Boundedness and growth of ranks

Héctor Pastén
Pontificia Universidad Católica de Chile (Chile)
I'll explain how classical conjectures of Lang on diophantine approximation of algebraic points imply Honda's conjecture on ranks of elliptic curves over number fields. This suggests that ranks are bounded over quadratic twists of elliptic curves over the rationals (unlike global function fields). On the other hand, I'll also explain some recent work with Garcia-Fritz where we show that certain patterns on elliptic curves give rise to large ranks, confirming a conjecture of Bremner.