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

TL;DR

This paper proves the non-triviality of the $p$-adic Abel-Jacobi images of generalized Heegner cycles modulo $p$ over certain extensions, providing evidence for deep conjectures in number theory related to elliptic curves and modular forms.

Contribution

It establishes the non-triviality of these images in a new setting, extending previous results on Heegner points and contributing to the understanding of Bloch-Beilinson and Bloch-Kato conjectures.

Findings

Non-triviality of $p$-adic Abel-Jacobi images modulo $p$

Extension of results on Heegner points over $ ext{Z}_l$-anticylotomic extensions

Evidence supporting Bloch-Beilinson and Bloch-Kato conjectures

Abstract

Generalised Heegner cycles are associated to a pair of an elliptic Hecke eigenform and a Hecke character over an imaginary quadratic extension $K/\Q$. Let $p$ be an odd prime split in $K/\Q$ and $l\neq p$ 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 an evidence for the refined Bloch-Beilinson and the Bloch-Kato conjecture. In the case of two, it provides a refinement of the results of Cornut and Vatsal on the non-triviality of Heegner points over the $\Z_l$-anticylotomic extension of $K$.