Number theory seminar.

Speaker: Asbjørn Nordentoft, Université Paris-Saclay

Title: Horizontal p-adic L-functions

Abstract: 

Goldfeld’s Conjecture predicts that exactly 50% of quadratic twists of a fixed elliptic curve will have L-function non-vanishing at the central point. When considering the non-vanishing of higher order twists of elliptic curve L-functions, a conjecture of David-Fearnly-Kisilevsky predicts that 100% are non-vanishing. Very little was previously known beyond the quadratic case as the problem lies beyond the current technology of e.g. analytic number theory. In this talk I will present a p-adic approach relying on the construction of a ‘horizontal p-adic L-function’. The main technical proven is a structure theorem for the character zeros of such horizontal measures. We use our construction to obtain strong quantitative non-vanishing results for general order twists. In particular, we obtain the best bound towards Goldfeld's Conjecture for one hundred percent of elliptic curves (improving on a result of Ono). I will also present applications to simultaneous non-vanishing and Diophantine stability. 

This is joint work with Daniel Kriz