On the non-triviality of the p-adic Abel–Jacobi image of generalised Heegner cycles modulo p, I: Modular curves

Abstract

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.

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