Horizontal $p$-adic $L$-functions

Abstract

We define new objects called 'horizontal $p$-adic $L$-functions' associated to $L$-values of twists of elliptic curves over $\mathbb{Q}$ 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 $\mathbb{Q}$ we obtain strong quantitative lower bounds on the number of non-vanishing central $L$-values of twists by Dirichlet characters of fixed order $d\equiv 2\mathrm{mod}4$ greater than two. We also obtain non-vanishing results for general $d$, including $d=2$, under mild assumptions. In particular, for elliptic curves with $E[2](\mathbb{Q})=0$ 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.

Minor changes to the exposition, removed conditions in Corollary 5.17 and changed the terminology of 'Taylor-Wiles primes' to 'orderly primes'. 46 pages