On the mod p cohomology for GL2

Abstract

Let p be a prime number and F a totally real number field unramified at places above p. Let r‾:Gal(F‾/F)→GL2(Fp‾) be a modular Galois representation which satisfies the Taylor-Wiles hypothesis and some technical genericity assumptions. For v a fixed place of F above p, we prove that many of the admissible smooth representations of GL2(Fv) over Fp‾ associated to r‾ in the corresponding Hecke-eigenspaces of the mod p cohomology have Gelfand–Kirillov dimension [Fv:Qp]. This builds on and extends the work of Breuil-Herzig-Hu-Morra-Schraen in [2] and Hu-Wang in [12], giving a unified proof in all cases (r‾ either semisimple or not at v).