The Torsion Anomalous Conjecture

Let $G$ be a semi-abelian variety. A subvariety $V\subseteq G$ is called transverse if it is not contained in any translate, and weak-transverse if it is not contained in any torsion variety. An irreducible subvariety $Y$ of $V$ is $V$-torsion anomalous if it is an irreducible component of $V\cap(B+\zeta)$, where $B+\zeta$ is an irreducible torsion variety of $G$, and

$$\operatorname{codim}Y<\operatorname{codim}V+\operatorname{codim}B.$$

It is maximal if it is not contained in a $V$-torsion anomalous variety of strictly larger dimension.

Torsion Anomalous Conjecture. An irreducible subvariety $V$ of a semi-abelian variety contains only finitely many maximal $V$-torsion anomalous varieties.

The conjecture is a natural variant of a conjecture of Bombieri, Masser and Zannier and is related to the Zilber–Pink, Manin–Mumford and Mordell–Lang conjectures. It is known in several special cases, including curves in products of elliptic curves, abelian varieties with complex multiplication, tori, and certain codimension-two settings; it remains open in general.