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We prove a $p$-converse theorem for elliptic curves $E/\mathbb{Q}$ withcomplex multiplication by the ring of integers $\mathcal{O}_K$ of an imaginaryquadratic field $K$ in which $p$ is ramified. Namely, letting $r_p =\mathrm{corank}_{\mathbb{Z}_p}\mathrm{Sel}_{p^{\infty}}(E/\mathbb{Q})$, we showthat $r_p \le 1 \implies \mathrm{rank}_{\mathbb{Z}}E(\mathbb{Q}) =\mathrm{ord}_{s = 1}L(E/\mathbb{Q},s) = r_p$ and $\#\mathrm{Sha}(E/\mathbb{Q})< \infty$. In particular, this has applications to two classical Diophantineproblems. 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 withwork of Smith, it resolves the congruent number problem in 100\% of cases andestablishes Goldfeld's conjecture on ranks of quadratic twists for thecongruent number family. The method for showing the above $p$-converse theoremrelies on new interplays between Iwasawa theory for imaginary quadratic fieldsat nonsplit primes and relative $p$-adic Hodge theory. In particular, we showthat a certain de Rham period $q_{\mathrm{dR}}$ can be used to constructanticyclotomic $p$-adic $L$-functions for Hecke characters and newforms,interpolating anticyclotomic twists of positive Hodge-Tate weight in thecentral critical range. Moreover, one can relate the Iwasawa module of ellipticunits to these anticyclotomic $p$-adic $L$-functions via a new "Coleman map",which is, roughly speaking, the $q_{\mathrm{dR}}$-expansion of the Colemanpower series map. Using this, we formulate and prove a new Rubin-type mainconjecture for elliptic units, which is eventually related to Heegner points inorder to prove the $p$-converse theorem.

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