Daniel Kriz (Princeton)
Generalized Heegner cycles and congruences of p-adic L-functions

Abstract: I will discuss my result establishing a congruence between the Bertolini-Darmon-Prasanna anticyclotomic p-adic L-function attached to a newform f with reducible residual p-adic Galois representation and the Katz p-adic L-function. From this, there follows a congruence between p-adic Abel-Jacobi images of certain generalized Heegner cycles and products of certain Bernoulli numbers and Euler factors. As an application, one can show that when a semistable elliptic curve E/Q has reducible mod 3 Galois representation, quadratic twists with algebraic and analytic rank equal to r occur with positive proportion (ordering by absolute value of twisting discriminant) for r = 0 or 1. This in particular verifies the weak Goldfeld conjecture for these curves. The main result also has higher-dimensional applications to the Beilinson-Bloch conjecture for Rankin-Selberg motives. If time permits I will also discuss recent work with Chao Li pertaining to relationships between ranks within quadratic twist families of elliptic curves.