# Algebraic Geometry Seminar

#### Vojta’s conjecture and uniform boundedness of full-level structures on abelian varieties over number fields

In 1977, Mazur proved that the torsion subgroup of an elliptic curve over $\mathbb{Q}$ is, up to isomorphism, one of only 15 groups. Before Merel gave a qualitative generalization of this result to arbitrary number fields, it was known that variants of the $abc$ conjecture would imply uniform boundedness of torsion on elliptic curves over number fields of bounded degree. In this talk, I will explain how, using Vojta’s conjecture as a higher-dimensional generalization of the abc conjecture, one can deduce similar uniform boundedness statements for full-level structures on abelian varieties of fixed dimension over number fields.  This is joint work with Dan Abramovich and Keerthi Madapusi-Pera.