Modularity of Noether-Lefschetz cycles and geometric theta lifting for Siegel modular
Ryan Tamura (UW Madison)

I will discuss joint work with Timothy Cheek and Jiacheng Xia that resolves a cohomological version of a conjecture of Greer and Lian on the modularity of Noether–Lefschetz cycles on Siegel modular varieties. Our proof uses Kudla–Millson theory and the Mathai–Quillen description of the Kudla–Millson Schwartz form obtained by Brancherau. This result was also obtained independently by García, Greer, and Iribar López. I will then introduce our theory of geometric theta lifting for Siegel modular varieties, including the case of level structure. This construction gives rise to lifts from cusp forms to cycles in the span of Noether–Lefschetz cycles. Crucially, nonzero geometric theta lifts lie outside the span of the Chern classes of the Hodge bundle. Time permitting, I will discuss further consequences of our theory for the geometry of Noether–Lefschetz cycles on Siegel modular varieties.

Zoom (680 4828 0736, the password is the first Fourier coefficient of the modular j-function (as digits)).