Heegner cycles and p-adic L-functions

Francesc Castella, Ming Lun Hsieh

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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.

Original languageEnglish (US)
Pages (from-to)567-628
Number of pages62
JournalMathematische Annalen
Volume370
Issue number1-2
DOIs
StatePublished - Feb 1 2018

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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