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

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

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 $\ell\equiv 4,7,8 \pmod 9$. parent of this event

VibeMathed record: The p-converse to Gross-Zagier-Kolyvagin in Selmer corank at most one, at every prime evidence for this event

Revised edition (October 7, 2026): The Selmer converse for elliptic curves at every prime evidence for this event

Input (family 006): Goldfeld's analytic density conjecture and the 2-converse evidence for this event

Earlier edition (October 3, 2026) evidence for this event

Keller and Yin, p-converse theorems (Conjecture A), arXiv 2410.23241 evidence for this event

Goldfeld's conjecture entry, an input to this result evidence for this event

The Birch-Swinnerton-Dyer formula in Selmer corank at most one evidence for this event

Revised edition (October 7, 2026): Goldfeld's analytic density conjecture and the 2-converse for elliptic curves evidence for this event

Companion: The Selmer converse for elliptic curves at every prime evidence for this event

This event attributed to Unreleased internal OpenAI model