Browse › Number theory › 015 Torus-packet equidistribution in prime, quartic, and sextic degrees
Equidistribution of Primitive Sextic Torus Packets
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Abstract
We prove that the complete ideal-class packets of maximal orders in totally real sextic fields with no proper intermediate fields become equidistributed, with no escape of mass, in the space of unimodular lattices as their field discriminants tend to infinity. The limit is Haar probability measure. Each packet includes all coordinate-sign translates and is weighted by diagonal-orbit volume.
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- OpenAI math release, Lean scope note 015: Torus-packet equidistribution in prime, quartic, and sextic degrees (lean/docs/015.md, release of 2026-10-06). The family's Lean scope note lists only the prime-degree companion paper as formalized and gives no Lean statement for the sextic theorem. [family-level]
Related prior literature mentioned by others
- Manfred Einsiedler, Elon Lindenstrauss, Philippe Michel and Akshay Venkatesh, Distribution of periodic torus orbits and Duke’s theorem for cubic fields (Annals of Mathematics 173, 2011). The VibeMathed family entry identifies this cubic equidistribution theorem and its higher-degree discussion as background to the family’s packet-equidistribution claims. [family-level]
Possibly relevant literature (found by the agent, not checked)
- Ilya Khayutin (joint work with A. Wieser), Two-step equidistribution for bi-quadratic torus packets (New England Dynamics and Number Theory Seminar talk, 2021-12-02). The talk treats composite-degree torus packets whose limits meet intermediate homogeneous subspaces, the obstruction the paper's intermediate-resolution argument is designed to exclude in degree six.
- Ilya Khayutin, Non-vanishing of Class Group L-functions for Number Fields with a Small Regulator (arXiv:1901.06710, 2019). Shows escape of mass for the trivial-class torus orbit when the regulator is small, which bears on the paper's restriction to complete volume-weighted packets.
Digests and expositions
- Latent Seminar, Arithmetic torus packets and equidistribution (2026-10-06). This exposition distinguishes the three packet theorems and outlines tightness, entropy, and the additional quartic and sextic arguments, explicitly without independently verifying the complete proofs. [family-level]
Elsewhere
- Emergent Mind: family 015
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026,
author = {{OpenAI}},
title = {{Equidistribution of Primitive Sextic Torus Packets}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/primitive-sextic-torus-packets.pdf}{OAI:Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/015.md. The Lean statements have not been checked by people against the paper's claims.
Family 015: Torus-packet equidistribution in prime, quartic, and sextic degrees
Proves Haar equidistribution without escape of mass for complete volume-weighted torus packets from totally real fields: arbitrary lattices and prescribed local types in fixed prime degree at least five, arbitrary-order Picard packets in primitive quartic fields, and maximal-order ideal-class packets in primitive sextic fields. Here primitive means having no proper intermediate field; the relevant order or field discriminant tends to infinity.
Other papers in this family:
- Equidistribution of Prime-Degree Torus Packets with Arbitrary Local Type · September 24, 2026
- Equidistribution of primitive quartic torus packets for arbitrary orders · October 5, 2026