Browse › Number theory › 018 The Margulis–Platonov conjecture over global fields
The Margulis–Platonov conjecture over number fields
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Abstract
We prove the Margulis–Platonov conjecture over number fields. For an absolutely almost simple simply connected group, every noncentral abstract normal subgroup of its rational points is the inverse image of an open normal subgroup of the product of its anisotropic nonarchimedean local groups.
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Possibly relevant literature (found by the agent, not checked)
- Andrei S. Rapinchuk and Wojciech Tralle, On Almost Strong Approximation in Reductive Algebraic Groups (arXiv:2503.00696, v7 7 Sep 2026; accepted in Algebra & Number Theory per arXiv comments). Develops almost strong approximation for sets of valuations of positive Dirichlet density, and its Theorem 1.5 and Corollary 1.6 give congruence-kernel triviality conditional on Margulis–Platonov, the setting of the paper's congruence-subgroup corollary; not in this paper's bibliography, though the companion function-field paper cites it.
- Gopal Prasad and Andrei S. Rapinchuk, On the congruence kernel for simple algebraic groups (arXiv:1503.03713, March 2015). Gives triviality and centrality criteria for congruence kernels over global fields for infinite sets of places, relevant context for the paper's congruence-subgroup corollary and its stated limits.
Digests and expositions
- Platzi ("Platzi Team"), The Margulis–Platonov conjecture over number fields (Platzi course "OpenAI Math Preprints", lesson 43; undated). Short video lesson with written summary of this preprint, including the reduction to a finite obstruction and the $D_4$/$E_6$ cases; the page does not state how it was produced.
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 018
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:The-Margulis-Platonov-conjecture-over-number-fields-September-23-2026,
author = {{OpenAI}},
title = {{The Margulis--Platonov conjecture over number fields}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/The-Margulis-Platonov-conjecture-over-number-fields-September-23-2026/paper.pdf}{OAI:The-Margulis-Platonov-conjecture-over-number-fields-September-23-2026}},
year = {2026}
}Family 018: The Margulis–Platonov conjecture over global fields
Proves the Margulis–Platonov conjecture over every global field, including function fields of characteristic two. For an absolutely almost simple simply connected algebraic group G over k, every noncentral abstract normal subgroup of $G(k)$ is the inverse image of an open normal subgroup in the finite product of its anisotropic nonarchimedean local groups.
Other papers in this family:
- The Margulis–Platonov conjecture over global function fields · October 5, 2026