p-adic and motivic measure on Artin n-stacks

TL;DR

This paper develops a p-adic measure theory for Artin n-stacks over p-adic integers, proving rationality of associated series and demonstrating uniformity across primes using motivic integration.

Contribution

It introduces a new p-adic measure framework for Artin n-stacks and establishes the uniform rationality of Serre series via motivic integration.

Findings

Proved rationality of Poincare and Serre series for Artin n-stacks.

Established uniform rationality of Serre series across different primes.

Developed a measure theory for Artin n-stacks over p-adic integers.

Abstract

We construct a notion of p-adic measure on Artin n-stacks which are strongly of finite type over the ring of p-adic integers. We also prove the rationality of of the Poincare series and the Serre series associated with such stacks. Finally, we use motivic integration to show that the rationality of the Serre series is uniform with respect to p.