Institute of Mathematics

Talk

Modul:   MAT770  Oberseminar: Algebraische Geometrie

On the Zilber-Pink Conjecture for complex abelian varieties

Gabriel Dill talk

Date: 11.11.19  Time: 13.15 - 14.45  Room: Y27H25

In 2016, Habegger and Pila proved the Zilber-Pink conjecture for a curve in an abelian variety, both defined over the field of algebraic numbers; i.e. they proved that if the intersection of the curve with the union of all algebraic subgroups of codimension at least 2 is infinite, then the curve must be contained in a proper algebraic subgroup. In joint work with Fabrizio Barroero, we deduce from their theorem the same result in the case when the curve and the abelian variety are defined over the complex numbers, using a recent result of Gao. More generally, we reduce the full conjecture for abelian varieties to the case when everything is defined over the field of algebraic numbers. We also show that the two slightly different formulations of the conjecture due to Zilber on one hand and Pink on the other hand are actually equivalent (for abelian varieties).