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Math-2777 · mathematics · Number theory

The exact BSD formula at low 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, if its p-primary full Selmer group has $Z_p$ corank zero or one for at least one prime p, must analytic rank equal Mordell-Weil rank, must the entire Tate-Shafarevich group be finite, and must the leading coefficient of L(E,s) at $s=1$ equal the BSD expression using the Neron period, regulator, full Sha order, finite Tamagawa factors and rational torsion order?

Short English statement

For every elliptic curve E over Q, if its p-primary full Selmer group has $Z_p$ corank zero or one for at least one prime p, must analytic rank equal Mordell-Weil rank, must the entire Tate-Shafarevich group be finite, and must the leading coefficient of L(E,s) at $s=1$ equal the BSD expression using the Neron period, regulator, full Sha order, finite Tamagawa factors and rational torsion order?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2777-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2777

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; 2026-10-03).

The exact BSD formula at low 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; 2026-10-03), the asserted result is that a rational elliptic curve satisfying a Selmer-corank condition of zero or one at one prime satisfies the exact BSD leading-coefficient identity, with every primary factor included.

Mathematical statement of the claimed result

Manuscript 1.
For an elliptic curve $E/\mathbb Q$ and any prime $q$, write $s_q(E)=\operatorname{corank}_{\mathbb Z_q}\operatorname{Sel}_{q^{\infty}}(E/\mathbb Q)$, $r(E)=\operatorname{rank}E(\mathbb Q)$ and $a(E)=\operatorname{ord}_{s=1}L(E,s)$. If $s_q(E)\in\{0,1\}$, the manuscript claims \[ r(E)=a(E)=s_q(E),\qquad\#\operatorname{Sha}(E/\mathbb Q)<\infty, \] and, with $r=r(E)$, \[ \frac{L^{(r)}(E,1)}{r!} =\frac{\Omega_E\operatorname{Reg}_E\,\#\operatorname{Sha}(E/\mathbb Q)\prod_{\ell\text{ finite}}c_{\ell}(E)} {(\#E(\mathbb Q)_{\rm tors})^2}. \] Here $\Omega_E$ is the real Néron period, $\operatorname{Reg}_E$ the Néron--Tate regulator (one in rank zero) and $c_{\ell}$ the Tamagawa numbers. No reduction, torsion, CM or residual-representation conditions are added. The low-Selmer-corank premise remains essential; this is not a claim of unrestricted BSD.

Outline of the argument

For manuscript 1 (family 002; 2026-10-03), the argument tracks the discrepancy between analytic periods and arithmetic determinant volumes in integral cohomology. Residual obstruction tests control vertical prime factors, while character specialization and theta-series comparisons remove horizontal factors; Heegner heights treat the rank-one case.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 002; 2026-10-03).

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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