Chow Groups of Abelian Varieties and Beilinson's Conjecture

TL;DR

This paper introduces a property for subsemigroups of endomorphisms of abelian varieties and proves that orbits of cycles under such semigroups have finite-dimensional spans in the Chow group.

Contribution

It defines the property AA for subsemigroups of endomorphisms and establishes finite-dimensionality of cycle orbits in the Chow group for abelian varieties over number fields.

Findings

Property AA holds for endomorphism semigroups over number fields.

Orbits of cycles under semigroups with property AA are finite-dimensional.

Results relate to Beilinson's conjecture on algebraic cycles.

Abstract

In the present paper we introduce the property AA of a subsemigroup of the endomorphism semigroup of an abelian variety, which holds for semigroup of endomorphisms of an abelian variety defined over a number field, and show that the orbit of any cycle under a semigroup with property AA in the Chow group $\otimes\Q$ has finite dimensional span.