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Automorphic realization of Langlands parameters for split groups over global function fields, III. Classical groups

Primarily AI-generated textHuman understanding: some partsmath.NT — Number Theorymath.AG — Algebraic Geometrymath.RT — Representation Theory

Contributed by Boyang Hu ↗

Version 1 / Nov 12, 2026 / CC BY 4.0

Abstract

Let GG be a split semisimple group whose simple factors are of classical type, over a global function field FF of arbitrary positive characteristic. We prove that every semisimple elliptic Langlands parameter ΓF→G^(Q‾ℓ)\Gamma_F\to\widehat G(\overline{\mathbb Q}_\ell) is the parameter of a cusp form on GG, in the sense of V. Lafforgue's excursion operators; we call such a form an automorphic realization of the parameter. Neither FF nor its field of constants is extended. We start from L. Lafforgue's theorem for general linear groups and use automorphic descent and the theta correspondence. The work is to prove that the descent does not vanish over FF, to carry out the computations with root subgroups in small characteristic, in particular in characteristic two, and to identify the parameter itself and not only its unramified Hecke eigenvalues. This is the third of three papers. Together with the first, which contains the arithmetic foundations, and the second, which treats the exceptional types by a geometric method, it gives automorphic realizations of semisimple parameters on all split reductive groups, with a restriction on the characteristic for the exceptional types only.

Author comments: Part III of a series of three papers; Parts I (Arithmetic foundations and the main theorems) and II (The geometric construction) are deposited separately. The folder anc/ contains the Python programs described in the appendix on computer checks.

Provenance statement

The mathematical arguments of the three papers of this series were developed with GPT-6 Astra Pro. The papers were then organized and rewritten with Claude Fable 5.2, and in the course of this several arguments were replaced or expanded. The three papers were then reviewed in detail, line by line, by GPT and Claude models, in sessions which had not taken part in writing them, and the errors found in this way were corrected. Boyang Hu organized the work and is responsible for the manuscript.

Tools used

OpenAI
ChatGPTVersion GPT-6 Pro
Anthropic
Claude FableVersion Claude Fable 5.2
Python
SymPyVersion 1.14.0
Python
NumPyVersion 2.5.3

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Version history

  1. v1 · 68 pagesSubmitted by Boyang HuPart III of a series of three papers; Parts I (Arithmetic foundations and the main theorems) and II (The geometric construction) are deposited separately. The folder anc/ contains the Python programs described in the appendix on computer checks.CurrentNov 12, 2026