The zeta function of stacks of $G$-zips and truncated Barsotti-Tate groups

TL;DR

This paper computes the zeta functions of stacks of G-zips and truncated Barsotti-Tate groups, linking their structure to Weyl groups and root systems, advancing understanding of their automorphisms over finite fields.

Contribution

It provides explicit formulas for zeta functions of these stacks using Weyl group actions, based on classification results by Pink, Wedhorn & Ziegler and Gabber & Vasiu.

Findings

Zeta functions expressed via Weyl group actions

Classification of G-zips over algebraically closed fields

Automorphism groups of truncated Barsotti-Tate groups

Abstract

We study stacks of truncated Barsotti-Tate groups and the $G$-zips defined by Pink, Wedhorn & Ziegler. The latter occur naturally when studying truncated Barsotti-Tate groups of height 1 with additional structure. By studying objects over finite fields and their automorphisms we determine the zeta functions of these stacks. These zeta functions can be expressed in terms of the Weyl group of the reductive group $G$ and its action on the root system. The main ingredients are the classification of $G$-zips over algebraically closed fields and their automorphism groups by Pink, Wedhorn & Ziegler, and the study of truncated Barsotti-Tate groups and their automorphism groups by Gabber & Vasiu.