Rational curves on elliptic surfaces
Given a non-isotrivial elliptic curve E over K=Fq(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=C(t), the rank E(L)
is zero for any rational function field L=C(u). The yoga that
suggests this theorem leads to other remarkable statements about
rational curves on surfaces generalizing a conjecture of Lang.