Noether-Lefschetz cycles on the moduli space of abelian varieties

The locus of non-simple abelian varieties in the moduli space of principally polarized abelian varieties gives rise to Noether-Lefschetz cycles. We study their intersection theoretic properties using the tautological projection constructed in [CMOP24], and show that projection defines a homomorphism when restricted to cycles supported on that locus. Using Hecke correspondences and the pullback by Torelli we prove that $[{\mathcal{A}}_1\times {\mathcal{A}}_{g-1}]$ is not tautological in the sense of [vdG99] for $g=12$ and $g\ge 16$ even. We also explore the connections between Noether-Lefschetz cycles and the Gromov-Witten theory of a moving elliptic curve.