Action of Hecke Correspondences on Heegner Curves on a Siegel Threefold

✍ Scribed by Dmitry Logachev

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 273 KB
• Volume: 236
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.2000.8490

✦ Synopsis

We get an analog of Kolyvagin's trace relations for a Siegel threefold X. Ž Namely, let V ; X be a Heegner curve points of V correspond to Abelian . surfaces with some fixed multiplication ring and let T be a Hecke corresponp Ž . dence on X, so T V is a codimension 2 cycle on X. We describe the set of p Ž .

Ž . irreducible components of the support of T V in terms of geometry of T t , p p Ž . where t g X is a generic point. T t is a three-dimensional quadric hypersurface p over ‫ކ‬ . We find also some equivalence relations on this set of irreducible p components.

Ž The same method can be applied to other pairs V ; X or more generally chains . X ; X ; иии ; X of Shimura varieties, and other Hecke correspondences.

n n y1 1 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.