Arithmetic volumes of moduli stacks of Shtukas

TL;DR

This paper introduces the concept of tautological classes in the cohomology of moduli stacks of shtukas, linking their arithmetic volumes to derivatives of Artin L-functions and exploring the structure of the phantom tautological ring for applications in function field analogs of Colmez's Conjecture.

Contribution

It establishes a formula connecting arithmetic volumes of tautological classes to derivatives of Artin L-functions and analyzes the structure of the phantom tautological ring with applications to function field conjectures.

Findings

Arithmetic volume of tautological classes relates to higher derivatives of Artin L-functions.

Structure of the phantom tautological ring is characterized using Hecke correspondences and Vinberg's degeneration.

Applications include a function field analog of Colmez's Conjecture.

Abstract

We define and study "tautological classes" in the cohomology of moduli stacks of shtukas, pursuing two directions of applications. First, we prove a formula relating the "arithmetic volume" of tautological classes to higher derivatives of Artin $L$-functions, which can be viewed as an arithmetic analog of Hirzebruch's Proportionality principle. Second, we define and analyze the structure of the "phantom tautological ring", using a general relation between Hecke correspondences and Vinberg's degeneration, and give applications to a function field analog of Colmez's Conjecture.