Number theory·OpenAI Math·Result 030
Modularity of elliptic curves over imaginary quadratic fields
One of the 372 results in OpenAI's public mathematics catalogue, filed under number theory: 1 preprint, the latest dated 4 October 2026. This page quotes OpenAI's own summary of the claim, links the paper and abstract, and embeds the animation.
Chapter 10 of 10 in Large Prime Gaps Have Positive Density, Artin's Primitive Root Conjecture and 8 More · starts at 3:09 · watch from there on YouTube →
What OpenAI claims
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.
Paper
Modularity of elliptic curves over imaginary quadratic fields
We prove the modularity conjecture for elliptic curves over imaginary quadratic fields: every such curve is modular, with matching local parameters at every place.
Status
Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.
Quick answers
- Is it proven?
- Not independently yet. OpenAI posted a preprint on 4 October 2026; it has not been peer reviewed, so treat the result as OpenAI's claim until other mathematicians have checked it.
- What exactly does OpenAI claim?
- 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.
- Where is the proof?
- In the paper, Modularity of elliptic curves over imaginary quadratic fields (PDF).
- Is there a video?
- Yes, on YouTube (3:29, chapter 10).
Also in
- Chapter 66 of 84 in Every OpenAI Math Result, Part 3 of 4: Operator Paving, Naimark's Problem and 82 More · starts at 22:41 · watch from there on YouTube →
The video
Large Prime Gaps Have Positive Density, Artin's Primitive Root Conjecture and 8 More · Compilation · 3:29 · chapter 10 from 3:09 · description, sources and chapters →
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