Bhargava: elliptic curves have cyclic reduction at a positive proportion of primes (claim)
Awaiting review
Importance: major (4 of 5)The takeaway
Serre showed under the generalized Riemann hypothesis that an elliptic curve over ℚ without full rational 2-torsion has cyclic reduction E(F_p) at a positive proportion of primes; Murty removed GRH for CM curves, and Gupta–Murty proved infinitely many such primes in general. On 31 Oct Manjul Bhargava (Princeton) posted on Hexagon an unconditional positive lower proportion for every such curve. The argument uses Cureton–Thorner’s zero-free half-plane and ‘the marked Type II estimates of OpenAI’. Per the provenance statement, handwritten notes were turned into LaTeX ‘with additional writing and details added by both the author and GPT-6’. GPT-6 also helped formalise the deduction in Lean, with the analytic inputs left as hypotheses. Only some parts have been checked by a human, and it is not refereed.
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Unverified claim
- Importance
- Major (4 of 5)
- Last verified
- 31 October 2026
Your AI and this story
- GPT-6 Astra184 days after its cutoff
- Claude Opus 5.5123 days after its cutoff
- Gemini 4 Argon214 days after its cutoff
- Grok 4.7153 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 123 days before it.
Key facts
- Claim (abstract): ‘We prove that every elliptic curve over ℚ whose points of order two are not all rational has cyclic reduction at a positive lower density of primes. This is the only necessary restriction … the theorem of Cureton and Thorner supplies N=32. The proof uses the varying-cofactor sieve of our earlier works on primitive roots and primitive points, in conjunction with the marked Type II estimates of OpenAI’
- Provenance (Hexagon, verbatim): ‘The argument was worked out in handwritten notes, which were then converted into LaTeX, with additional writing and details added by both the author and GPT-6. The deduction of Theorems 1 and 2 from specified analytic and arithmetic inputs was then formalized in Lean with the help of GPT-6.’ The Cureton–Thorner hypothesis, the candidate-prime estimate (Proposition 6), the smoothed prime-ideal bound and the prime number theorem ‘remain explicit hypotheses … their proofs were not formalized here’. Text declaration: ‘A mix of human-written and AI-generated’; human understanding: ‘Some parts … but not all’
- History (paper, §1): Borosh–Moreno–Porta’s numerical study; Serre’s 1977–78 density prediction and its proof under GRH (published by Murty); Murty removed GRH for CM curves, Cojocaru simplified it; Gupta and Murty proved ≫ X/(log X)² cyclic primes unconditionally; Cojocaru showed the asymptotic follows from zero-freeness in Re s > 3/4, which the available Re s > 31/32 does not give
- Chain of AI-era inputs: OpenAI’s 6 Oct release (Artin primitive-root family, marked Type II estimates) → Cureton–Thorner’s quasi-Riemann hypothesis for every Dedekind zeta function (Re s > 31/32, Hexagon, 12 Oct) → Bhargava’s positive-density Artin result (Hexagon 2610.00228, 14 Oct) and primitive points paper → this cyclicity theorem
- Contributor: the Hexagon record lists Manjul Bhargava, Princeton University (Fields Medal 2014) as sole author and sponsor; arXiv copy: none on 31 Oct
- Verification status: Hexagon deposit (Hexagon does not judge correctness), not refereed; Lean covers the deduction from stated inputs only; the unconditional status rests on OpenAI’s and Cureton–Thorner’s unrefereed results; no reaction found on 31 Oct
What happened
For an elliptic curve E over ℚ and a prime p of good reduction, the group E(F_p) is either cyclic or not. Serre predicted the proportion of primes where it is cyclic and proved the prediction under the generalized Riemann hypothesis. Without GRH, a positive proportion was known only for curves with complex multiplication.
On 31 Oct Manjul Bhargava posted on Hexagon a proof of a positive lower proportion for every curve whose 2-torsion is not fully rational, which is the necessary condition. The argument rests on two recent unrefereed inputs from the AI era: OpenAI’s marked Type II estimates and Cureton and Thorner’s zero-free half-plane for Dedekind zeta functions. The provenance statement says GPT-6 added writing and details to Bhargava’s handwritten argument and helped formalise the deduction in Lean.
Why it matters
It is a classical problem, and a Fields medallist is extending the OpenAI release step by step. The whole chain still needs refereeing.
What is disputed or not yet verified
| Verification | Hexagon deposit, not refereed; partial Lean (deduction only); depends on unrefereed AI-era inputs |
|---|
Sources
3 sources from 1 site. Numbers match the chips in the text.
3 sources: 3 primary
Primary
- Hexagon 2610.00699: Cyclic reductions of elliptic curves at a positive proportion of primes (Bhargava)hexagonmath.org, paper
- Hexagon 2610.00699 PDFhexagonmath.org, paper
- Hexagon 2610.00228: positive lower density for Artin’s primitive roots (Bhargava)hexagonmath.org, paper
Changes
- Filed (science agent; Hexagon API record, provenance statement and PDF introduction read)