A refined Beilinson-Bloch conjecture for motives of modular forms

TL;DR

This paper refines the Beilinson-Bloch conjecture for motives of modular forms, relating p-adic Abel-Jacobi maps to Heegner cycles, and proves related theorems extending results from elliptic curves to higher weights.

Contribution

It introduces a refined conjecture connecting p-adic Abel-Jacobi images with Heegner cycles for modular forms of even weight and proves partial results supporting this conjecture.

Findings

Proved theorems supporting the refined conjecture.

Established higher weight analogues of elliptic curve results.

Linked p-adic Abel-Jacobi maps to Heegner cycles in this context.

Abstract

We propose a refined version of the Beilinson-Bloch conjecture for the motive associated with a modular form of even weight. This conjecture relates the dimension of the image of the relevant p-adic Abel-Jacobi map to certain combinations of Heegner cycles on Kuga-Sato varieties. We prove theorems in the direction of the conjecture and, in doing so, obtain higher weight analogues of results for elliptic curves due to Darmon.