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Math-2805 · mathematics · Number theory

The two-primary BSD formula at low two-Selmer corank

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every elliptic curve E over Q with full 2-power Selmer $Z_2-corank$ at most one, must analytic and Mordell-Weil ranks agree, must whole Sha be finite, and must the BSD-normalized leading L-value be a positive rational number whose 2-adic valuation equals that of the Sha order? No rational two-torsion or reduction-type hypothesis is imposed.

Short English statement

For every elliptic curve E over Q with full 2-power Selmer $Z_2-corank$ at most one, must analytic and Mordell-Weil ranks agree, must whole Sha be finite, and must the BSD-normalized leading L-value be a positive rational number whose 2-adic valuation equals that of the Sha order? No rational two-torsion or reduction-type hypothesis is imposed.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2805-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2805

The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 002; 2026-09-24).

The two-primary BSD formula at low two-Selmer corank

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 002; 2026-09-24), the asserted result is that the two-adic part of the BSD leading-coefficient identity holds for rational elliptic curves of two-power Selmer corank at most one, and for almost every quadratic twist.

Mathematical statement of the claimed result

Manuscript 1.
For any elliptic curve $E/\mathbb Q$, let $s_2(E)=\operatorname{corank}_{\mathbb Z_2}\operatorname{Sel}_{2^{\infty}}(E/\mathbb Q)$ and assume $s_2(E)\le1$. The claim is \[ r=\operatorname{rank}E(\mathbb Q)=\operatorname{ord}_{s=1}L(E,s)=s_2(E), \qquad |\operatorname{Sha}(E/\mathbb Q)|<\infty. \] With $T_E=E(\mathbb Q)_{\rm tors}$, real Néron period $\Omega_E$, Néron--Tate regulator $\operatorname{Reg}_E$ and finite-prime Tamagawa numbers $c_{\ell}(E)$, it further claims \[ Q_E=\frac{L^{(r)}(E,1)|T_E|^2}{r!\Omega_E\operatorname{Reg}_E\prod_{\ell} c_{\ell}(E)} \in\mathbb Q_{>0},\qquad v_2(Q_E)=v_2(|\operatorname{Sha}(E/\mathbb Q)|). \] There is no additional restriction on $E[2]$ or reduction at two. The formula determines the two-primary valuation; it does not state equality $Q_E=|\operatorname{Sha}|$ at all primes or address larger Selmer corank.

Outline of the argument

For manuscript 1 (family 002; 2026-09-24), the proof compares integral determinant lattices with Siegel-unit periods and central Kummer lattices. Separate residual and character-division arguments establish primitivity at two, and the rank-one comparison uses a squared Heegner logarithm rather than only rational period identities.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 002; 2026-09-24).

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

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