Shimura varieties with ${\Gamma }_1\left(p\right)$-level via Hecke algebra isomorphisms: the Drinfeld case

Haines, Thomas J.; Rapoport, Michael

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 45 (2012) no. 5, pp. 719-785.

On étudie le facteur local en $p$ de la fonction zêta semi-simple d’une variété de Shimura du type de Drinfeld, où la structure de niveau en $p$ est donnée par le radical pro-unipotent d’un sous-groupe d’Iwahori. La méthode suivie est une adaptation à ce cas de la méthode de comptage de Langlands-Kottwitz. On détermine de façon explicite la fonction test dans l’algèbre de Hecke correspondante  ; puis on démontre que c’est un élément central en déterminant ses images sous des isomorphismes d’algèbres de Hecke dus à Goldstein, Morris et Roche.

We study the local factor at $p$ of the semi-simple zeta function of a Shimura variety of Drinfeld type for a level structure given at $p$ by the pro-unipotent radical of an Iwahori subgroup. Our method is an adaptation to this case of the Langlands-Kottwitz counting method. We explicitly determine the corresponding test functions in suitable Hecke algebras, and show their centrality by determining their images under the Hecke algebra isomorphisms of Goldstein, Morris, and Roche.

Classification : 14G35, 11F70
Keywords: Shimura varieties, Hasse-Weil zeta functions, automorphic $L$-functions
Mot clés : variétés de Shimura, fonctions zêta de Hasse-Weil, fonctions $L$ automorphes

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