2025 Theses Doctoral

Towards generic modularity of higher theta series over global function fields and p-adic local fields

Zeff, Chaim Avram Bettigole

Recent work of Feng-Yun-Zhang constructs higher theta series for unitary groups valued in cycles on the moduli stack of Hermitian shtukas, and proves using Fourier-theoretic methods that these theta series are modular after restriction to the generic fiber and on l-adic cohomology.

In the first part of this thesis, motivated by ideas from the relative Langlands program we generalize their construction to a natural class of reductive dual pairs, which contains almost all dual pairs for which such a construction should be expected, and prove generic modularity on l-adic cohomology. In characteristic zero, no analogue of moduli stacks of shtukas with more than one leg exists globally, but such analogues do exist locally over p-adic fields, as defined by Scholze--Weinstein.

In the second part of this thesis, we construct special cycles on these stacks, which under certain assumptions should be directly analogous to the higher theta series, and in this case prove generic modularity. Our main tool is a Fourier transform for cohomological correspondences, which we develop in the setting of p-adic geometry, building on ideas of Anschütz-Le Bras and using the power of recent six-functor formalisms due to Scholze and Heyer-Mann.