Let E/Q be an elliptic curve and fix a positive integer D. A classical question in arithmetic statistics asks to understand the rank of the F-rational points of E as F varies over cyclic degree D extensions of Q. Via the Birch—Swinnerton-Dyer Conjecture this is (believed to be) equivalent to understanding the non-vanishing of the central values of the L-series of E twisted by Dirichlet characters of order D.

In this talk, I will give a general introduction to the non-vanishing problem mentioned above and explain a new p-adic approach developed by Daniel Kriz and myself. More precisely, we associate to an elliptic curve E/Q a measure (the horizontal p-adic L-function) interpolating L-values of E twisted by Dirichlet characters of p-power order and conductor prime to p. This can be seen a "horizontal" version of the classic theory of "vertical" p-adic L-functions. For D=2 we use our methods to obtain the best results towards a conjecture of Goldfeld (in rank 0 and 1) for 100 % of elliptic curves.

This is joint work with Daniel Kriz.