Horizontal $p$-adic $L$-functions

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 $d\equiv 2\mathrm{m}\mathrm{o}\mathrm{d}4$ 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.