GAAG seminar - Horizontal anticyclotomic twists of L-functions of elliptic curves

This talk will discuss non-vanishing at the critical center of dihedral twists of L-functions (or their derivatives, depending on the root number) of elliptic curves E over Q by anticyclotomic characters \chi of prime conductor P (for varying P) of an imaginary quadratic field K. There are consequences for the rational points and the Tate-Shafarevic group along \chi.This work is joint with Philippe Michel and was inspired by a wonderful (vertical) result in anticyclotomic Z_p-extensions of K by Cornut and Vatsal, establishing a conjecture of Mazur.