Parity conjecture for elliptic curves

Parity conjecture for elliptic curves

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

• OpenAI, Exact Birch–Swinnerton-Dyer Formula from Low Selmer Corank, 2026-10-03; citation key OAI:Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-3-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main, eq:bsd-main.
• OpenAI, The Selmer converse for elliptic curves at every prime, 2026-09-24; citation key OAI:The-Selmer-converse-for-elliptic-curves-at-every-prime-September-24-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main.
• OpenAI, The two-primary Birch–Swinnerton-Dyer formula in Selmer corank at most one, 2026-09-24; citation key OAI:The-two-primary-Birch-Swinnerton-Dyer-formula-in-Selmer-corank-at-most-one-September-24-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main.
• OpenAI, A pointwise 2-converse for elliptic curves with rational two-torsion, 2026-09-24; citation key OAI:A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main.
• OpenAI, Goldfeld's analytic density conjecture and the 2-converse for elliptic curves, 2026-09-23; citation key OAI:Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-September-23-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main-converse.
• OpenAI catalogue entry.

Parity conjecture for elliptic curves

Definitions and mathematical statement

By the Mordell--Weil rank of $E/K$ mean the integer $r$ in $E(K)\cong E(K)_{\mathrm{tors}}\oplus{\mathbb{Z}}^r$. The global root number is $w(E/K)=\prod_v w(E/K_v)\in\{\pm1\}$, the product of normalized local epsilon-factor signs; almost every finite-place factor is one and each archimedean factor is $-1$. The target is \[ w(E/K)=(-1)^r \] for every elliptic curve over every number field. Root numbers are defined locally without presuming a global analytic continuation theorem. The claim concerns the algebraic rank, not merely parity of a conjectural analytic order of vanishing or a Selmer-group rank.

Formulation and concept references: [S1].

Short English statement

Does the product of an elliptic curve's local root-number signs always determine whether its rational-point rank over the number field is even or odd?

Research attempt

Current frontier.

The formulation source proves $p$-parity when $E$ has a $p$-isogeny, including an odd-degree Galois extension variant; its Corollary 1.5 adds finiteness of the $p$-primary Tate--Shafarevich group to deduce Mordell--Weil parity [S1], Theorems 1.4--1.4$'$ and Corollary 1.5. It also proves the full parity assertion when complex multiplication is defined over the ground field. Those precise statements must not be converted into unconditional Mordell--Weil parity for every $E/K$. The attempt targets the divisible Tate--Shafarevich contribution, which a superficial use of the Cassels--Tate pairing can miss.

Attack route.

One route proves finiteness of the Tate--Shafarevich group, which is stronger than needed. Another proves just evenness of its divisible $p$-corank. A third combines quadratic extension and twist identities to propagate signs, but a product of two unknown parities does not determine either individually. We pursue the weaker evenness requirement and test a proposed pairing proof. The result is an exact reduction on the known $p$-parity locus and an explicit abstract model refuting the pairing shortcut.

Attempt.

Fix a prime $p$. Write \[ r=\operatorname{rank}E(K),\quad s_p=\operatorname{corank}_{{\mathbb{Z}}_p} \operatorname{Sel}_{p^\infty}(E/K),\quad d_p=\operatorname{corank}_{{\mathbb{Z}}_p}\operatorname{Sha}(E/K)[p^\infty]. \] The Kummer exact sequence is \begin{equation}\label{wsref-0-65-71-31-39-36-3a-73-65-6c-6d-65-72} 0\longrightarrow E(K)\otimes{\mathbb{Q}}_p/{\mathbb{Z}}_p \longrightarrow\operatorname{Sel}_{p^\infty}(E/K) \longrightarrow\operatorname{Sha}(E/K)[p^\infty]\longrightarrow0. \end{equation} These cofinitely generated groups have additive corank, hence $s_p=r+d_p$; this is the source's distinction between the two ranks [S1], equations $4$--$5$.

On any locus where the established $p$-parity theorem gives $w(E/K)=(-1)^{s_p}$, \begin{equation}\label{wsref-0-65-71-31-39-36-3a-64-65-66-65-63-74} w(E/K)(-1)^r=(-1)^{d_p}. \end{equation} Thus full parity on that locus is equivalent to evenness of $d_p$, not to finiteness of the whole group. In particular $d_p=0$ is sufficient but stronger. No equality between algebraic and analytic rank has been used, so the root number retains the local definition in the target.

The proposed proof is to use the Cassels--Tate pairing. Its radical on the relevant primary part is the maximal divisible subgroup; on the finite quotient it gives the expected alternating duality for an elliptic curve. The source explicitly separates that finite quotient from the divisible contribution [S1], discussion preceding Theorem 1.10. It therefore proves an evenness statement for the finite paired part, not automatically for $d_p$.

Abstract countermodel to the shortcut. For any prime $p$ and integer $a\ge1$, let \[ A=({\mathbb{Q}}_p/{\mathbb{Z}}_p)\oplus({\mathbb{Z}}/p^a{\mathbb{Z}})^2. \] Define a bilinear pairing into ${\mathbb{Q}}/{\mathbb{Z}}$ by \[ \langle(t,x,y),(t',x',y')\rangle=\frac{xy'-yx'}{p^a}\pmod{\mathbb{Z}}. \] It is alternating, its radical is exactly the first, divisible summand, and its quotient pairing is perfect. The finite quotient has square order $p^{2a}$ and even $p$-rank, while the divisible corank is one. Every abstract property just cited is compatible with odd $d_p$. This is not claimed to be the Tate--Shafarevich group of an elliptic curve; it disproves only the logical inference from those pairing properties.

What additional structure would suffice? Let \[ D_p=\operatorname{Sha}(E/K)[p^\infty]_{\rm div},\qquad V_p(D_p)=\left(\varprojlim_m D_p[p^m]\right)\otimes_{{\mathbb{Z}}_p}{\mathbb{Q}}_p, \] where transition maps are multiplication by $p$. Since $D_p\simeq({\mathbb{Q}}_p/{\mathbb{Z}}_p)^{d_p}$, the vector space $V_p(D_p)$ has dimension $d_p$. A nondegenerate alternating ${\mathbb{Q}}_p$-bilinear form on it would force $d_p$ even. For odd $d_p$, its matrix $B$ would satisfy \[ \det B=\det B^{\mathsf T}=\det(-B)=(-1)^{d_p}\det B=-\det B, \] so $\det B=0$ in the characteristic-zero field ${\mathbb{Q}}_p$, including $p=2$. This is a sufficient missing pairing lemma. The existing Cassels--Tate pairing cannot supply it by restriction: it vanishes identically on $D_p$. A new derived or height-theoretic construction would need both a definition on this vector space and a proof of nondegeneracy.

Even that hypothetical lemma would settle the general target only after $p$-parity is available for a suitable prime for every $E/K$. We have not established that global coverage. Within the $p$-isogeny cases covered by the source, however, the single even-corank statement would suffice by (eq196:defect).

Stress test.

Finite-level groups can have even-dimensional nondegenerate quotients while their compatible divisible kernels have odd corank; the countermodel shows this without a limiting ambiguity. Therefore taking an inverse limit of quotient pairings and declaring it nondegenerate is unjustified. Nor does a product of local root numbers supply a pairing on $D_p$ by formal analogy. For curves with ground-field complex multiplication the source gives additional arithmetic structure and actual parity; that special input cannot be extended to a general elliptic curve by ignoring endomorphism hypotheses. The finite alternating-matrix checks in the script test the countermodel, not existence of a curve with odd $d_p$.

Outcome.

Outcome: Conditional reduction. On the established $p$-parity locus, the exact remaining defect is $(-1)^{d_p}$. A nondegenerate alternating form on the divisible Tate module would remove it, but the usual pairing does not provide such a form, and a concrete abstract example explains why. No finiteness of $\operatorname{Sha}$, general even-corank theorem, or unconditional parity theorem beyond the cited cases is proved.

Sources

• S1 Kęstutis Česnavičius, The p-parity conjecture for elliptic curves with a p-isogeny; arXiv1207.0431v3 April8,2014; PDF internal date May22,2018. https://arxiv.org/pdf/1207.0431. Catalogue formulation source.