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.