On the Explicit Torsion Anomalous Conjecture

TL;DR

This paper proves explicit bounds on torsion anomalous points in varieties within products of elliptic curves, confirming their finiteness and deriving new effective results related to the Mordell-Lang Conjecture.

Contribution

It provides the first explicit bounds for torsion anomalous points in non-CM and CM cases, advancing the understanding of their distribution and finiteness.

Findings

Explicit height bounds for torsion anomalous points in non-CM case

Effective finiteness results for torsion anomalous points

New bounds for rational points on certain families of curves

Abstract

The Torsion Anomalous Conjecture states that an irreducible variety $V$ embedded in a semi-abelian variety contains only finitely many maximal $V$-torsion anomalous varieties. In this paper we consider an irreducible variety embedded in a product of elliptic curves. Our main result provides a totally explicit bound for the N\'eron-Tate height of all maximal $V$-torsion anomalous points of relative codimension one, in the non CM case, and an analogous effective result in the CM case. As an application, we obtain the finiteness of such points. In addition, we deduce some new explicit results in the context of the effective Mordell-Lang Conjecture; in particular we bound the N\'eron-Tate height of the rational points of an explicit family of curves of increasing genus.