Horizontal p-adic L-functions

In this talk I will explain how to associate to an elliptic curve E/Q a measure (i.e. the horizontal p-adic L-function) interpolating L-values of E twisted by Dirichlet characters of p-power order and conductor prime to p. It turns out that there is a general structure result for the character zeroes of such measures, reminiscent of the Weierstrass preparation theorem. This allows for progress to be made towards conjectures of David-Fearnley-Kisilevsky. In particular, obtaining the best results towards Goldfeld’s conjecture (in rank 0 and 1) for 100 % of elliptic curves.

 This is joint work with Daniel Kriz.