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Arithmetic volume of Shtukas and Langlands duality

Zeyu Wang, Wenqing Wei

Abstract

We extend the work of Feng--Yun--Zhang relating the arithmetic volume of Shtukas with derivatives of zeta functions by allowing arbitrary coweights for split semisimple algebraic groups. As in their original work, the formula involves some numbers called eigenweights. We obtain uniform formulas for the eigenweights in terms of the Langlands dual group, marking the first structural role for the dual group in such formulas governing derivatives of L-functions.

Arithmetic volume of Shtukas and Langlands duality

Abstract

We extend the work of Feng--Yun--Zhang relating the arithmetic volume of Shtukas with derivatives of zeta functions by allowing arbitrary coweights for split semisimple algebraic groups. As in their original work, the formula involves some numbers called eigenweights. We obtain uniform formulas for the eigenweights in terms of the Langlands dual group, marking the first structural role for the dual group in such formulas governing derivatives of L-functions.

Paper Structure

This paper contains 62 sections, 23 theorems, 224 equations, 7 tables.

Table of Contents

  1. Introduction
  2. Motivation
  3. Main result
  4. Moduli stack of Shtukas
  5. Arithmetic volume of the moduli of Shtukas
  6. Main result: same modification type
  7. Main result: different modification types
  8. New ingredients
  9. Notations
  10. General formalism
  11. Volume data
  12. Local volume data
  13. Global volume data
  14. Local-to-global construction
  15. Relative Hecke operators
  16. ...and 47 more sections

Key Result

Theorem 1.2

For $\lambda_I=(\lambda,\lambda,\cdots,\lambda)$ where $\lambda\in X_*(T)_+$, assuming $\mathop{\mathrm{Sht}}\nolimits_{G,\lambda_I}\neq \varnothing$, we have Here, $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (52)

  • Example 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Definition 2.1
  • Example 2.2: Local volume datum for determinant line bundle
  • Example 2.3: Cup product with a cohomology class
  • Definition 2.4
  • Example 2.5
  • ...and 42 more