On torsion anomalous intersections

TL;DR

This paper advances the understanding of torsion anomalous varieties in abelian varieties, proving cases of a deep conjecture and providing explicit bounds, with implications for the effective Mordell-Lang Conjecture.

Contribution

It proves certain cases of the torsion anomalous conjecture for varieties in CM elliptic curves and provides explicit bounds, extending previous theoretical results.

Findings

Torsion anomalous varieties of codimension one are non-dense in certain cases.

Explicit uniform bounds are established depending on the variety.

The conjecture is proved for codimension two varieties in CM elliptic curves.

Abstract

A deep conjecture on torsion anomalous varieties states that if $V$ is a weak-transverse variety in an abelian variety, then the complement $V^{ta}$ of all $V$-torsion anomalous varieties is open and dense in $V$. We prove some cases of this conjecture. We show that the $V$-torsion anomalous varieties of relative codimension one are non-dense in any weak-transverse variety $V$ embedded in a product of elliptic curves with CM. We give explicit uniform bounds in the dependence on $V$. As an immediate consequence we prove the conjecture for $V$ of codimension two in a product of CM elliptic curves. We also point out some implications on the effective Mordell-Lang Conjecture.