Modularity of Elliptic Curves over Number Fields (End_K(E)=Z)

Modularity of Elliptic Curves over Number Fields (End_K(E)=Z)

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 030; 2026-10-04), the asserted result is that all elliptic curves over imaginary quadratic fields are modular with the stated local-parameter compatibility at every place.

Mathematical statement of the claimed result

Manuscript 1.

For every imaginary quadratic number field $K$ and elliptic curve $E/K$, the manuscript claims an isobaric automorphic $\pi_E$ on $\operatorname{GL}_2(\mathbb A_K)$ of trivial central character and weight zero with, for every prime $\ell$ and $\iota:\overline{\mathbb Q}_{\ell}\simeq\mathbb C$, \[ r_{\iota}(\pi_E)\simeq H^1_{\rm et}(E_{\overline K},\overline{\mathbb Q}_{\ell}),\qquad L(E/K,s)=L(\pi_E,s-1/2). \] For every finite $v\nmid\ell$, the Frobenius-semisimplified Weil--Deligne representation is claimed to equal local Langlands for $\pi_{E,v}\otimes|\det|_v^{-1/2}$. The complex-place unitary parameter is $\operatorname{diag}(z/|z|,\bar z/|z|)$. When $E$ has no CM the asserted representation is cuspidal. For CM curves it is induced automorphically, with a sum of characters instead when $K$ already contains the CM field. All local factors, including bad-reduction factors, are included.

Outline of the argument

For manuscript 1 (family 030; 2026-10-04), cyclic covers have one degeneration whose Jacobian splits into copies of the target curve and another that makes monodromy explicit. A geometric switch of coefficient primes and three automorphic lifts propagate automorphy through the family before descent to the original field.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 030; 2026-10-04).

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, Modularity of elliptic curves over imaginary quadratic fields, 2026-10-04; citation key OAI:Modularity-of-elliptic-curves-over-imaginary-quadratic-fields-October-4-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main.
• OpenAI catalogue entry.

Modularity of Elliptic Curves over Number Fields (End_K(E)=Z)

Definitions and mathematical statement

For $E/K$ with $\operatorname{End}_K(E)={\mathbb{Z}}$, let $(r_v,N_v)$ be its finite-place Weil--Deligne representations. Use Thorne's geometric-Frobenius convention to form \[ \pi(E)=\bigotimes_{v\nmid\infty}' \operatorname{rec}_{K_v}^{-1}(r_v\otimes|\cdot|^{1/2},N_v),\qquad A=\varinjlim_U H^*(Y_U,{\mathbb{Q}}). \] Here $Y_U$ are the adelic locally symmetric spaces in the source, and the transition maps give the finite-adelic group action. The assertion is the existence of a $\operatorname{GL}_2({\mathbb{A}}_K^\infty)$-equivariant injection $\pi(E)\hookrightarrow A\otimes_{{\mathbb{Q}}}{\mathbb{C}}$. The condition concerns endomorphisms over $K$; replacing it by $\operatorname{End}_{\bar K}(E)={\mathbb{Z}}$ would narrow the target.

Formulation and concept references: [S1].

Short English statement

Does every elliptic curve in the specified number-field class occur in the automorphic cohomology predicted by Thorne's modularity formulation?

Research attempt

Current frontier.

The incorporated formulation was read in Thorne, pp. 646--650, especially Definition 2.3 and Conjecture 2.6. It is a statement about an injection of the entire finite-adelic representation, not just equality of good-prime traces; Thorne explicitly distinguishes it from his Conjecture 2.5. Geometric Frobenius and the twist $|\cdot|^{1/2}$ are retained. His Section 4 explains residual modularity and modularity lifting. [S1]. Caraiani--Newton establish modularity over infinitely many imaginary quadratic fields, including ${\mathbb{Q}}(\sqrt{-d})$ for $d=1,2,3,5$, under their precise $X_0(15)$ criterion; their more general CM-field results retain image hypotheses. This does not quantify over every number field. [E1]. Dummigan--Tiwari's August 2026 version proves an implication from arithmetic hyperbolic uniformization to geometric modularity for totally real fields of odd degree, not existence of that uniformization for every curve. [S4], [E2]. These source statements were inspected; their complete proofs were not independently reverified here.

Attack route.

Potential automorphy followed by descent would need a descent theorem over the original field, not an averaging operation on cohomology: changing the base number field is not the same as passing to a finite covering level $Y_V\to Y_U$. A second route is to build the exact characteristic-zero eigensystem from congruences of increasing precision. The attractive feature is that a fixed level has finite-dimensional cohomology; the difficulty is that neither bounded level, compatible precision, nor survival beyond torsion comes for free. I investigate this second route and isolate exactly what compactness can and cannot supply.

Attempt.

Let $\mathcal O$ be the integer ring of a finite extension of ${\mathbb{Q}}_p$, with uniformizer $\varpi$ and finite residue field. Let $M$ be a finite free $\mathcal O$-module, let $\mathcal T$ be any family of endomorphisms of $M$, and prescribe scalars $a_T\in\mathcal O$ for $T\in\mathcal T$. No simultaneous diagonalization is assumed.

Lemma 39.1 (fixed-level, all-precision lifting). Suppose that for every $r\ge1$ there is a vector $u_r\in M/\varpi^rM$ whose reduction modulo $\varpi$ is nonzero and such that \[ (T-a_T)u_r=0\quad\text{in }M/\varpi^rM\qquad(T\in\mathcal T). \] Then there is $u\in M\setminus\varpi M$ satisfying $(T-a_T)u=0$ for every $T\in\mathcal T$. In particular the prescribed simultaneous eigenspace in $M\otimes_{\mathcal O}\operatorname{Frac}(\mathcal O)$ is nonzero.

Proof. At level $r$ take the finite set of all primitive simultaneous solutions. Reduction maps a solution to a primitive solution at every smaller positive level. Join each solution to its reduction at the preceding level, adjoining a root at level zero. The resulting finitely branching tree has vertices at arbitrarily large depths. To construct an infinite path, choose at each step a child with descendants at arbitrarily large depths; one exists because there are only finitely many children. This gives compatible $v_r$ without assuming that the originally chosen $u_r$ were compatible. Completeness gives $u\in\varprojlim M/\varpi^rM=M$. For each fixed $T$, $(T-a_T)u$ belongs to every $\varpi^rM$, whose intersection is zero by freeness. The first reduction is nonzero, so $u$ is primitive and remains nonzero after inverting $\varpi$. $\square$

For a possible application, fix one compact open level $U$ and degree $q$, and take the torsion-free quotient of an integral cohomology lattice, completed at $p$, as $M$. At good places set $a_{T_v}=a_v(E)$ in the convention of the source. The lemma shows that primitive realizations of this exact system at every precision, in that one lattice, would give a characteristic-zero system. It gives no way to manufacture those realizations from a single residual modularity statement. Even obtaining the trace eigensystem would only reach the weaker stage identified by Thorne; all finite local factors, including their monodromy, still need to be identified before Conjecture 2.6 has been addressed. [S1], pp. 648--650.

The torsion issue can be exposed before using any automorphic theorem. For a bounded complex $C^\bullet$ of finite free $\mathcal O$-modules, the long exact sequence associated to multiplication by $\varpi^r$ gives \[ 0\longrightarrow H^q(C)/\varpi^rH^q(C) \longrightarrow H^q(C\otimes\mathcal O/\varpi^r) \longrightarrow H^{q+1}(C)[\varpi^r]\longrightarrow0. \] Here exactness follows by taking the kernel and cokernel of multiplication in the adjacent terms of that long exact sequence. Therefore residual cohomology can consist entirely of classes arising from torsion in the next degree. For the two-term complex $\mathcal O\xrightarrow{\varpi}\mathcal O$ in degrees $q,q+1$, the rationalized complex is acyclic whereas its reduction modulo $\varpi$ has one-dimensional cohomology in both degrees. A nonzero mod-$p$ class alone thus does not establish a nonzero rational class.

There is also a prescribed-eigenvalue obstruction, even with no torsion. Take $M={\mathbb{Z}}_p$, $T=0$, and prescribe $a_T=p^s$ for a positive integer $s$. A primitive solution exists modulo $p^r$ precisely for $r\le s$, and there is no nonzero solution over ${\mathbb{Q}}_p$. Indeed $p^su\equiv0\pmod{p^r}$ for a unit $u$ holds exactly when $r\le s$. Thus an arbitrarily impressive finite precision computation cannot replace the all-precision quantifier. The accompanying script checks this equivalence for small primes and precisions by exact integer arithmetic, as a test of the example rather than of modularity.

Stress test.

The proof uses finite residue fields, fixed finite rank, fixed level, and primitivity in a torsion-free lattice. Letting $U=U_r$ vary destroys the single compact tree. Permitting $u_r$ to become more divisible allows all normalized mass to vanish. Working in torsion cohomology can yield an inverse-limit vector that dies in characteristic zero. Each would invalidate a proposed application. Furthermore, potential automorphy over an extension does not identify a $\operatorname{GL}_2({\mathbb{A}}_K^\infty)$-subrepresentation over $K$ by elementary Galois averaging. A full proof must preserve the ramified Weil--Deligne operators $N_v$, not merely Frobenius semisimplifications at good places. The hypothesis $\operatorname{End}_K(E)={\mathbb{Z}}$ is unchanged: curves acquiring complex multiplication only after extending $K$ have not been removed.

One possible next lemma would assert, for every curve in the entry, a single level and an integral local-type condition admitting primitive eigensystems modulo every $p^r$, with characteristic-zero local-global compatibility at all finite places. That is a substantial automorphy assertion, not a consequence of Lemma 39.1. It is intentionally left unproved rather than hidden in a claim that compatible lifts are automatic.

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Outcome.

Outcome: Exploratory approach with unresolved gap.

A fixed-level lifting lemma and two explicit obstructions to naive residual lifting have been proved. Historical novelty of this elementary lemma is not claimed. The attempt has not constructed the required primitive systems, nor identified all finite local factors; consequently it has not supplied the requested injection. The useful reduction is that compactness itself is not the obstacle once those exact hypotheses are met: the obstacle is producing them while preventing torsion, level growth, and loss of local information.

Sources

• S1 J. A. Thorne, Elliptic curves and modularity, 8ECM proceedings, EMS Press, 2023, DOI 10.4171/8ECM/12. https://ems.press/content/book-chapter-files/28276. Catalogue formulation source.
• S2 Elliptic curves and modularity. https://ems.press/content/book-chapter-files/28276?nt=1. Catalogue research/status reference; not a proof endorsement.
• S3 Modularity of elliptic curves over cyclotomic Z_p-extensions of real quadratic fields. https://arxiv.org/html/2206.12860v1. Catalogue research/status reference; not a proof endorsement.
• S4 Modular elliptic curves and hyperbolic uniformization. https://arxiv.org/abs/2607.03830. Catalogue research/status reference; not a proof endorsement.
• E1 Ana Caraiani and James Newton, On the modularity of elliptic curves over imaginary quadratic fields, arXiv:2301.10509, abstract and main theorem statements. https://arxiv.org/abs/2301.10509. Consulted 27 September 2026; the field and residual-image restrictions are retained.
• E2 Neil Dummigan and Devendra Tiwari, Modular elliptic curves and hyperbolic uniformization, arXiv:2607.03830v2, 27 August 2026, abstract and stated implication. https://arxiv.org/abs/2607.03830v2. Consulted 27 September 2026; conditional uniformization implication, not universal modularity.