Higher theta series for unitary groups over function fields

TL;DR

This paper constructs generalized virtual fundamental classes for special cycles on Hermitian shtukas, forming higher theta series conjectured to be modular, supported by structural and modularity evidence.

Contribution

It extends the construction of virtual fundamental classes to broader settings, linking them to higher derivatives of Fourier coefficients and proposing their modularity.

Findings

Cycle classes behave as conjectured under intersection operations.

Verification of modularity in specific cases.

Introduction of derived algebraic geometry for special cycles.

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.