Intersecting subvarieties of abelian varieties with algebraic subgroups of complementary dimension

Abstract

Listen

Translate article

In fact both authors stated stronger conjectures. Zilber conjectured a finiteness statement for subvarieties that are contained in an improper intersection of X with an algebraic subgroup of S. Pink was motivated by a more general conjecture in the context of mixed Shimura varieties. He also showed that Conjecture 1 implies the celebrated Mordell–Lang conjecture. An important precursor was set in 1999 by Bombieri, Masser, and Zannier [3] who proved the following result for an algebraic curve C embedded in the algebraic torus Gm , not contained in the translate of a proper algebraic subgroup, and defined over Q, an algebraic closure of Q: the set of algebraic points of C which are contained in the union of all algebraic subgroups of codimension at least 1 has bounded absolute Weil height. They then used their result to show that there are only finitely many points on C contained in the union of all algebraic subgroups of codimension at least 2.