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Math-23 · mathematics · Number theory

Birch & Swinnerton-Dyer conjecture

Status Open · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Let $K$ be a number field, a finite extension of $\mathbb Q$, and let $E/K$ be an elliptic curve: a smooth projective genus-one curve with a specified $K$-rational point. Its rational points form a finitely generated abelian group $E(K)\cong\mathbb Z^r\oplus T$, with $T$ finite; $r$ is the Mordell--Weil rank.

For a finite place $v$ of good reduction, let $q_v$ be the size of its residue field and put $a_v=q_v+1-\#E(\mathbb F_{q_v})$. The Hasse--Weil $L$-function, initially defined in a right half-plane, is \[ L(E/K,s)=\prod_{v\text{ good}} (1-a_vq_v^{-s}+q_v^{1-2s})^{-1} \prod_{v\text{ bad}}(1-a_vq_v^{-s})^{-1}. \] At bad places, $a_v$ is $1$ for split multiplicative reduction, $-1$ for nonsplit multiplicative reduction, and $0$ for additive reduction. The rank form of the conjecture asserts analytic continuation to a neighborhood of $s=1$ and \[ \operatorname{ord}_{s=1}L(E/K,s)=r. \] The order is the integer $m\ge0$ for which $L(E/K,s)=(s-1)^m g(s)$ with $g$ holomorphic and $g(1)\ne0$. The quantifier covers every elliptic curve over every number field. This record concerns the rank equality; the stronger formula for the leading Taylor coefficient is not included in its source statement.

Short English statement

For every elliptic curve over a number field, does its number of independent rational points equal the order of vanishing of its L-function at one?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-23-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-23

The arithmetic conclusions require a rational elliptic curve with Selmer corank zero or one at the specified prime; the rational-two-torsion companion has that additional hypothesis. Rank equality, finite Tate--Shafarevich group, or the stated leading-coefficient identities in those cases do not extend to all ranks and number fields in this problem. Average conclusions for quadratic twists also leave individual exceptional curves untreated.

Birch & Swinnerton-Dyer conjecture

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; In manuscript 2 (family 002; 2026-09-24), the asserted result is that for any prime, a rational elliptic curve of Selmer corank zero or one has the corresponding analytic and rational-point ranks, together with finite Tate--Shafarevich group; the application includes primes congruent to $4$, $7$, or $8$ modulo $9$ as sums of two rational cubes; In manuscript 3 (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; In manuscript 4 (family 004; 2026-09-24), the asserted result is that a rational elliptic curve with a nonidentity rational point of order two satisfies the low-corank two-adic Selmer converse even at arbitrary reduction at two; In manuscript 5 (family 006; 2026-09-23), the asserted result is that among signed squarefree twists of each rational elliptic curve, analytic ranks zero and one each occur with density one half; the paper also claims the low-corank two-Selmer converse.

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.
Manuscript 2.
For every elliptic curve $E/\mathbb Q$, every prime $p$ and $r\in\{0,1\}$, let $s_p(E)$ be the $\mathbb Z_p$-corank of its $p^{\infty}$ Selmer group. The claimed converse is \[ s_p(E)=r\ \Longrightarrow\ \operatorname{ord}_{s=1}L(E,s)=\operatorname{rank}E(\mathbb Q)=r, \qquad\#\operatorname{Sha}(E/\mathbb Q)<\infty. \] No additional reduction, CM, residual-representation, rational-torsion or isogeny hypothesis is imposed. Coranks greater than one and an exact BSD leading coefficient are not concluded by this theorem.
Manuscript 3.
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.
Manuscript 4.
For an elliptic curve $E/\mathbb Q$, put $s_2(E)=\operatorname{corank}_{\mathbb Z_2}\operatorname{Sel}_{2^{\infty}}(E/\mathbb Q)$. If $E(\mathbb Q)[2]\ne0$ and $s_2(E)\in\{0,1\}$, the manuscript claims \[ \operatorname{ord}_{s=1}L(E,s)=\operatorname{rank}_{\mathbb Z}E(\mathbb Q)=s_2(E),\qquad \#\operatorname{Sha}(E/\mathbb Q)<\infty. \] Both rational two-torsion and the low-corank condition belong to this theorem; it does not cover arbitrary elliptic curves or arbitrary Selmer corank.
Manuscript 5.
For $E/\mathbb Q$, write $a(E)=\operatorname{ord}_{s=1}L(E,s)$, $r(E)=\operatorname{rank}E(\mathbb Q)$ and $c_2(E)=\operatorname{corank}_{\mathbb Z_2}\operatorname{Sel}_{2^{\infty}}(E/\mathbb Q)$. The first claimed theorem is \[ c_2(E)\in\{0,1\}\Rightarrow a(E)=r(E)=c_2(E),\quad\#\operatorname{Sha}(E/\mathbb Q)<\infty. \] Let $\mathcal D(X)$ be the nonzero signed squarefree integers with $|d|\le X$ and $E^{(d)}$ the quadratic twist. The analytic Goldfeld assertion is, for each $j=0,1$, \[ \lim_{X\to\infty}\frac{|\{d\in\mathcal D(X):a(E^{(d)})=j\}|}{|\mathcal D(X)|}=\tfrac12. \] It further claims that a density-one set of twists has equal analytic and Mordell--Weil rank and finite whole Tate--Shafarevich group. This density statement does not assert those properties for every twist.

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. For manuscript 2 (family 002; 2026-09-24), a tame deformation gives an upper bound on an Artin-module length. Integral theta-series coefficients and Heegner logarithms produce an incompatible lower bound unless the low-corank converse holds; auxiliary quadratic fields separate the two root-number cases.

For manuscript 3 (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. For manuscript 4 (family 004; 2026-09-24), binary coefficient systems detect central values or Heegner points. Integral interpolation and arithmetic contraction identities isolate one address in a network of twists, transferring a nonvanishing witness back to the original curve.

For manuscript 5 (family 006; 2026-09-23), finite cubes of twists preserve the required local classes. A central determinant or Heegner coordinate is nonzero at the noncentral cube vertices, and bounded integral interpolation recovers the missing vertex; the Selmer distribution then supplies the density assertion.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 002; 2026-10-03), manuscript 2 (family 002; 2026-09-24), manuscript 3 (family 002; 2026-09-24), manuscript 4 (family 004; 2026-09-24), manuscript 5 (family 006; 2026-09-23).

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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