On the cohomology of simple Shimura varieties with non quasi-split local groups

TL;DR

This paper investigates the cohomology of simple Shimura varieties with non quasi-split local groups, establishing vanishing properties of test functions and their base change expressions, with applications to trace formulas and zeta functions.

Contribution

It extends the analysis of Scholze test functions to non quasi-split local groups and proves their pseudostabilization and explicit distribution expressions.

Findings

Vanishing of twisted orbital integrals for these test functions

Existence of pseudostabilization base changes for non quasi-split groups

Applications to stable trace formula and Hasse-Weil zeta functions

Abstract

We study the Scholze test functions for bad reduction of simple Shimura varieties at a prime where the underlying local group is any inner form of a product of Weil restrictions of general linear groups. Using global methods, we prove that these test functions satisfy a vanishing property of their twisted orbital integrals, and we prove that the pseudostabilization base changes of such functions exist (even though the local group need not be quasi-split) and can be expressed in terms of explicit distributions in the stable Bernstein center. We then deduce applications to the stable trace formula and local Hasse-Weil zeta functions for these Shimura varieties.