Non-dense sets of subvarieties in a power of an elliptic curve

TL;DR

This paper proves that certain algebraic points of bounded height satisfying specific conditions form a non-dense subset in varieties within powers of elliptic curves, impacting conjectures like Pink-Zilber and Mordell-Lang.

Contribution

It establishes non-density results for algebraic points with bounded height on subvarieties of elliptic curve powers, advancing understanding in unlikely intersections.

Findings

Algebraic points of bounded height are non-dense in the subvarieties.

Results support conjectures related to unlikely intersections.

Implications for Pink-Zilber and Mordell-Lang conjectures.

Abstract

Let V be an algebraic variety embedded in a power of an elliptic curve, both defined over the algebraic numbers. We show that the set of algebraic points of V which are of bounded height and which satisfy certain algebraic conditions are a non-dense subset of V. This result has implications in the context of the Pink-Zilber Conjecture and Mordel-Lang plus Bogomolov Theorem.