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

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 $\geq 2$.

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 $\geq 2$.