Openai 2026 modularity elliptic curves imaginary quadratic fields

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theorem_1_1: The manuscript's main claim: every elliptic curve over every imaginary quadratic field K is realized by an isobaric weight-zero automorphic representation of GL_2 over K, cuspidal without CM, with local-global compatibility at every place and L(E/K,s) = L(pi_E,s-1/2). Claims checked only.

OpenAI, Modularity of elliptic curves over imaginary quadratic fields, OpenAI Math Release preprint, October 4, 2026. Released under the Apache License 2.0 at https://github.com/openai/math (revision fd4aeeb2e), folder preprints/Modularity-of-elliptic-curves-over-imaginary-quadratic-fields-October-4-2026; the held PDF, paper.pdf in the release, is retained as openai_2026_modularity_elliptic_curves_imaginary_quadratic_fields.pdf, and the release's TeX bundle in that folder is the TeX source cited on this card. The license term was read in the release's root LICENSE file at that revision on 2026-10-12.

@misc{OAI:Modularity-of-elliptic-curves-over-imaginary-quadratic-fields-October-4-2026,
  author = {{OpenAI}},
  title = {{Modularity of elliptic curves over imaginary quadratic fields}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Modularity-of-elliptic-curves-over-imaginary-quadratic-fields-October-4-2026/paper.pdf}{OAI:Modularity-of-elliptic-curves-over-imaginary-quadratic-fields-October-4-2026}},
  year = {2026}
}

The held PDF has 30 pages with a text layer; its printed page numbers equal the PDF's, its title page is dated October 4, 2026, and its metadata records a creation date of 5 October 2026. The TeX source is the text read here: build/main.tex (89 lines) inputs nine section files under build/sections/ (2,100 lines in all), two of which input TikZ figure files under build/figures/, and the bibliography is the 41-entry build/references.bib.

Attestation. The release README states that the manuscripts were "produced by an internal OpenAI model", that the collection "includes results at different stages of verification", that "Not all have accompanying Lean formalizations" and that "Some of the unformalized results could have issues". Its account of the method says that "The vast majority of results were obtained with the same procedure"; the two exceptions it names concern other subjects (a zero-free region for the Riemann zeta function and the Hodge conjecture for CM abelian varieties), and it does not say per manuscript which procedure produced it. The manuscript's own README carries only the title, the author line "OpenAI", the date and the citation block, and adds no sentence about authorship or assistance; the title page names OpenAI as the sole author, and the TeX source carries no statement on how the text was produced. These are the source's own statements, recorded as attestations and not as this corpus's review. No refereed publication, arXiv version or independent review of the manuscript is recorded here, and nothing on this card is independently reviewed.

Formalization. The release lists no formalization of this manuscript: its catalog lean/formalization.yaml has no entry for it and its family has no page under lean/docs/ at revision fd4aeeb2e.

Companions. The release files this manuscript alone in its family ("Modularity over imaginary quadratic fields" in the release overview), and it holds no earlier dated version of it; there is no companion to read or link.

Read status: claims checked for Theorem 1.1, together with the statements of Corollary 5.4, Definition 6.1, Theorems 6.2--6.4, Proposition 8.4, Proposition 9.3 and Lemma 9.4, read clause by clause in the release's TeX source (build/sections/introduction.tex label thm:main at lines 17--43; systems.tex label sys:congruences at lines 256--267; automorphy-inputs.tex labels def:decomposed-generic at lines 11--21, aut:lift at lines 35--57, aut:seed at lines 65--71 and aut:elliptic-compat at lines 80--93; residual.tex label res:specialization at lines 199--218; automorphy.tex labels aut:over-F at lines 94--98 and aut:descent at lines 153--160). The rest of the TeX was read once, and all proofs were read for their structure only; no step was checked, and the specialized forms of Caraiani and Newton's theorems (Theorems 6.2--6.4) were not compared with the preprint they cite. Nothing here is independently reviewed.

Contents

Theorems, lemmas, definitions and remarks share one counter within each section; the numbers below are the PDF's.

• Section 1, Introduction (pp. 1--3). Describes the automorphy conjecture for an elliptic curve $E$ over an imaginary quadratic field $K$: an automorphic representation of $\mathrm{GL}_2(\mathbb A_K)$ with local parameters matching the cohomology of $E$ at every place, including inertia and monodromy at bad places and the Hodge types $(1,0),(0,1)$ at infinity. States Theorem 1.1 (p. 1). Section 1.1 (p. 2) surveys the background: Bianchi groups (Bianchi, Swan, Harder), Langlands's problems paper, the computations of Grunewald--Helling--Mennicke, Cremona and Cremona--Whitley; Wiles, Taylor--Wiles and Breuil--Conrad--Diamond--Taylor over $\mathbb Q$ and Wiles's $3$--$5$ switch; Galois representations for Bianchi forms (Harris--Soudry--Taylor, Taylor, Berger--Harcos), Harris--Lan--Taylor--Thorne and Scholze; Calegari--Geraghty's patching; potential modularity over CM fields (Allen et al., Theorem 1.0.1) and Boxer--Calegari--Gee--Pilloni; Allen--Khare--Thorne; and Caraiani--Newton, whose results include modularity of every elliptic curve over an imaginary quadratic $K$ for which $X_0(15)(K)$ has rank zero. The manuscript says Theorem 1.1 removes the remaining restrictions and that Caraiani--Newton's lifting theorem and residual criterion "remain essential inputs" (p. 2). Section 1.2 (pp. 2--3) outlines the "geometric prime switch": for non-CM $E$, a large prime $p$ with $\mathrm{SL}_2(\mathbb F_p)$ in the image on $E[p]$, $n=3p$, $F=K(\mu_n)$, a genus-$n$ cyclic cover $C_n(t)$ of a quadratic twist $B_t$ of $E$ with a dihedral involution, and rank-two pieces $V_\theta$ of its cohomology indexed by characters $\theta$ of $\mu_n$, linked by congruences modulo $p$, $3$ and $p$ from $H^1$ of an auxiliary elliptic curve $A$ to $H^1(B_t)$, the prime $5$ serving only to start from $A$. It names antecedents (Tautz--Top--Verberkmoes, Ellenberg, Mestre, Darmon, Darmon--Mestre, Taylor) and says the local-constancy and two-conjugate specialization arguments are stated so they can be used apart from the automorphy application.
• Section 2, Galois and automorphic conventions (p. 4). Étale $H^1$ with $\mathbb Q_l$ coefficients, the cyclotomic character of Hodge--Tate weight $-1$ (so an elliptic curve has weights $0,1$ and determinant $\epsilon_l^{-1}$), bars for semisimplified reductions; $r_\iota(\pi)$ for regular algebraic cuspidal $\pi$ over a CM field in the normalization of Harris--Lan--Taylor--Thorne, Scholze and Caraiani--Newton, with the Tate-normalized reciprocity map $\mathrm{rec}^T(\pi_v)=\mathrm{rec}(\pi_v\otimes|\det|_v^{-1/2})$ and geometric Frobenius; "weight zero"; the complex-place parameter; the shift $L(E/L,s)=L(\pi,s-1/2)$; and the definitions of potentially crystalline and potentially ordinary of weight zero.
• Section 3, Cyclic covers and two degenerations (pp. 4--8). For $E:y^2=h(x)$ over a number field and odd $n\ge3$, the twist $B_t:h(t)y^2=h(x)$ with the point $R_t=(t,1)$ and $S_t=[n]R_t$. Lemma 3.1 (p. 5): for each divisor $m>1$ of $n$, a smooth proper family of genus-$m$ curves, the Kummer cover $Y^m=f_t$ with $\operatorname{div}f_t=(S_t)-(-S_t)-n(R_t)+n(-R_t)$, with an involution inverting the rotations and quotient maps to smaller covers and to $B_t$. Lemma 3.2 (p. 6): regular semistable models at two boundary parameters, with nodes of smoothing order one. Proposition 3.3 (p. 6): at the first boundary point the fiber is of compact type, a rational curve carrying $m$ copies of $E$, so the Jacobian family extends to an abelian scheme with fiber $E^m$ (Bosch--Lütkebohmert--Raynaud). Remark 3.4 (p. 7) confines this to characteristic zero. Lemma 3.5 (p. 7): at the second point the fiber is an unramified degree-$m$ cover of $E$ joined to a rational curve at $m$ nodes, with explicit local node equations. Figure 1 (p. 8) draws the two dual graphs.
• Section 4, Integral homology and explicit monodromy (pp. 8--11). Over $\mathbb C$, a slit model (Lemma 4.1, p. 8) and an integral symplectic basis permuted by the deck group, Proposition 4.2 (p. 9): $H_1$ and $H^1$ of each smooth fiber are free of rank two over the group ring $\mathbb Z[\Gamma_m]$, and étale $H^1$ with $\mathbb Z_\ell$ coefficients is free of rank two over $\mathbb Z_\ell[\Gamma_m]$ for every $\ell$, including $\ell\mid m$. Lemma 4.4 (p. 9) computes the vanishing classes at the second degeneration (Figure 2, p. 10), and Theorem 4.5 (p. 10), by the Picard--Lefschetz formula (SGA 7 II, Exposé XV), gives two monodromy operators in one basis whose off-diagonal entries over the group ring are $\pm(2-u-u^{-1})$.
• Section 5, Rank-two systems and integral congruences (pp. 11--14). For non-CM $E/K$, chooses $p>100$ with $\mathrm{SL}_2(\mathbb F_p)$ in the mod-$p$ image (Serre's open image theorem), sets $n=3p$ and $F=K(\mu_n)$, a CM field abelian over $\mathbb Q$ that does not contain $\zeta_5$, and defines the character spaces $V_{\theta,\ell}$ in $H^1(C_n(t))$. Proposition 5.1 (p. 12): each is semisimple of dimension two with determinant $\epsilon_\ell^{-1}$ and Hodge--Tate weights $\{0,1\}$, potentially semistable, and the family over $\ell$ is a compatible system (Tsuji's comparison, Faltings's semisimplicity). Proposition 5.2 (p. 13): the trivial character gives $H^1(B_t)$, and an order-three character gives $H^1$ of the elliptic curve $A_t=\operatorname{Jac}(C_3(t)/\langle w\rangle)$ over $F$. Lemma 5.3 (p. 13): evaluating the free group-ring module at a character gives a stable lattice whose reduction depends only on the reduced character. Corollary 5.4 (p. 14): for every parameter, with $\chi=\psi\eta$ of order $3p$, the congruences $\overline V_{\eta,p}\simeq\overline V_{\chi,p}$, $\overline V_{\chi,3}\simeq\overline V_{\psi,3}$ and $\overline V_{\psi,p}\simeq\overline V_{1,p}$.
• Section 6, Automorphy inputs (pp. 14--16). Definition 6.1 (p. 14): decomposed genericity, as in Caraiani--Newton's Definition 2.1.27. Theorem 6.2 (p. 15), a specialized form of Caraiani--Newton's Theorem 5.2 (their third arXiv version), the automorphy lifting theorem used three times; Theorem 6.3 (p. 15), their Theorem 6.1 with Lemma 6.1.3, the mod-$5$ starting point for a non-CM elliptic curve over a CM field without $\zeta_5$; Theorem 6.4 (p. 15), their Lemma 6.1.3, local-global compatibility for a modular non-CM elliptic curve. Also cited: Harris--Lan--Taylor--Thorne and Scholze for the Galois representations and Varma's Theorem 1 for local-global compatibility away from $l$. The section lists the conditions still to be arranged at one parameter $t$.
• Section 7, Prescribing the local reduction type (pp. 16--20). Lemma 7.1 (p. 16): finite torsion of an abelian scheme over a local field is locally constant in the parameter. Lemma 7.2 (p. 16): semistable Frobenius polynomials of fixed dimension over a fixed residue field form a finite set and determine the toric rank, and the Newton polygon when that rank is zero. Proposition 7.3 (p. 17): hence the semistable reduction type is locally constant after a fixed finite extension; near a fiber $E_0^g$, nearby fibers are ordinary, have all slopes $1/2$, or are totally toric, as $E_0^g$ is. Lemma 7.4 (p. 18): a potentially totally toric abelian variety of dimension $g$ has monodromy of rank $g$ at every coefficient prime, the residue characteristic included (through Bloch--Kato's crystalline subspace and the valuation matrix of the periods). Proposition 7.5 (p. 19) gives open sets of parameters at the places above $3p$ on which $\operatorname{Jac}(C_n(t))$ has the reduction type of $E^n$, and Corollary 7.6 (p. 20) transfers it to every character piece at once.
• Section 8, One specialization with all residual properties (pp. 20--25). Lemma 8.1 (p. 21): for $(m,\ell)=(3,p),(3,5),(p,3)$ the geometric monodromy on every nontrivial character space mod $\ell$ is absolutely irreducible, contains opposite unipotents, and has one element regular semisimple on all of them, for $E$ and its conjugate. Lemma 8.2 (p. 22): $\overline V_{1,p}$ is absolutely irreducible over $F(\zeta_p)$ and decomposed generic, by a count of trace-$\pm2$ elements in $\mathrm{SL}_2(\mathbb F_p)$ applied to $E$ and its conjugate together. Lemma 8.3 (p. 23): Hilbert irreducibility with weak approximation (Serre's Topics in Galois Theory) for a finite quotient of the fundamental group of an open subset of affine space, with constant field. Proposition 8.4 (p. 23): one $t\in U(K)$, inside prescribed local opens, at which every needed residual representation is absolutely irreducible over the cyclotomic extension and decomposed generic and $A_t$ has no CM; the proof specializes the restriction of scalars $\operatorname{Res}_{K/\mathbb Q}U$ so that both conjugate families are controlled.
• Section 9, Propagation of automorphy and descent (pp. 25--28). Lemma 9.1 (p. 25): automorphy of one member of a compatible system gives all members. Lemma 9.2 (p. 25): the local automorphic conditions at $3$ and $p$ (zero monodromy in the potentially good case; a quadratic or trivial twist of Steinberg, hence ordinary, in the totally toric case), from Varma and Caraiani--Newton's Lemma 6.1.3(4) with Geraghty's Lemma 5.6. Proposition 9.3 (p. 26): $B_t$ is modular over $F$, by the seed Theorem 6.3 and three applications of Theorem 6.2. Lemma 9.4 (p. 27): descent along a cyclic prime-degree extension of CM fields (strong multiplicity one, cyclic base change of Langlands and Arthur--Clozel, a character twist). The section then descends from $F$ to $K$, untwists from $B_t$ to $E$, matches the local parameters, and treats CM curves through Hecke characters and the theory of complex multiplication of Shimura and Taniyama (pp. 27--28).
• References (pp. 28--30): 41 items; the one unpublished item is Caraiani--Newton's preprint, arXiv:2301.10509v3 (27 March 2025), whose third version's numbering the manuscript uses. No release manuscript is cited.

The manuscript has no numerical or computer-assisted component and flags no conditional step; its external inputs are cited at statement level, the Caraiani--Newton theorems in specialized forms the manuscript states itself. The release folder holds only the PDF, the TeX build directory and the README; there is no verification/ folder.

Bears on

The manuscript names no Erdős problem, and no problem page is identified as using it. Where a problem page, one of its claim pages or a library card it links records an elliptic-curve argument, for example for Problems 157, 158, 364, 368, 672, 829, 937, 939, 960 and 976, the curves are defined over $\mathbb Q$, or over a finite field for Problems 157 and 158. For Problems 157 and 158 the curve is the one in Hasse's bound on the linked Cilleruelo card; for Problem 368 the claim page records Pasten's modular approach through Shimura curves; for Problem 672 the Frey-curve arguments are on the linked Bennett--Bruin--Győry--Hajdu and Bennett--Siksek cards; and for Problems 829 and 976 the curves are on the linked Stewart and Cuevas Barrientos--Pasten cards. Over $\mathbb Q$ modularity is already known by Breuil--Conrad--Diamond--Taylor and is not the open point; Problems 157, 937 and 960 are solved. No page records a route through elliptic curves over an imaginary quadratic field; Problem 1214's claim page notes an elliptic-curve analogue over number fields only as not needed for that solved question.