Non-vanishing of finite order twists of L-values via horizontal p-adic L-functions

Goldfeld’s Conjecture predicts that exactly 50% of quadratic twists of a fixed elliptic curve have L-function non-vanishing at the central point. Correspondingly, when considering twists by higher order Dirichlet characters, it has been predicted by David-Fearnly-Kisilevsky that 100% should be 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 ‘horizontal p-adic L-functions’. This yields strong quantitative non-vanishing results for general order twists. In particular, we obtain the best result 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 talk is based on joint work with Daniel Kriz.
In the introductory talk, I will give an introduction to the concept in arithmetic statistics known as 'Diophantine stability' and the connections to the Birch—Swinnerton-Dyer Conjecture. In particular, I will discuss Goldfeld's Conjecture, as well as the case of the cyclotomic $p$-tower studied in Iwasawa theory.