Openai 2026 selmer converse elliptic curves at prime

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corollary_10_1: The claimed Sylvester cube-sum cases: for every prime l = 4, 7, 8 mod 9 the curve X^3 + Y^3 = l Z^3 has analytic and Mordell–Weil rank one and finite Sha; the positive-rank input Walsh's construction for Problem 939 needs.

theorem_1_1: The claimed unrestricted low-corank Selmer converse for elliptic curves over Q at every prime, argued by contradiction from a torsion Heegner point; nothing here is independently reviewed.

OpenAI, The Selmer converse for elliptic curves at every prime, OpenAI Math Release preprint, September 24, 2026. Released under the Apache License 2.0 at https://github.com/openai/math (revision adc7f1241), folder preprints/The-Selmer-converse-for-elliptic-curves-at-every-prime-September-24-2026; the held PDF, main.pdf in the release, is retained as openai_2026_selmer_converse_elliptic_curves_at_prime.pdf, and the release's TeX bundle sits beside main.pdf in that folder.

@misc{OAI:The-Selmer-converse-for-elliptic-curves-at-every-prime-September-24-2026,
  author = {{OpenAI}},
  title = {{The Selmer converse for elliptic curves at every prime}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-Selmer-converse-for-elliptic-curves-at-every-prime-September-24-2026/main.pdf}{OAI:The-Selmer-converse-for-elliptic-curves-at-every-prime-September-24-2026}},
  year = {2026}
}

Attestation, recorded from the release's own statements and not as a review by this corpus: the release README says the manuscripts were "produced by an internal OpenAI model", that the collection "includes results at different stages of verification", that not all have Lean formalizations and that "Some of the unformalized results could have issues". The manuscript's own README adds nothing beyond the title, author "OpenAI", the date and the citation block; the TeX source carries no statement on AI use or human assistance. The title page names no individual author. No refereed publication, arXiv version or independent review of the manuscript is recorded here and nothing on this card is independently reviewed.

The release's Lean catalog (lean/formalization.yaml) lists no formalization for this manuscript, and its family has no page under lean/docs/; nothing here is a formal proof of any statement.

Companions. The release groups this manuscript in the family "The full BSD formula from low Selmer corank", with Exact Birch--Swinnerton-Dyer Formula from Low Selmer Corank (October 3, 2026), which the family description says proves the full leading-term formula under the same corank hypothesis, and The two-primary Birch--Swinnerton-Dyer formula in Selmer corank at most one (September 24, 2026), its $2$-primary version; this manuscript supplies the rank-equality and finiteness statement those formulas build on. Neither companion is held in this library, so neither is linked. The proof also imports two theorems from a third release manuscript, Goldfeld's analytic density conjecture and the $2$-converse for elliptic curves (September 23, 2026), which is not held in this library either.

Read status: claims checked for Theorem 1.1, Theorems 2.1--2.2, Lemma 2.3, Proposition 2.4 and Corollary 10.1, read clause by clause in the TeX source (sections/01-introduction.tex label thm:main, sections/02-reduction.tex labels thm:two-converse-input, thm:twist-density-input, lem:auxiliary-fields, prop:arithmetic-reduction, sections/10-sylvester-cube-sums.tex label cor:sylvester) on 2026-10-07; the statements of the intermediate propositions of Sections 3--9 were read, and all proofs were read for their structure only; no step was checked; nothing here is independently reviewed.

Contents

The manuscript is 82 PDF pages; main.tex inputs ten section files and one appendix file. Throughout, $A_0/\mathbb Q$ is an elliptic curve, $p$ a prime, $s_p(A_0)$ the $\mathbb Z_p$-corank of the full $p$-power Selmer group and $a(A_0)$ the order of vanishing of $L(A_0,s)$ at $s=1$.

• Section 1, Introduction (pp. 2--6): states Theorem 1.1, the converse for every curve and every prime in coranks $0$ and $1$, with finiteness of the whole Tate--Shafarevich group and no hypothesis on reduction, complex multiplication, the residual representation, torsion or isogenies; it says the theorem gives rank equality and finiteness only, not the Birch--Swinnerton-Dyer leading term. It announces the cube-sum application at $p=3$ and says those cases "have also been treated by direct Heegner-point methods" (p. 3) by Yin and by Burungale--Tian (2026 preprints). A survey of the converse problem follows (Skinner--Urban, Skinner, W. Zhang, Burungale--Castella--Skinner, Castella--Grossi--Lee--Skinner, Keller--Yin, Castella--Wan, Castella, Burungale--Skinner--Tian--Wan, Burungale--Castella--Skinner--Tian, Burungale--Tian, Kriz). The strategy paragraph (pp. 4--6, with the roadmap Figure 1 on p. 6): assume the conductor-one Heegner trace $y$ is torsion; build a tame character $\psi$ with values in $R_b=k[t]/(t^b)$ from cyclic $p$-quotients of ring class groups at a sequence of split primes $q_i$, taken as a $p$-adic ultralimit; bound a one-sided Greenberg cohomology module above by $b_*-1$, with $b_*=1$ or $3$ according to whether the Selmer line of $A_0/K$ localizes nontrivially at $p$ (the "strict case" is $b_*=3$); build theta forms on a unitary group of signature $(3,1)$ whose constant terms vanish modulo $t^{b_*}$ because of the torsion assumption; lift them to cusp forms, attach Galois representations and extract extensions giving length at least $b_*$; the two bounds contradict.
• Section 2, Arithmetic reduction and conventions (pp. 6--9): defines the compact Selmer groups $H^1_f(M,V_pA_0)$ and records the two imported results, Theorem 2.1 (the $2$-converse: $2^\infty$-Selmer corank $r\in\{0,1\}$ gives analytic rank, Mordell--Weil rank $r$ and finite Sha) and Theorem 2.2 (quadratic twists of analytic rank $0$ and $1$ each have density $1/2$ among squarefree $d$ ordered by $|d|$), both cited to Theorems 1.1 and 1.2 of the release manuscript on Goldfeld's conjecture; Theorem 2.1 disposes of $p=2$. For odd $p$, Lemma 2.3 chooses coprime negative odd fundamental discriminants $D,D'$ splitting every prime of $2Np$, with $(D/|D'|)=1$ and the twists $A_0^D,A_0^{D'},A_0^{DD'}$ of minimal analytic rank for their sign, so that $K=\mathbb Q(\sqrt D)$, $F=\mathbb Q(\sqrt{DD'})$, $L=K(\sqrt{D'})$ make $L/F$ CM and finitely unramified. Proposition 2.4: $\dim H^1_f(K,V)=\dim H^1_f(K,V^{DD'})=1$, the second Selmer line has nonzero localization at each of the two places of $K$ above $p$, and if the conductor-one Heegner trace $y\in A_0(K)$ is non-torsion then Theorem 1.1 follows by Gross--Zagier and Kolyvagin. The rest of the paper assumes $y$ torsion and derives a contradiction. Inputs named: the $p$-parity theorem (Dokchitser--Dokchitser), modularity (Breuil--Conrad--Diamond--Taylor), Gross--Zagier, Kolyvagin.
• Section 3, A tame deformation and its Selmer length (pp. 9--20): Lemma 3.1 computes the one-sided Greenberg spaces of the base representation and isolates the strict case ($r=1$, Selmer line spanned by a conjugation-invariant, everywhere locally trivial class). Proposition 3.2 constructs the primes $q_i$ and characters $\ell_i:G_K\to\mathbb Z/p^{n_i}$ by Chebotarev, with a limiting Frobenius $\gamma$ satisfying $\det(V(\gamma)-1)\neq0$. Definition 3.3 and Lemma 3.4 set up admissible cochains (coefficientwise ultralimits over a nonprincipal ultrafilter on the prime indices, with one common denominator). Lemmas 3.5--3.6 give the first-order obstruction via a Heisenberg commutator; Proposition 3.7 is the upper bound: the Greenberg module over $R_b$ has length $0$ outside the strict case and at most $2$ in it. Lemma 3.8 is a uniform local logarithm estimate; Proposition 3.9 shows the weighted Heegner logarithm sums vanish modulo $t^{b_*}$ under the torsion assumption.
• Section 4, The holomorphic theta family (pp. 20--29): theta lifts from the definite unitary group $U(A)$ carrying the Jacquet--Langlands transfer of the weight-two form of $A_0$ (base-changed to $F$) to $U(3,1)$ at both real places, with character $\eta=\mathcal C^m\nu_i$; Lemma 4.1 computes the two parabolic operators $U_2,U_3$ (eigenvalue $p^2$); Lemma 4.2 the spherical parameters; Proposition 4.3 records the family; Lemma 4.5 (uniform unary moments) and Lemma 4.6 (integrality of mixed Fourier--Jacobi expansions) give the uniform denominator bound, using Katz's expansion principle, Andreatta--Goren and Hida--Tilouine.
• Section 5, The constant terms and a square-root period comparison (pp. 29--38): Proposition 5.1, every constant Fourier--Jacobi coefficient tends to zero modulo $t^{b_*}$; proved through Lemma 5.2, a cleared identity $A_iS_i^2=B_iP_{0,i}^2P_{1,i}^2$ relating the binary theta scalar to two raw toric periods, by the Rallis inner product formula (via the anisotropic Siegel--Weil identity of Yamana and Gan--Qiu--Takeda) and Waldspurger's formula, with the local gamma factors matched; the smallness of $P_{0,i}$ comes from the Heegner logarithm of Proposition 3.9 through the Bertolini--Darmon--Prasanna method extended to bad reduction by Berkovich--Coleman integration (Katz--Rabinoff--Zureick-Brown).
• Section 6, A positive Fourier--Jacobi coefficient (pp. 38--45): Lemmas 6.1--6.4 and Proposition 6.5 exhibit a nonconstant coefficient of bounded valuation, so that the theta forms do not vanish; the key nonvanishing input is the characteristic-zero nonvanishing theorem (Theorem 1.1 and Corollary 4.20) of Burungale--He--Tian--Ye (arXiv 2508.19706v2); the manuscript states that only characteristic-zero nonvanishing is used (PDF p. 42). It is applied together with the line-period criterion of Borade et al. (2025).
• Section 7, Cusp lifting on the ordinary locus (pp. 45--53): on the ordinary locus of the target PEL variety (Lan's compactifications, Katz's Serre--Tate theory), Lemmas 7.1--7.5 lift the theta sections to genuine cusp forms of a Hasse-shifted weight modulo $p^{M_i}$ and Proposition 7.6 gives a homomorphism from the cusp Hecke order to $(\mathcal O/p^{M_i})[t]/(t^{b_*})$ sending every operator to its theta value.
• Section 8, Characteristic-zero parameters and local control (pp. 53--61): Lemmas 8.1--8.2 and Proposition 8.3 transfer the cusp forms to $\mathrm{GL}_4(\mathbb A_L)$ by Labesse's CM base change, with the sign property checked by pseudocoefficients (Clozel--Delorme, Salamanca-Riba, Adams--Johnson); the text says this argument "does not use the weighted fundamental lemma" (p. 54). Theorem 8.4 is the imported Galois input (Barnet-Lamb--Gee--Geraghty--Taylor, Chenevier--Harris, Barnet-Lamb--Geraghty--Harris--Taylor); Proposition 8.5 attaches four-dimensional representations; Lemmas 8.6--8.8 give the local flags at $p$ and the inertia control away from $p$; Proposition 8.9 removes nilpotents in the Hecke order.
• Section 9, Extraction of the extension space (pp. 61--68): a finite limiting matrix algebra over the ultraproduct of the Hecke orders (Lemma 9.1 uses Shirshov and Amitsur--Levitzki for a uniform generator count), corner modules in the style of Bellaïche--Chenevier, Proposition 9.4 eliminating the cyclotomic corner, Lemma 9.5 and Corollary 9.6 giving a vanishing Fitting ideal, and Proposition 9.7, the lower bound: an injection of a module of $R_{b_*}$-length at least $b_*$ into the Greenberg module. The closing paragraph (p. 68) combines Propositions 3.7 and 9.7 to conclude Theorem 1.1.
• Section 10, Sylvester's positive prime cube-sum cases (pp. 68--70): Corollary 10.1, for every prime $\ell\equiv4,7,8\pmod 9$ the curve $E_\ell:X^3+Y^3=\ell Z^3$ has analytic and Mordell--Weil rank one and finite Sha, so $\ell$ is a sum of two rational cubes; deduced from Theorem 1.1 at $p=3$ by bounding $s_3(E_\ell)\le1$ through Satgé's paired $3$-isogeny descent (1987) with the Sha terms retained, and the root number $-1$ from Dasgupta--Voight (2018). The section credits Dasgupta--Voight, Yin (2026), Burungale--Tian (2026) and Kriz (2022) with earlier treatments of these cases.
• Appendix A, Measure comparison and change of characteristic (pp. 70--75): Lemma A.1 compares normalized global torus factors over function fields and Lemma A.2 transfers a nonstandard weighted fundamental lemma from large positive characteristic to characteristic zero by the Cluckers--Hales--Loeser transfer principle. It is written against an undated author-hosted draft (Halleck-Dubé, The Weighted Fundamental Lemma for Non-Split Groups, cited by section and theorem numbers) and Ngô's fundamental lemma paper. The introduction calls the appendix "a separate normalized global measure comparison" (p. 6); the text does not say whether any step of the main proof depends on it, and Section 8 says the transfer it uses avoids the weighted fundamental lemma.
• Bibliography (pp. 76--82): 80 printed entries, [1]--[80]; the 2026 items cited are the release manuscript on Goldfeld's conjecture, the preprints of Yin, of Burungale--Tian and of Burungale--He--Tian--Ye, and Burungale--Tian's Annals paper on the rank-zero converse for CM curves; two undated author-hosted items, the Halleck-Dubé draft and the Castella--Liu--Wan addendum, were both accessed. The TeX references.bib holds 96 entries, with locator notes naming the versions consulted, 16 of them uncited, among them an Atobe--Gan--Ichino--Kaletha--Mínguez--Shin preprint that the PDF does not print.

Conditional and external components flagged here: Theorems 2.1 and 2.2 are imported from another release manuscript that is itself unreviewed; the nonvanishing input of Section 6 and the Galois input of Theorem 8.4 are cited at statement level; the appendix rests on an unpublished draft. The manuscript contains no numerical or computer-assisted component and the release folder holds no verification/ directory (only build, main.pdf and README.md). The manuscript names no Erdős problem.

Bears on

• Problem 939: claimed input for the third part only. Corollary 10.1 would give $\operatorname{rank}E(\mathbb Q)>0$ for $E:Y^2=X^3-432\ell^2$ (the Weierstrass model of $X^3+Y^3=\ell Z^3$) for every prime $\ell\equiv4,7,8\pmod 9$, which is the hypothesis of Walsh's Theorem 1.1 and would make that construction yield infinitely many pairwise coprime $3$-powerful $a+b=c$ with $c=\ell^4z^3$ for each such prime. That part is already answered yes by Nitaj and Cohn, so the manuscript adds a further source of solutions and nothing on the open cases $r=4$ and $r=5$. The manuscript itself credits Yin and Burungale--Tian with prior proofs of the same cases. The claim is unverified here; the page's status rests on its acceptance evidence.
• Walsh 2024: Corollary 10.1 is a claimed proof of the positive-rank hypothesis of Walsh's Theorem 1.1 for every odd prime $p\equiv4,7,8\pmod 9$, and addresses the card's remark that a Selmer computation suggests rank one there; unverified here, and the same hypothesis is also claimed by the 2026 preprints of Yin and of Burungale--Tian that the manuscript cites.