summer 2000 



Matteo Longo, Università degli Studi di Padova 15.01.2020 12:00, B137 Title: The Kolyvagin Conjecture for modular forms Abstract:
Kolyvagin conjecture for elliptic curves was formulated by V.Kolyvagin in the nineties, and predicts the nontriviality of certain cohomology classes costructed from Heegner points. Consequences of this conjecture include the ppart of the Birch and SwinnertonDyer conjecture, parity results, and a precise description of the structure of the TateShafarevich group of the elliptic curve. The original conjecture is now a theorem by the work of many people, including W.Zhang, C.Skinner and R.Venerucci. We investigate an analogue of Kolyvagin Conjecture for higher weight modular forms in which Heegner points are replaced by Heegner cycles on KugaSato varieties. As a consequence, we obtain some results on the structure of the TateShafarevich group attached to the modular form, and a ppart of a BlochKato conjecture in analytic rank 1. This is a work in collaboration with Stefano Vigni. 