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Higher theta series for unitary groups over function fields

Tony Feng, Zhiwei Yun, Wei Zhang

Abstract

In previous work, we defined certain virtual fundamental classes for special cycles on the moduli stack of Hermitian shtukas, and related them to the higher derivatives of non-singular Fourier coefficients of Siegel-Eisenstein series. In the present article, we construct virtual fundamental classes in greater generality, including those expected to relate to the higher derivatives of singular Fourier coefficients. We assemble these classes into "higher" theta series, which we conjecture to be modular. Two types of evidence are presented: structural properties affirming that the cycle classes behave as conjectured under certain natural operations such as intersection products, and verification of modularity in several special situations. One innovation underlying these results is a new approach to special cycles in terms of derived algebraic geometry.

Higher theta series for unitary groups over function fields

Abstract

In previous work, we defined certain virtual fundamental classes for special cycles on the moduli stack of Hermitian shtukas, and related them to the higher derivatives of non-singular Fourier coefficients of Siegel-Eisenstein series. In the present article, we construct virtual fundamental classes in greater generality, including those expected to relate to the higher derivatives of singular Fourier coefficients. We assemble these classes into "higher" theta series, which we conjecture to be modular. Two types of evidence are presented: structural properties affirming that the cycle classes behave as conjectured under certain natural operations such as intersection products, and verification of modularity in several special situations. One innovation underlying these results is a new approach to special cycles in terms of derived algebraic geometry.

Paper Structure

This paper contains 108 sections, 80 theorems, 325 equations.

Table of Contents

  1. Introduction
  2. The modularity conjecture
  3. Main results
  4. Linear invariance
  5. A refinement of the main result of FYZ
  6. The case $n=m=1$
  7. Intersection with "CM cycles"
  8. Geometric properties of special cycles
  9. Some remarks on the derived algebro-geometric method
  10. Notation
  11. Notation for cycle classes
  12. Derived notation
  13. Some notational departures from FYZ
  14. Formulation of the conjecture
  15. Some (more) special cycles on moduli of shtukas
  16. ...and 93 more sections

Key Result

Theorem 1.3

Given a decomposition $\mathcal{E} \approx \mathcal{E}_1 \oplus \mathcal{E}_2 \oplus \ldots \oplus \mathcal{E}_j$, and $a_i \in \mathcal{A}_{\mathcal{E}_i}(k)$, the intersection product coincides with the sum of $[\mathcal{Z}_{\mathcal{E}}^r(a)]$ over all $a \colon \mathcal{E} \rightarrow \sigma^*\mathcal{E}^{\vee}$ satisfying the condition that

Theorems & Definitions (235)

  • Conjecture 1.1: Modularity conjecture
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Lemma 2.6
  • ...and 225 more