TY - JOUR
T1 - Heegner cycles and p-adic L-functions
AU - Castella, Francesc
AU - Hsieh, Ming Lun
N1 - 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
SN - 0025-5831
VL - 370
SP - 567
EP - 628
JO - Mathematische Annalen
JF - Mathematische Annalen
IS - 1-2
ER -