Noether-Lefschetz cycles on the moduli space of abelian varieties

The locus of nonsimple 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 [4], 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 [A₁ × Ag-1] is not tautological in the sense of [38] for $g=12$ and g≥ 16 even. We also explore the connections between Noether-Lefschetz cycles and the Gromov-Witten theory of a moving elliptic curve.