The Birch-Swinnerton-Dyer formula in Selmer corank at most one

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Clay Mathematics Institute: Birch and Swinnerton-Dyer conjecture evidence for this event

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

The Birch-Swinnerton-Dyer formula in Selmer corank at most one 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

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

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

Revised edition (October 6, 2026): The two-primary Birch-Swinnerton-Dyer formula in Selmer corank at most one evidence for this event

For every $E/\mathbb{Q}$ and prime $q$ with $s_q(E)=\mathrm{corank}\,\mathrm{Sel}_{q^\infty}(E/\mathbb{Q})\in\{0,1\}$: $\mathrm{rank}\,E(\mathbb{Q})=\mathrm{ord}_{s=1}L(E,s)=s_q(E)$, $Sha(E/\mathbb{Q})$ is finite, and the full BSD leading-term formula holds at every prime, with no hypotheses on reduction, torsion, isogenies, CM or residual representations. By Gross-Zagier-Kolyvagin this covers every curve of analytic rank at most one (curator's reading). The new step is the exact $p$-adic valuation at every odd $p$; $p=2$ comes from the two-primary companion. With family 006 it gives full BSD for a density-one set of quadratic twists of each curve. It says nothing about analytic rank $\geq 2$. parent of this event

Companion: The two-primary 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