Selmer groups of Hecke characters and Chow groups of self products of CM elliptic curves

TL;DR

This paper establishes bounds on Selmer groups of Hecke characters associated with CM elliptic curves using explicit reciprocity laws and Iwasawa theory, and applies these bounds to Chow groups of self-products of such curves.

Contribution

It introduces a novel approach combining Kato's reciprocity law, the Main Conjecture, and Nekovar's work to bound torsion in Chow groups of CM elliptic curves.

Findings

Bounded Selmer groups in terms of $L$-values.

Bound torsion in Chow groups of self-products.

Results valid for almost all primes $p$.

Abstract

We bound Selmer groups attached to Grossencharacters of CM elliptic curves by the appropriate $L$-value. Our method is to use Kato's explicit reciprocity law and the Main Conjecture as proved by Rubin. These results are then used together with work of Han and fundamental results of Nekovar to bound the torsion in the image of the $p$-adic cycle class map for self products of CM elliptic curves. Our results are valid for Chow groups of arbitrary codimension for almost all $p$.