Heegner cycles and $p$-adic $L$-functions

TL;DR

This paper proves the vanishing of Selmer groups for certain automorphic forms by linking Heegner cycles and $p$-adic $L$-functions, confirming cases of the Bloch-Kato and parity conjectures.

Contribution

It extends Kolyvagin's Euler system method to higher weights and anticyclotomic settings, establishing new cases of the Bloch-Kato and parity conjectures.

Findings

Vanishing of Selmer groups in specific cases

Higher weight analogue of Mazur's conjecture proven

Parity conjecture confirmed in the studied setting

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 Nekovar's work a proof of the parity conjecture in this setting.