Enrico Da Ronche: Kolyvagin's conjecture for non-ordinary modular forms

Number theory seminar: Enrico Da Ronche (University of Genova)

Kolyvagin's conjecture for non-ordinary modular forms

Kolyvagin's conjecture predicts the existence of a non-trivial 
class in the system built by Kolyvagin starting from Heegner points on
elliptic curves. Thanks to Heegner cycles built by Besser on Kuga-Sato
varieties over Shimura curves, it is possible to build a system of
Kolyvagin classes attached to a modular form of any weight, whose
non-triviality was proved by Longo, Pati and Vigni in the ordinary case.
In this talk we will see how these classes are built and we will give 
a sketch of the proof of the conjecture in the non-ordinary setting.