Shimura varieties with $\Gamma_1(p)$-level via Hecke algebra isomorphisms: the Drinfeld case

TL;DR

This paper investigates the local factors of the semi-simple zeta function for Drinfeld-type Shimura varieties with specific level structures, employing Hecke algebra isomorphisms and the Langlands-Kottwitz method.

Contribution

It introduces a novel approach to determine test functions at p for Shimura varieties with (p)-level, extending the Langlands-Kottwitz counting method to this setting.

Findings

Explicit determination of test functions at p using Hecke algebra isomorphisms

Extension of the Langlands-Kottwitz method to Drinfeld-type Shimura varieties

Enhanced understanding of local factors of the semi-simple zeta function

Abstract

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 use Hecke algebra isomorphisms to determine the test functions at p.