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The Margulis-Platonov conjecture on normal subgroups of groups of rational points

Let GG be an absolutely almost simple simply connected algebraic group over a global field kk, let AA be the finite set of nonarchimedean places vv with rankkvG=0\mathrm{rank}_{k_v}G=0, and let δA:G(k)→∏v∈AG(kv)\delta_A:G(k)\to\prod_{v\in A}G(k_v) be the diagonal map. The Margulis-Platonov conjecture asserts that the compact anisotropic local groups account for all abstract normal subgroups: every noncentral normal subgroup NN of G(k)G(k) is δA−1(W)\delta_A^{-1}(W) for an open normal subgroup WW. It was known for isotropic groups, inner type A (Segev-Seitz, Rapinchuk-Segev-Seitz) and most anisotropic types, leaving anisotropic outer A, trialitarian D4D_4 and E6E_6 cases over number fields. Is every noncentral normal subgroup of G(k)G(k) of this form?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Algebraic groups over global fields; arithmetic groups
Posed by
Grigory Margulis and Vladimir Platonov
Year posed
1979
Years open
47y
Solved
2026-09-23
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
42 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1 (number fields): for GG absolutely almost simple and simply connected over a number field kk, every noncentral normal subgroup N◃G(k)N\triangleleft G(k) equals δA−1(W)\delta_A^{-1}(W) for an open normal WW of ∏v∈AG(kv)\prod_{v\in A}G(k_v); when AA is empty, N=G(k)N=G(k). With Rapinchuk's theorem it gives the congruence subgroup property for SS of positive Dirichlet density in the split places. The October 5 companion proves the same over every global function field, including characteristic two. It does not settle Serre's congruence subgroup conjecture for finite SS or arbitrary infinite SS.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family has two manuscripts: number fields (September 23, 2026, this entry's principal) and global function fields including characteristic two (October 5, 2026).

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the number-field manuscript was read against the conjecture; it states the full conjecture over number fields with abstract normal subgroups and no finite-generation assumption, using the established reduction (cases already known) and supplying the remaining anisotropic outer A, trialitarian and E6E_6 cases. The function-field companion's abstract states the same over every global function field. Neither proof was refereed; there is no Lean formalization. The number-field paper relies on cited prior work (Segev-Seitz, Rapinchuk-Potapchik, Prasad-Rapinchuk's metaplectic kernel) as inputs.

Sources

Changelog1 change

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