Non-divisible cycles on products of very general Abelian varieties

TL;DR

This paper develops a method to generate infinitely many non-divisible codimension 2 cycles on products of very general Abelian varieties, using a new notion of the cycle's field of definition and ramification properties.

Contribution

It introduces the concept of 'field of definition' for cycles modulo a prime and applies ramification to produce infinitely many non-divisible cycles, extending previous results.

Findings

Infinite non-divisible cycles on products of Abelian varieties.

Field of definition is a ramified extension of the modular variety's function field.

Chow group modulo a prime is infinite for products of three or more elliptic curves.

Abstract

In this paper, we give a recipe for producing infinitely many non-divisible codimension $2$ cycles on a product of two or more very general Abelian varieties. In the process, we introduce the notion of "field of definition" for cycles in the Chow group modulo (a power of) a prime. We show that for a quite general class of codimension $2$ cycles we call "primitive cycles," the field of definition is a ramified extension of the function field of a modular variety. This ramification allows us to use Nori's isogeny method \cite{N} (modified by Totaro \cite{T}) to produce infinitely many non-divisible cycles. As an application, we prove the Chow group modulo a prime of a product of $3$ or more very general elliptic curves is infinite, generalizing work of Schoen.