Lubin-Tate moduli space of semisimple mod p Galois representations for GL₂ and Hecke modules

2023/06/20 by Cédric Pépin, Tobias Schmidt, Pépin, Cédric +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2306.11863

openalex publication_date 2023/06/20 · openalex created_date 2023/06/24 · openalex updated_date 2026/07/28

Abstract

Let p be an odd prime. Let F be a non-archimedean local field of residue characteristic p, and let \mathbbF_q be its residue field. Let H⁽¹⁾_\mathbbF_q be the pro-p-Iwahori-Hecke algebra of the p-adic group \textrm GL₂(F) with coefficients in \mathbbF_q, and let Z(H⁽¹⁾_\mathbbF_q) be its center. We define a scheme X(q)_\mathbbF_q whose geometric points parametrize the semisimple two-dimensional Galois representations of \textrm Gal(F/F) over \mathbbF_q. Then we construct a morphism from the spectrum of Z(H⁽¹⁾_\mathbbF_q) to X(q)_\mathbbF_q generalizing the morphism appearing in \citePS2 for F=ℚ_p. In the case F/ℚ_p, we show that the induced map from Hecke modules to Galois representations, when restricted to supersingular modules, coincides with Grosse-Klönne's bijection \citeGK18. For this, we determine the Lubin-Tate (φ,Γ)-modules associated to absolutely irreducible Galois representations.

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