Result 030, Number theory

Modularity of elliptic curves over imaginary quadratic fields

Proves the modularity conjecture for elliptic curves over imaginary quadratic fields: every elliptic curve over every imaginary quadratic field is modular, with matching local parameters at every place.

Proof

The bigger picture

Why it matters

Elliptic curves encode arithmetic questions in the geometry of cubic equations. This manuscript claims that every such curve over an imaginary quadratic field has an analytic counterpart, linking two different ways of studying its arithmetic.

What changes?

The unreviewed manuscript reports modularity for every elliptic curve over every imaginary quadratic field, with no exceptions stated. An elliptic curve is a smooth cubic curve with a chosen origin; an imaginary quadratic field extends the rational numbers by adjoining the square root of a negative integer. Modularity associates the curve with an automorphic form, a highly structured analytic object. The claimed correspondence matches local parameters everywhere: at every prime of the field and at its complex place.

What does that help mathematicians do?

The everywhere-local matching makes the claim stronger than agreement at almost all primes. It identifies the local ingredients used to build the curve's L-function, an object that packages arithmetic information, with those of its automorphic counterpart. Researchers could therefore translate questions about these ingredients between the geometric and analytic descriptions without leaving exceptional primes outside the correspondence. That completeness is the concrete mathematical payoff.

Are there practical applications?

Its immediate value is foundational: the claimed theorem would place an entire class of arithmetic curves within the automorphic framework, rather than requiring modularity as an additional assumption for each curve. This supplies a bridge for further number-theoretic arguments. The supplied abstract does not describe a computational method or a practical deployment.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Modularity of elliptic curves over imaginary quadratic fields

October 4, 2026 30 pages

We prove the modularity conjecture for elliptic curves over imaginary quadratic fields: every such curve is modular, with matching local parameters at every place.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.