Modularity of $d$-elliptic loci with level structure

TL;DR

This paper studies the modularity of generating series of special cycles on moduli spaces with level structure, providing new results on their non-vanishing for certain parameters and connecting to conjectures about elliptic loci.

Contribution

It proves the modularity of generating series of special cycles with level structure and demonstrates non-vanishing of associated modular forms for specific cases.

Findings

Generating series are modular due to Kudla-Millson theorem.

Non-vanishing of modular forms for genus 2 when level N ≥ 11 and N ≠ 12.

Connection between special cycles and elliptic Noether-Lefschetz loci.

Abstract

We consider the generating series of special cycles on $\mathcal{A}_1(N)\times \mathcal{A}_g(N)$, with full level $N$ structure, valued in the cohomology of degree $2g$. The modularity theorem of Kudla-Millson for locally symmetric spaces implies that these series are modular. When $N=1$, the images of these loci in $\mathcal{A}_g$ are the $d$-elliptic Noether-Lefschetz loci, which are conjectured to be modular. In the appendix, it is shown that the resulting modular forms are nonzero for $g=2$ when $N\geq 11$ and $N\neq 12$.