Short··0:21·Number theory
Fontaine–Mazur Conjecture at the Prime 2: OpenAI Claims Modularity
Watch on YouTubeAbout the problem →
About this video
Modularity says the point counts of a curve modulo every prime are secretly the coefficients of a modular form; for elliptic curves this was proved between 1995 and 2001. OpenAI claims the same at the prime 2 for every odd, irreducible two-dimensional 2-adic Galois representation that is unramified outside finitely many primes and de Rham at 2 with distinct Hodge–Tate weights (modular up to a Tate twist, with no residual restrictions), along with a pro-modularity result and the claim that every component of the 2-adic Hecke algebra at odd level has dimension 4. These are OpenAI's preprints, not yet peer reviewed.
Papers (OpenAI, catalogue result 010):
- Unrestricted pro-modularity at the prime two: github.com/openai/math/blob/main/preprints/Unrestricted-pro-m…
- The Dimension of the Two-Adic Hecke Algebra at Odd Level: github.com/openai/math/blob/main/preprints/The-Dimension-of-t…
- Fontaine–Mazur modularity at the prime 2: github.com/openai/math/blob/main/preprints/Fontaine-Mazur-mod…
The animation illustrates the result; it is not a proof. The curve and modular form shown (level 11) are a classical example of the matching.
Gradient Damage: mathematics you can see, equations you can hear.
youtube.com/@gradientdamage
Every result explained: gradientdamage.com/p/010
About the problem
Number theory·OpenAI Math·Result 010
Unrestricted pro-modularity at the prime two
Every continuous odd absolutely irreducible two-dimensional 2-adic representation of Gℚ unramified outside finitely many primes occurs in a completed Hecke algebra at some odd tame level. Also proves classical modularity up to Tate twist for irreducible odd representations with these finiteness conditions that are de Rham at 2 with distinct Hodge–Tate weights, resolving the dyadic Fontaine–Mazur case without residual restrictions.
Topics
Fontaine-Mazur conjectureFontaine Mazurmodularitymodularity theoremGalois representations2-adicp-adicprime twoHecke algebraEmerton dimension conjecturemodular formselliptic curvesLanglands programnumber theoryOpenAIOpenAI mathAI mathematicsAI proofmathsmathematicsmath shorts3D animation
