Modularity of higher theta series I: cohomology of the generic fiber

TL;DR

This paper proves the modularity of higher theta series in cohomology for unitary groups over function fields, introducing new Fourier transform techniques and sheaf-cycle correspondences within derived algebraic geometry.

Contribution

It establishes the generic modularity of higher theta series' cohomological realizations, employing novel arithmetic Fourier transform and sheaf-cycle methods in derived algebraic geometry.

Findings

Proves modularity of higher theta series in cohomology.

Introduces arithmetic Fourier transform on moduli spaces.

Develops a sheaf-cycle correspondence extending classical methods.

Abstract

In a previous paper we constructed $\textit{higher}$ theta series for unitary groups over function fields, and conjectured their modularity properties. Here we prove the generic modularity of the $\ell$-adic realization of higher theta series in cohomology. The proof debuts a new type of Fourier transform, occurring on the Borel-Moore homology of moduli spaces for shtuka-type objects, that we call the $\textit{arithmetic Fourier transform}$. Another novelty in the argument is a $\textit{sheaf-cycle correspondence}$ extending the classical sheaf-function correspondence, which facilitates the deployment of sheaf-theoretic methods to analyze algebraic cycles. Although the modularity property is a statement within classical algebraic geometry, the proof relies on derived algebraic geometry, especially a nascent theory of $\textit{derived Fourier analysis}$ on derived vector bundles, which we develop.