The elliptic torsion anomalous conjecture in codimension 2

TL;DR

This paper proves the torsion anomalous conjecture for codimension 2 varieties in products of elliptic curves, extending previous results known only for CM elliptic curves, and provides effective height bounds.

Contribution

It establishes the conjecture for a broader class of elliptic curves and offers explicit bounds on the heights of maximal torsion anomalous varieties.

Findings

Proved the torsion anomalous conjecture for codimension 2 in E^N.

Extended known results from CM to non-CM elliptic curves.

Provided effective bounds on the heights of anomalous varieties.

Abstract

The torsion anomalous conjecture states that for any variety V in an abelian variety there are only finitely many maximal V-torsion anomalous varieties. We prove this conjecture for V of codimension 2 in a product E^N of any elliptic curve E. This was known only when E has CM. We also give an effective upper bound for the normalized height of these maximal V-torsion anomalous varieties.