Twisted Gross-Zagier theorems and central derivatives in Hida families

Given a Hida family of modular forms, a conjecture of Greenberg predicts that L-functions of forms in the family should generically vanish to order 0 or 1 at the center of the functional equation. Similarly the Selmer groups of forms in the family should generically be of rank 0 or 1. In this talk I will prove a generalization of the Gross-Zagier theorem, relating Neron-Tate heights of special points on the modular Jacobian J_1(N) to derivatives of L-functions, and explain how this generalization can be used to verify Greenberg's conjecture for any particular Hida family.