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

Abstract

In a previous paper we constructed 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 arithmetic Fourier transform. Another novelty in the argument is a 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 derived Fourier analysis on derived vector bundles, which we develop.