Congruences of algebraic automorphic forms and supercuspidal\n representations

2020/09/17 by Jessica Fintzen, Sug Woo Shin, Fintzen, Jessica +1 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Finite Group Theory Research #Advanced Topics in Algebra

paper · pdf · doi:10.48550/arxiv.2009.08476

Abstract

Let G be a connected reductive group over a totally real field F which is\ncompact modulo center at archimedean places. We find congruences modulo an\narbitrary power of p between the space of arbitrary automorphic forms on\nG( mathbb A_F) and that of automorphic forms with supercuspidal components at\np, provided that p is larger than the Coxeter number of the absolute Weyl group\nof G. We illustrate how such congruences can be applied in the construction\nof Galois representations.\n Our proof is based on type theory for representations of p-adic groups,\ngeneralizing the prototypical case of GL(2) in [arXiv:1506.04022, Section 7] to\ngeneral reductive groups. We exhibit a plethora of new supercuspidal types\nconsisting of arbitrarily small compact open subgroups and characters thereof.\nWe expect these results of independent interest to have further applications.\nFor example, we extend the result by Emerton--Pa vsk =unas on density of\nsupercuspidal points from definite unitary groups to general G as above.\n

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• An introduction to representations of p-adic groups

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