states that there exists a (non-effective) constant c(n), depending only on n, such that if p>c(n) and G acts semisimply on F_pⁿ, then H¹(G,F_pⁿ)=0. We solve the long-standing problem, also considered by Serre of giving an effective proof of Nori's Theorem. Our approach yields the optimal constant c(n)=n+2. We also prove a more general version of Nori's theorem, namely, that for all powers q of p, if G acts semisimply on F_qⁿ and p>n+2, then H¹(G,F_qⁿ) is trivial. We apply these results to refine a criterion, proved by Çiperiani and Stix, which gives sufficient conditions for an affirmative answer to a classical question posed by Cassels in the case of abelian varieties over number fields.

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