Abstract 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 A₆-₁ 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.
Aitor Iribar López (Thu,) studied this question.