Horizontal pp-adic LL-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 $d\equiv 2 \text{ mod } 4$ 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, as well as results on $p$-adic valuations of moments of $L$-functions.Comment: 52 page

Supersingular main conjectures, Sylvester's conjecture and Goldfeld's conjecture

We prove a $p$-converse theorem for elliptic curves $E/\mathbb{Q}$ with complex multiplication by the ring of integers $\mathcal{O}_K$ of an imaginary quadratic field $K$ in which $p$ is ramified. Namely, letting $r_p = \mathrm{corank}_{\mathbb{Z}_p}\mathrm{Sel}_{p^{\infty}}(E/\mathbb{Q})$, we show that $r_p \le 1 \implies \mathrm{rank}_{\mathbb{Z}}E(\mathbb{Q}) = \mathrm{ord}_{s = 1}L(E,s) = r_p$ and $\#\mathrm{Sha}(E/\mathbb{Q}) < \infty$. In particular, this has applications to two classical problems of arithmetic. First, it resolves Sylvester's conjecture on rational sums of cubes, showing that for all primes $\ell \equiv 4,7,8 \pmod{9}$, there exists $(x,y) \in \mathbb{Q}^{\oplus 2}$ such that $x^3 + y^3 = \ell$. Second, combined with work of Smith, it resolves the congruent number problem in 100\% of cases and establishes Goldfeld's conjecture on ranks of quadratic twists for the congruent number family. The method for showing the above $p$-converse theorem relies on new interplays between Iwasawa theory for imaginary quadratic fields at nonsplit primes and relative $p$-adic Hodge theory. In particular, we show that a certain de Rham period $q_{\mathrm{dR}}$ introduced by the author in previous work can be used to construct analytic continuations of ordinary Serre-Tate expansions on the Igusa tower to the infinite-level overconvergent locus. Using this coordinate, one can construct 1-variable measures for Hecke characters and CM-newforms satisfying a new type of interpolation property. Moreover, one can relate the Iwasawa module of elliptic units to these anticyclotomic measures via a new "Coleman map", which is roughly the $q_{\mathrm{dR}}$-expansion of the Coleman power series map. Using this, we formulate and prove a new Rubin-type main conjecture for elliptic units, which is eventually related to Heegner points in order to prove the $p$-converse theorem.Comment: some correction