The p-converse to Gross-Zagier-Kolyvagin in Selmer corank at most one, at every prime

For every elliptic curve $E/\mathbb{Q}$, every prime $p$ and $r\in\{0,1\}$: $s_p(E)=r$ implies $\mathrm{ord}_{s=1}L(E,s)=\mathrm{rank}\,E(\mathbb{Q})=r$ and the whole of $Sha(E/\mathbb{Q})$ is finite, with no restriction on reduction type, CM, residual representation, torsion or isogenies. It does not address coranks $\geq 2$ and does not evaluate the BSD leading term (that is the companion entry). As an application it rederives that $X^3+Y^3=\ell Z^3$ has rank one for primes $\ell\equiv 4,7,8 \pmod 9$.

For every elliptic curve $E/\mathbb{Q}$, every prime $p$ and $r\in\{0,1\}$: $s_p(E)=r$ implies $\mathrm{ord}_{s=1}L(E,s)=\mathrm{rank}\,E(\mathbb{Q})=r$ and the whole of $Sha(E/\mathbb{Q})$ is finite, with no restriction on reduction type, CM, residual representation, torsion or isogenies. It does not address coranks $\geq 2$ and does not evaluate the BSD leading term (that is the companion entry). As an application it rederives that $X^3+Y^3=\ell Z^3$ has rank one for primes $\ell\equiv 4,7,8 \pmod 9$.