$H^0$ of Igusa varieties via automorphic forms

Arno Kret; Sug Woo Shin

doi:10.5802/jep.246

H^{0}

of Igusa varieties via automorphic forms
[

H^{0}

des variétés d’Igusa via les formes automorphes]

Arno Kret ¹ ; Sug Woo Shin ²

¹ Korteweg-de Vries Institute Science Park 105, 1090 GE Amsterdam, Netherlands
² Department of Mathematics, UC Berkeley Berkeley, CA 94720, USA

Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 1299-1390.

Résumé
Abstract

Notre théorème principal décrit la cohomologie en degré $0$ des variétés d’Igusa non basiques en termes de représentations automorphes de dimension $1$ dans le cadre des réductions modulo $p$ des variétés de Shimura de type Hodge avec niveau hyper-spécial en $p$ . Nous obtenons comme application une approche complètement nouvelle de deux questions géométriques. Premièrement, nous déduisons l’irréductibilité de la tour d’Igusa et sa généralisation aux variétés d’Igusa non basiques dans la même généralité, ce qui étend des résultats d’Igusa, Ribet, Falting-Chai, Hida, et d’autres. Deuxièmement, nous vérifions la partie discrète de la conjecture des orbites de Hecke, qui revient à l’assertion que les composantes irréductibles d’une feuille centrale non basique appartiennent à une unique orbite sous l’action de l’algèbre de Hecke première à $p$ , ce qui généralise des travaux de Chai, Oort, Yu, entre autres. Nous démontrons aussi des critères purement locaux d’irréductibilité de la feuille centrale. Notre preuve est basée sur une formule de type Langlands-Kottwitz pour les variétés d’Igusa due à Mack-Crane, sur une étude asymptotique de la formule des traces, et sur une estimée pour les représentations unitaires et leurs modules de Jacquet en théorie des représentations des groupes $p$ -adiques due à Howe-Moore et à Casselman.

Our main theorem describes the degree $0$ cohomology of non-basic Igusa varieties in terms of one-dimensional automorphic representations in the setup of mod $p$ Hodge-type Shimura varieties with hyperspecial level at $p$ . As an application we obtain a completely new approach to two geometric questions. Firstly, we deduce irreducibility of Igusa towers and its generalization to non-basic Igusa varieties in the same generality, extending previous results by Igusa, Ribet, Faltings–Chai, Hida, and others. Secondly, we verify the discrete part of the Hecke orbit conjecture, which amounts to the assertion that the irreducible components of a non-basic central leaf belong to a single prime-to- $p$ Hecke orbit, generalizing preceding works by Chai, Oort, Yu, et al. We also show purely local criteria for irreducibility of central leaves. Our proof is based on a Langlands–Kottwitz type formula for Igusa varieties due to Mack-Crane, an asymptotic study of the trace formula, and an estimate for unitary representations and their Jacquet modules in representation theory of $p$ -adic groups due to Howe–Moore and Casselman.

Reçu le : 2022-04-06
Accepté le : 2022-11-16
Publié le : 2023-10-20

DOI : 10.5802/jep.246

Classification : 11G18
Keywords: Shimura varieties, Igusa varieties, central leaves, automorphic representations, trace formula
Mot clés : Variétés de Shimura, variétés d’Igusa, feuilles centrales, représentations automorphes, formule des traces

Affiliations des auteurs :

Arno Kret ¹ ; Sug Woo Shin ²

¹ Korteweg-de Vries Institute Science Park 105, 1090 GE Amsterdam, Netherlands
² Department of Mathematics, UC Berkeley Berkeley, CA 94720, USA

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {$H^0$ of {Igusa} varieties via automorphic forms},
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Arno Kret; Sug Woo Shin. $H^0$ of Igusa varieties via automorphic forms. Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 1299-1390. doi : 10.5802/jep.246. https://jep.centre-mersenne.org/articles/10.5802/jep.246/

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