Heegner cycles and p-adic L-functions
Journal
Mathematische Annalen
Journal Volume
370
Journal Issue
44563
Pages
567-628
Date Issued
2018
Author(s)
Abstract
In this paper, we deduce the vanishing of Selmer groups for the Rankin–Selberg convolution of a cusp form with a theta series of higher weight from the nonvanishing of the associated L-value, thus establishing the rank 0 case of the Bloch–Kato conjecture in these cases. Our methods are based on the connection between Heegner cycles and p-adic L-functions, building upon recent work of Bertolini, Darmon and Prasanna, and on an extension of Kolyvagin’s method of Euler systems to the anticyclotomic setting. In the course of the proof, we also obtain a higher weight analogue of Mazur’s conjecture (as proven in weight 2 by Cornut–Vatsal), and as a consequence of our results, we deduce from Nekovs work a proof of the parity conjecture in this setting. © 2017, Springer-Verlag Berlin Heidelberg.
Type
journal article