On the Torsion Anomalous Conjecture in CM abelian varieties

TL;DR

This paper proves new cases of the Torsion Anomalous Conjecture for CM abelian varieties using effective methods, extending previous results and deriving implications for the effective Mordell-Lang Conjecture.

Contribution

It provides the first effective proofs of certain cases of the TAC in CM abelian varieties, generalizing prior work on CM elliptic curves.

Findings

Proved new cases of the TAC for CM abelian varieties.

Derived implications for the effective Mordell-Lang Conjecture.

Extended previous results from CM elliptic curves to higher-dimensional CM abelian varieties.

Abstract

The Torsion Anomalous Conjecture (TAC) states that a subvariety V of an abelian variety A has only finitely many maximal torsion anomalous subvarieties. In this work we prove, with an effective method, some cases of the TAC when the ambient variety A has CM, generalising our previous results in products of CM elliptic curves. When V is a curve, we give new results and we deduce some implications on the effective Mordell-Lang Conjecture.