On the non-triviality of the p-adic Abel–Jacobi image of generalised Heegner cycles modulo p, I: Modular curves
Generalised Heegner cycles are associated with a pair of an elliptic newform and a Hecke character over an imaginary quadratic extension K/Q. Let p be an odd prime split in K/Q and let l≠p be an odd unramified prime. We prove the non-triviality of the p-adic Abel-Jacobi image of generalised Heegner cycles modulo p over the Z_l-anticylotomic extension of K. The result is evidence for the refined Bloch-Beilinson and the Bloch-Kato conjecture. In the case of weight two and l an ordinary prime, it provides a non-trivial refinement of the results of Cornut and Vatsal on Mazur's conjecture regarding the non-triviality of Heegner points over the Z_l -anticylotomic extension of K. In the case of weight two and l a supersingular prime, it settles Mazur's conjecture earlier known just for l ordinary.
Additional Information© 2020 University Press, Inc. Received by editor(s): July 9, 2017; Received by editor(s) in revised form: July 29, 2018, and June 25, 2019; Published electronically: January 8, 2020. We are grateful to our advisor Haruzo Hida for persistent guidance and encouragement. During the preparation of the first draft, the author was a graduate student in UCLA. We are grateful to friends back then for the distinctive atmosphere. We thank Ming-Lun Hsieh for numerous instructive comments on the previous versions of the article and also for his assistance. We thank Francesc Castella for helpful conversations, particularly regarding the p-adic Waldspurger formula. We thank Barry Mazur, Dinakar Ramakrishnan, Christopher Skinner and Burt Totaro for instructive comments and encouragement. We also thank Miljan Brakocevic, Brian Conrad, Henri Darmon, Samit Dasgupta, Chandrashekhar Khare, Jan Nekovář, Ye Tian, Vinayak Vatsal, Kevin Ventullo and Xinyi Yuan for helpful conversations about the topic. Finally, we are indebted to the referee. The current form of the article owes much to the detailed comments and suggestions of the referee.
Submitted - 1410.0300.pdf