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Modularity of elliptic curves over totally real field and CM fields not containing \zeta_5

A mix of human-written and AI-generated textHuman understanding: some partsmath.NT — Number Theorymath.AG — Algebraic Geometry

Contributed by Bao Le Hung ↗

Version 1 / Oct 08, 2026 / CC BY 4.0

Abstract

This notes tries to parse the basic structure of OpenAI's new argument proving modularity of elliptic curves over imaginary quadratic fields. The argument also proves modularity over all totally real field, and in the original form, modularity over CM fields not containing \zeta_5. The general CM case has also now been resolved by an independent repository which improves the Caraiani-Newton modularity lifting theorem for CM field at the prime 5, which was the main reason for the extra hypothesis. I kept the argument in the totally real case separate to highlight the main new idea in the OpenAI proof, which is orthogonal to modularity lifting theorems.

Provenance statement

First draft generated by Astra GPT 6.1, introduction written by the contributor, revisions by Claude Opus 5.5

Tools used

OpenAI
CodexVersion Astra 6.1
Anthropic
Claude OpusVersion 5.5

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Version history

  1. v1Submitted by Bao Le HungInitial depositCurrentOct 08, 2026