A note on the projection of the diagonal subvariety

TL;DR

This paper examines the conditions under which the diagonal subvariety of a product of two CM elliptic curves is pre-periodic, linking it to the pre-periodicity of its projection on a product of projective lines.

Contribution

It provides a specific example illustrating the relationship between pre-periodicity of a diagonal subvariety and its projection in the context of CM elliptic curves.

Findings

Diagonal subvariety is pre-periodic iff its projection is pre-periodic.

Connects pre-periodicity properties between higher-dimensional and lower-dimensional projections.

Highlights special behavior in the case of CM elliptic curves.

Abstract

The diagonal subvariety on a product of two CM elliptic curves, is presented as an example of a dimension one subvariety, that is pre-periodic only if the respective projection on the product of two projective lines is also pre-periodic.