Zilber–Pink for algebraic tori

Establish the Zilber–Pink statement for the algebraic torus (G_m)^r: given an irreducible closed subvariety Y ⊂ (G_m)^r that is not contained in any proper torsion translate of an algebraic subtorus, prove that only finitely many maximal strict atypical special subvarieties of Y occur; equivalently, show that there are only finitely many maximal irreducible components C of intersections Y ∩ (t + H) with codim_Y C < codim_{(G_m)^r}(t + H), where H ranges over algebraic subtori of (G_m)^r and t ranges over torsion points.

Background

The paper reviews the Zilber–Pink framework for unlikely intersections in a general quadruple (Z, S, W, Y). The Zilber–Pink statement asserts that the maximal-under-inclusion strict atypical special subvarieties of Y are finite. Here, “atypical” means the intersection has larger-than-expected dimension (codimension drop), and “special” refers to a specified class of distinguished subvarieties within the ambient Z.

For the case Z = (G_m)r (an algebraic torus), the natural choice of special subvarieties is the set of torsion translates (cosets) of algebraic subtori. The authors note that, although geometric Zilber–Pink is known in broad generality (notably for variations of mixed Hodge structures), the core Zilber–Pink finiteness statement remains unresolved even in this basic torus setting. Addressing this case would settle one of the simplest outstanding instances of the conjectural picture.

References

On the other hand the Zilber-Pink problem remains open even in some of the simplest cases, for example when $Z = \mathbb{G}{r}_{m}$ and the special subvarieties are given by torsion translates of algebraic subgroups.

— Intersections and the Bézout Range: Abelian Varieties  (2604.02186 - Baldi et al., 2 Apr 2026) in Section 2.1 (The language of unlikely intersection problems)

The conjecture is still open in full generality; however, for the case of curves, it is known due to works of Bombieri, Masser and Zannier and Maurin in the case of tori, of Habegger and Pila and Barroero and Dill in the case of abelian varieties, and of Barroero, Kuhne and Schmidt in the case of curves in semiabelian varieties, when everything is defined over $$.

— Singular intersections in split semiabelian varieties  (2609.35307 - Ballini et al., 28 Sep 2026) in Section 1, Introduction