# Heights in families of abelian varieties

 Joint IAS/Princeton University Number Theory Seminar Topic: Heights in families of abelian varieties Speaker: Ziyang Gao Affiliation: Princeton University Date: Thursday, April 27 Time/Room: 4:30pm - 5:30pm/Fine 214, Princeton University

Given an abelian scheme over a smooth curve over a number field, we can associate two height function: the fiberwise defined Neron-Tate height and a height function on the base curve. For any irreducible subvariety $X$ of this abelian scheme, we prove that the Neron-Tate height of any point in an explicit Zariski open subset of $X$ can be uniformly bounded from below by the height of its projection to the base curve. We use this height inequality to prove the Geometric Bogomolov Conjecture over characteristic $0$. This is joint work with Philipp Habegger.