The Margulis–Platonov conjecture

The Margulis–Platonov conjecture describes the abstract normal subgroups of the rational points of an absolutely almost simple simply connected algebraic group G over a global field k: every noncentral normal subgroup should be the inverse image of an open normal subgroup of the product of the groups G(k_v) over the finitely many nonarchimedean places v where G is anisotropic. Two manuscripts in OpenAI's public mathematics collection, dated 23 September and 5 October 2026, claim the conjecture over number fields and over global function fields of every characteristic. The collection lists no Lean formalization of these results.

Background

As recounted in the number-field manuscript's introduction:
- Origins. The conjecture grew from Kneser's question on the simplicity of quaternionic norm-one groups and Platonov's local-to-global simplicity proposal; Margulis formulated the description by open subgroups of the anisotropic local product. Strong approximation and Margulis's normal subgroup theorem give the arithmetic framework and the reduction to normal subgroups of finite index.
- Known cases. The isotropic case follows from the Kneser–Tits theorem over global fields, with the final outer E₆ cases in Gille's account using work of Garibaldi and of Chernousov and Timoshenko. For anisotropic groups, Chernousov proved projective simplicity when the absolute rank is at least two and the group splits over a quadratic extension, and Tomanov and Sury treated classical and low-index unitary cases.
- Inner type A. After work of Platonov and Rapinchuk and of Raghunathan, Rapinchuk and Potapchik reduced this case to finite simple quotients of the multiplicative group of a division algebra; Segev and Segev–Seitz completed it, and Rapinchuk, Segev and Seitz proved that every finite quotient of that multiplicative group is solvable. The remaining open cases were anisotropic outer type A, trialitarian D₄ and E₆.

The original results

Number fields, 23 September 2026. The main theorem of The Margulis–Platonov conjecture over number fields states that for a number field k and an absolutely almost simple simply connected k-group G, every normal subgroup of G(k) not contained in the center is the inverse image, under the diagonal map, of an open normal subgroup of the product of the anisotropic nonarchimedean local groups. When that set of places is empty, every noncentral normal subgroup is all of G(k).
Global function fields, 5 October 2026. The Margulis–Platonov conjecture over global function fields states the same conclusion over every global function field, including characteristic two.
Scope, as stated: the manuscripts use the previously established cases and supply the remaining arguments, for anisotropic outer type A and, over number fields, for trialitarian D₄ and E₆; the function-field manuscript builds on the number-field one.
Method, as described: for special unitary groups of odd-degree division algebras, a nontrivial finite defect would give either a finite abelian character or a homomorphism onto a finite nonabelian simple group. Characters are excluded through an algebraic difference function, the additive span of the unitary group and the inner type A theorem. For a simple quotient, each rational unitary element receives a colour; a translation-invariant probability law on colour patterns with uniform marginals, built from Langlands's Eisenstein-series convergence and Howe–Moore decay, and the Furstenberg–Katznelson density theorem lead to a centralizer double-coset condition that a lemma on finite simple groups excludes. Other degrees follow by induction; over number fields, trialitarian D₄ is treated through products of two involutions in type A₂ subgroups and E₆ is reduced to D₄ and type A pieces by a torsor argument. Over function fields, where Harder's results leave only type A among anisotropic groups, finite additive Fourier analysis replaces the Furstenberg–Katznelson step, and the power construction is adapted to exponents divisible by the characteristic.
Sources. Commit adc7f12 of the openai/math repository: The Margulis–Platonov conjecture over number fields and The Margulis–Platonov conjecture over global function fields.
Formalization. The collection lists no Lean scope note or Comparator challenge for this family.

Public follow-up work

Date	Work	Relation	Public claim	Public status (checked 31 October 2026)
2026-10-16	J. Jardine, Lattice24, referee reviews rank_138 of the number-field manuscript and rank_018 of the function-field manuscript in JJardine919/openai-math-audit	Review, AI-assisted audit	Both reviews report no gap. The number-field review read all eight sections, checked the main identities numerically and symbolically and the reductions by hand, and lists three minor points: a cohomological injectivity step that rests entirely on a cited proposition of Prasad and Rapinchuk, the use of a quoted form of the Hasse principle of Borovoi in the exceptional types, and a conjugacy-class labelling in the Mathieu group M₁₂ that does not affect the argument. The function-field review read all ten sections and the appendix, checked the identities of three sections, including the palette and factorization identities, by computation and the sporadic-group table against ATLAS data, and accepted cited inputs such as the classification of finite simple groups without re-deriving them.	Part of the completed Lattice24 audit; the reviews are published under CC BY 4.0.
2026-10-11	AI Proved What?, episode 018 A hidden clock: the Margulis–Platonov conjecture, remade version of 25 October 2026	Reading, exposition	Video explainer of the two preprints, 159 pages, starting from an index-three normal subgroup of the rational unit quaternions that comes from the prime 2. It recalls the history from Kneser, Platonov and Margulis to the known isotropic cases, most anisotropic types and inner type A, and states that over number fields the result makes a conditional theorem of A. Rapinchuk on the congruence subgroup property for sets of primes of positive density unconditional, while saying nothing about Serre's congruence subgroup conjecture for finite sets of primes.	Episode status, last checked 30 October 2026: posted as preprints on 23 September and 5 October 2026; under expert review, not yet; peer reviewed, not yet; checked in Lean, no; the claims have not been withdrawn and no public critique was found. It notes that the proofs rely on many deep theorems, including the classification of finite simple groups. The site states that it is made with Anthropic's Claude. Read on 31 October 2026.

Notes

The background summarizes the number-field manuscript's own account of earlier work.
The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.