en.wikipedia.org/wiki/Zilber%E2%80%93Pink_conjecture
1 correction found
For algebraic tori and semiabelian varieties it was proposed by Boris Zilber and independently by Enrico Bombieri, David Masser, Umberto Zannier in the early 2000's.
This overstates Bombieri–Masser–Zannier’s role. Standard sources credit Bombieri–Masser–Zannier with the torus case, while Zilber is credited with the broader semiabelian-variety formulation.
Full reasoning
The sentence conflates two different historical formulations.
Credible survey and research sources distinguish them as follows:
- Bombieri–Masser–Zannier formulated conjectures for tori (the multiplicative group (\mathbb G_m^n)).
- Zilber later formulated a conjecture for semiabelian varieties, of which the toric case is a consequence.
A 2022 paper in Annales Scientifiques de l’École Normale Supérieure states that in the same paper, Bombieri–Masser–Zannier only suggested that an analogue of their torus result might hold for abelian varieties, and that "A couple of years later, Zilber independently stated a conjecture for semiabelian varieties". That directly contradicts the article’s wording that the semiabelian-variety version was proposed independently by Bombieri, Masser, and Zannier as well.
Likewise, Habegger’s paper explicitly labels the Bombieri–Masser–Zannier conjecture as a conjecture for (G_m^n), then separately explains that Zilber’s conjecture concerns an irreducible subvariety of a semi-abelian variety. So the historically accurate split is: BMZ for the toric case; Zilber for the semiabelian case; Pink for the Shimura-variety generalization.
So the sentence is inaccurate because it attributes the semiabelian-variety proposal to Bombieri–Masser–Zannier, whereas the cited literature instead treats their conjecture as the torus version and Zilber’s as the semiabelian version.
2 sources
- On the Zilber–Pink conjecture for complex abelian varieties (sample PDF, Ann. Sci. ENS)
"Bombieri, Masser and Zannier suggest that a possible analogue of their result for curves in G_m^n could hold for (families of) abelian varieties... A couple of years later, Zilber independently stated a conjecture for semiabelian varieties of which the result of Bombieri, Masser and Zannier is a consequence."
- Intersecting subvarieties of G_m^n with algebraic subgroups
"Conjecture 2. (Bombieri, Masser, Zannier) Let X ⊂ G_m^n be an irreducible closed subvariety..." and later: "Zilber’s Conjecture states that" for an irreducible closed subvariety of a "semi-abelian variety" the relevant intersection is contained in a finite union of proper algebraic subgroups.