2023/10/31 by Daniel Kriz, Kriz, Daniel, Asbjørn Christian Nordentoft +1 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2310.20678
openalex publication_date 2023/10/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/03
We define new objects called 'horizontal p-adic L-functions' associated to L-values of twists of elliptic curves over ℚ by characters of p-power order and conductor prime to p. We study the fundamental properties of these objects and obtain applications to non-vanishing of finite order twists of central L-values, making progress toward conjectures of Goldfeld and David--Fearnley--Kisilevsky. For general elliptic curves E over ℚ we obtain strong quantitative lower bounds on the number of non-vanishing central L-values of twists by Dirichlet characters of fixed order d≡ 2 \mod 4 greater than two. We also obtain non-vanishing results for general d, including d = 2, under mild assumptions. In particular, for elliptic curves with E[2](ℚ) = 0 we improve on the previously best known lower bounds on the number of non-vanishing L-values of quadratic twists due to Ono. Finally, we obtain results on simultaneous non-vanishing of twists of an arbitrary number of elliptic curves with applications to Diophantine stability.