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Automorphic realization of Langlands parameters for split groups over global function fields, II. The geometric construction

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 simply connected semisimple group over a global function field FF whose characteristic does not divide the order of the Weyl group of GG. 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. The parameter may be ramified at finitely many places and may have finite monodromy, and neither FF nor its field of constants is extended. The starting point is the geometric Langlands correspondence in characteristic zero. It provides a sheaf, and the function which the trace of Frobenius attaches to a nonzero sheaf can vanish identically. The main point of the construction is to arrange that the trace in question is a determinant det⁡(1−ϕ)\det(1-\phi) with no zero factor, and that it stays nonzero under specialization to the given field. This is the second of three papers. Together with the first, which contains the arithmetic foundations, and the third, which treats classical groups in every characteristic, 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 II of a series of three papers; Parts I (Arithmetic foundations and the main theorems) and III (Classical groups) 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

References

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Version history

  1. v1 · 93 pagesSubmitted by Boyang HuPart II of a series of three papers; Parts I (Arithmetic foundations and the main theorems) and III (Classical groups) are deposited separately. The folder anc/ contains the Python programs described in the appendix on computer checks.CurrentNov 12, 2026