Zilber-Pink conjecture

In mathematics, the Zilber–Pink conjecture is a far-reaching generalisation of many famous Diophantine conjectures and statements, such as André-Oort, Manin–Mumford, and Mordell-Lang. 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. For semiabelian varieties the conjecture implies the Mordell-Lang and Manin-Mumford conjectures. Richard Pink proposed a more general conjecture for Shimura varieties which also implies the André-Oort conjecture. In the case of algebraic tori, Zilber called it the Conjecture on Intersection with Tori (CIT). The general version is now known as the Zilber-Pink conjecture. It states roughly that atypical or unlikely intersections of an algebraic variety with certain special varieties are accounted for by finitely many special varieties.


In mathematics, the Zilber–Pink conjecture is a far-reaching generalisation of many famous Diophantine conjectures and statements, such as André-Oort, Manin–Mumford, and Mordell-Lang. 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. For semiabelian varieties the conjecture implies the Mordell-Lang and Manin-Mumford conjectures. Richard Pink proposed a more general conjecture for Shimura varieties which also implies the André-Oort conjecture. In the case of algebraic tori, Zilber called it the Conjecture on Intersection with Tori (CIT). The general version is now known as the Zilber-Pink conjecture. It states roughly that atypical or unlikely intersections of an algebraic variety with certain special varieties are accounted for by finitely many special varieties.
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