Tate cohomology and local base change of generic representations of \rm GL₃ -- non-banal case

2023/10/31 by Dhar, Sabyasachi
#FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2310.20399

Abstract

Let F be a finite extension of ℚ_p, and let E be a finite Galois extension of F with degree of extension l, where l and p are distinct odd primes. Let π_F be an integral, l-adic generic representation of \rm GL₃(F), and let π_E be the base change lifting of π_F to \rm GL₃(E). Let J_l(π_F) (resp. J_l(π_E)) be the unique generic sub-quotient of the mod-l reduction of π_F (resp. π_E). In this article, using the local converse theorem over local Artinian \mathbbF_l-algebras, we prove that the Frobenius twist of J_l(π_F) is isomorphic to the Tate cohomology group \widehatH⁰(\rm Gal(E/F),J_l(π_E)). The result of this article removes the hypothesis that the prime l does not divide the pro-order of \rm GL₂(F).

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