Asbjørn Nordentoft, Université Paris 13
Goldfeld’s Conjecture predicts that exactly 50% of quadratic twists of a fixed elliptic curve will have L-function vanishing at the central point. When considering the non-vanishing of twists of elliptic curve L-functions by characters of (fixed) order greater than 2, it has been predicted by David-Fearnly-Kisilevsky that 100% should be non-vanishing. Very little was previously known outside the quadratic case as the problem lies beyond the current technology of analytic number theory. In this talk I will present a p-adic approach (from the point of view of an analytic number theorist) relying on the construction of a ‘horizontal p-adic L-function’. This approach yields 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 p-adic moments of L-functions.
This is joint work with Daniel Kriz.