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Intersection formulas on moduli spaces of unitary shtukas

Yongyi Chen, Benjamin Howard

TL;DR

The paper advances the program of relating arithmetic intersection data on moduli of unitary shtukas to automorphic derivatives by extending the FYZ arithmetic Siegel-Weil formula to a class of singular Fourier coefficients in a low-dimensional setting. It develops and exploits a unitary doubling framework, including a duplication formula, to connect cycles on shtuka spaces to derivatives of base-change $L$-functions. In a concrete low-rank case, the authors prove a version of the ASW conjecture and formulate a Gross-Zagier–style intersection formula, conditional on a modularity conjecture that links geometric Fourier coefficients to automorphic data. The work highlights how nonproper moduli can still yield meaningful height-like pairings through derived intersections and paves a route to GKZ-type relations via the arithmetic theta lift and doubling machinery, with potential generalization to higher rank under modularity.

Abstract

Feng-Yun-Zhang have proved a function field analogue of the arithmetic Siegel-Weil formula, relating special cycles on moduli spaces of unitary shtukas to higher derivatives of Eisenstein series. We prove an extension of this formula in a low-dimensional case, and deduce from it a Gross-Zagier style formula expressing intersection multiplicities of cycles in terms of higher derivatives of base-change $L$-functions.

Intersection formulas on moduli spaces of unitary shtukas

TL;DR

The paper advances the program of relating arithmetic intersection data on moduli of unitary shtukas to automorphic derivatives by extending the FYZ arithmetic Siegel-Weil formula to a class of singular Fourier coefficients in a low-dimensional setting. It develops and exploits a unitary doubling framework, including a duplication formula, to connect cycles on shtuka spaces to derivatives of base-change -functions. In a concrete low-rank case, the authors prove a version of the ASW conjecture and formulate a Gross-Zagier–style intersection formula, conditional on a modularity conjecture that links geometric Fourier coefficients to automorphic data. The work highlights how nonproper moduli can still yield meaningful height-like pairings through derived intersections and paves a route to GKZ-type relations via the arithmetic theta lift and doubling machinery, with potential generalization to higher rank under modularity.

Abstract

Feng-Yun-Zhang have proved a function field analogue of the arithmetic Siegel-Weil formula, relating special cycles on moduli spaces of unitary shtukas to higher derivatives of Eisenstein series. We prove an extension of this formula in a low-dimensional case, and deduce from it a Gross-Zagier style formula expressing intersection multiplicities of cycles in terms of higher derivatives of base-change -functions.
Paper Structure (17 sections, 17 theorems, 158 equations)

This paper contains 17 sections, 17 theorems, 158 equations.

Table of Contents

  1. Introduction
  2. The Feng-Yun-Zhang classes
  3. The modularity conjecture
  4. Intersections with arithmetic theta lifts
  5. Notation
  6. Acknowledgements
  7. Statements and declarations
  8. Eisenstein series on unitary groups
  9. The quasi-split unitary group
  10. Fourier coefficients
  11. Siegel Eisenstein series
  12. The duplication formula
  13. Degenerate coefficients in low rank
  14. Cycles on moduli spaces of shtukas
  15. The arithmetic Siegel-Weil formula
  16. ...and 2 more sections

Key Result

Theorem 1.2.2

Suppose $\mathcal{E}_2$ is a line bundle on $X'$, and $a_2\colon \mathcal{E}_2\cong \sigma^*\mathcal{E}_2^\vee$ is a hermitian isomorphism. The naive special cycle is proper over $k$, and there exists a $K_1$-fixed automorphic form $\mathscr{D} \in \mathcal{A}(H_1)$, depending on $(\mathcal{E}_2,a_2)$, whose geometric Fourier coefficients are given by for every line bundle $\mathcal{E}_1$ on $X'

Theorems & Definitions (42)

  • Conjecture 1.2.1: Feng-Yun-Zhang
  • Theorem 1.2.2
  • Conjecture 1.3.1
  • Remark 1.3.2
  • Theorem 1.3.3
  • Remark 2.1.1
  • Remark 2.2.1
  • Remark 2.3.1
  • Proposition 2.3.2
  • proof
  • ...and 32 more