Rational curves on elliptic surfaces

Joint IAS/Princeton University Number Theory Seminar
Topic:Rational curves on elliptic surfaces
Speaker:Douglas Ulmer
Affiliation:Georgia Institute of Technology
Date:Thursday, May 5
Time/Room:4:30pm - 5:30pm/S-101
Video Link:https://video.ias.edu/puias/2016/0504-Ulmer

Given a non-isotrivial elliptic curve $E$ over $K = \mathbb F_q(t)$, there is always a finite extension $L$ of $K$ which is itself a rational function field such that $E(L)$ has large rank. The situation is completely different over complex function fields: For "most" $E$ over $K = \mathbb C(t)$, the rank $E(L)$ is zero for any rational function field $L = \mathbb C(u)$. The yoga that suggests this theorem leads to other remarkable statements about rational curves on surfaces generalizing a conjecture of Lang.