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Arithmetic volumes of moduli stacks of Shtukas

Tony Feng, Zhiwei Yun, Wei Zhang

TL;DR

The paper develops a comprehensive framework for tautological classes on moduli stacks of shtukas over function fields, introducing a regularized arithmetic volume defined via traces of Hecke-induced endomorphisms on Bun_G and linking these volumes to higher derivatives of the Gross-motive L-function. It establishes a function-field analogue of Hirzebruch’s proportionality principle through volume identities and derives an Atiyah–Bott–type description for Bun_G that extends to general reductive groups, including non-split and torus cases. A key innovation is the construction of the phantom tautological ring, its Frobenius-dual structure, and its relation to Colmez-type conjectures in the function-field setting. The paper also provides explicit computations of eigenweights in classical groups and develops a robust machinery (nabla operators, Ran filtration, and local determinants) to compute arithmetic volumes in split and quasisplit scenarios, with broad implications for automorphic and G-bundle cohomology in the function-field realm.

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.

Arithmetic volumes of moduli stacks of Shtukas

TL;DR

The paper develops a comprehensive framework for tautological classes on moduli stacks of shtukas over function fields, introducing a regularized arithmetic volume defined via traces of Hecke-induced endomorphisms on Bun_G and linking these volumes to higher derivatives of the Gross-motive L-function. It establishes a function-field analogue of Hirzebruch’s proportionality principle through volume identities and derives an Atiyah–Bott–type description for Bun_G that extends to general reductive groups, including non-split and torus cases. A key innovation is the construction of the phantom tautological ring, its Frobenius-dual structure, and its relation to Colmez-type conjectures in the function-field setting. The paper also provides explicit computations of eigenweights in classical groups and develops a robust machinery (nabla operators, Ran filtration, and local determinants) to compute arithmetic volumes in split and quasisplit scenarios, with broad implications for automorphic and G-bundle cohomology in the function-field realm.

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 -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.
Paper Structure (140 sections, 67 theorems, 512 equations)

This paper contains 140 sections, 67 theorems, 512 equations.

Table of Contents

  1. Introduction
  2. Volume and Hirzebruch proportionality for locally symmetric spaces
  3. Arithmetic volumes of Shimura varieties
  4. Arithmetic volumes of moduli of Shtukas
  5. Moduli of shtukas
  6. Endomorphisms of cohomology
  7. Tautological classes
  8. Arithmetic Volume
  9. Main results
  10. The case $r=0$
  11. Calculation of eigenweights
  12. Tautological rings
  13. Structure of the phantom tautological ring
  14. The tautological ring of Shimura varieties
  15. An analog of the Colmez conjecture
  16. ...and 125 more sections

Key Result

Theorem 1.3.8

Let $\mu=(\mu_{1},\cdots, \mu_{r})$ be an admissible sequence of minuscule dominant coweights of $G$. Let $\eta = (\eta_1, \ldots, \eta_r)$, where $\eta_{j}\in \textup{H}^{2(D_{\mu_j}+1)}({\mathbb{B} P_{\mu_j}})$ satisfying the Commutativity Assumption assump:split-operators-commute. Then, with the

Theorems & Definitions (166)

  • Definition 1.3.5
  • Remark 1.3.6
  • Theorem 1.3.8
  • Theorem 1.3.9
  • Theorem 1.3.10
  • Remark 1.3.11
  • Remark 1.4.2
  • Example 2.3.1
  • Definition 3.1.1
  • Example 3.1.2
  • ...and 156 more