TY - JOUR
T1 - Heegner cycles and p-adic L-functions
AU - Castella, Francesc
AU - Hsieh, Ming Lun
N1 - Funding Information:
During the preparation of this paper, F. Castella was partially supported by Grant MTM2012-34611 and by Prof. Hida’s NSF Grant DMS-0753991. M.-L. Hsieh was partially supported by a MOST Grant 103-2115-M-002-012-MY5.
Funding Information:
Fundamental parts of this paper were written during the visits of the first-named author to the second-named author in Taipei during February 2014 and August 2014; it is a pleasure to thank NCTS and the National Taiwan University for their hospitality and financial support. We would also like to thank Ben Howard, Shinichi Kobayashi and David Loeffler for their comments and enlightening conversations related to this work.
Publisher Copyright:
© 2017, Springer-Verlag Berlin Heidelberg.
PY - 2018/2/1
Y1 - 2018/2/1
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=85009881914&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85009881914&partnerID=8YFLogxK
U2 - 10.1007/s00208-017-1517-3
DO - 10.1007/s00208-017-1517-3
M3 - Article
AN - SCOPUS:85009881914
VL - 370
SP - 567
EP - 628
JO - Mathematische Annalen
JF - Mathematische Annalen
SN - 0025-5831
IS - 1-2
ER -