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Automorphic realization of Langlands parameters for split groups over global function fields, I. Arithmetic foundations and the main theorems

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 connected reductive group over a global function field FF. V. Lafforgue decomposed the space of cusp forms on GG according to semisimple Langlands parameters. This is the first of three papers which prove a converse: every semisimple elliptic parameter is the parameter of a cusp form on GG over FF itself, which we call an automorphic realization of the parameter. This holds in every positive characteristic for classical groups, in characteristic greater than 33 for G2G_2 and F4F_4, greater than 55 for E6E_6, and greater than 77 for E7E_7 and E8E_8. A parameter that is not elliptic is the parameter of an automorphic form which is not cuspidal, under the bound for the smallest Levi subgroup through which the parameter factors. The form has the prescribed eigenvalue for every excursion operator, and we determine for which lattices in the centre it can be taken invariant. The present paper contains the arithmetic part of the proof. It defines excursion operators on functions that are not assumed cuspidal, proves their compatibility with constant terms, shows that a function on which they act through an elliptic parameter is a cusp form, and passes from simply connected groups to isogenous groups and from Levi subgroups to the group. This reduces the theorems to the existence of a cusp form with a given elliptic parameter on a simply connected simple group. That is proved in the second paper by a geometric method, when the characteristic does not divide the order of the Weyl group, and in the third paper for classical groups, by descent from general linear groups.

Author comments: Part I of a series of three papers; Parts II (The geometric construction) and III (Classical groups) are deposited separately. The theorems of this part depend on the two existence theorems proved in Parts II and III.

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
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Claude FableVersion Claude Fable 5.2

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Version history

  1. v1 · 70 pagesSubmitted by Boyang HuPart I of a series of three papers; Parts II (The geometric construction) and III (Classical groups) are deposited separately. The theorems of this part depend on the two existence theorems proved in Parts II and III.CurrentNov 12, 2026