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

In a previous paper we constructed $\mathit{\text{higher}}$ theta series for unitary groups over function fields, and conjectured their modularity properties. Here we prove the generic modularity of the $\mathrm{\ell }$-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 $\mathit{\text{arithmetic}}\mathit{\text{Fourier}}\mathit{\text{transform}}$. Another novelty in the argument is a $\mathit{\text{sheaf-cycle}}\mathit{\text{correspondence}}$ 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 $\mathit{\text{derived}}\mathit{\text{Fourier}}\mathit{\text{analysis}}$ on derived vector bundles, which we develop.