Action of Hecke Correspondences on Heegner Curves on a Siegel Threefold

We get an analog of Kolyvaginʹs trace relations for a Siegel threefold X. Namely, let V X be a Heegner curve (points of V correspond to Abelian surfaces with some fixed multiplication ring) and let Tp be a Hecke correspondence on X, so Tp(V) is a codimension 2 cycle on X. We describe the set of irreducible components of the support of Tp(V) in terms of geometry of Tp(t), where t X is a generic point. Tp(t) is a three-dimensional quadric hypersurface over p. We find also some equivalence relations on this set of irreducible components. The same method can be applied to other pairs V X (or more generally chains Xn Xn − 1 ••• X1) of Shimura varieties, and other Hecke correspondences. Finally, we discuss the possibility of finding reductions at p of these irreducible components, and of applying the Birch–Mazur–Bloch method to prove that the Abel–Jacobi image of some linear combination of V and other similar curves is not of torsion.