Hexagon

math.NT — Number Theory

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

Contributed by Boyang Hu

Let $G$ be a split semisimple group whose simple factors are of classical type, over a global function field $F$ of arbitrary positive characteristic. We prove that every semisimple elliptic Langlands parameter $\Gamma_F\to\widehat G(\overline{\mathbb Q}_\ell)$ is the parameter of a cusp form on $G$ , in the sense of V. Lafforgue's excursion operators; we call such a form an automorphic realization of the parameter. Neither $F$ 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 $F$ , 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.

Primarily AI-generated textHuman understanding: some partsEisenstein seriesLanglands parametersautomorphic descentcharacteristic twofunction fieldstheta correspondence

Nov 12, 202668 pages

PDF View work

hexagon:2611.00398v1

math.NT — Number Theory

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

Contributed by Boyang Hu

Let $G$ be a split connected reductive group over a global function field $F$ . V. Lafforgue decomposed the space of cusp forms on $G$ 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 $G$ over $F$ itself, which we call an automorphic realization of the parameter. This holds in every positive characteristic for classical groups, in characteristic greater than $3$ for $G_2$ and $F_4$ , greater than $5$ for $E_6$ , and greater than $7$ for $E_7$ and $E_8$ . 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.

Primarily AI-generated textHuman understanding: some partsLanglands parametersconstant termscusp formsexcursion operatorsfunction fieldsshtukas

Nov 12, 202670 pages

PDF View work

Matching works

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

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

Advanced search

math.AC — Commutative Algebra
math.AG — Algebraic Geometry
math.AP — Analysis of PDEs
math.AT — Algebraic Topology
math.CA — Classical Analysis and ODEs
math.CO — Combinatorics
math.CT — Category Theory
math.CV — Complex Variables
math.DG — Differential Geometry
math.DS — Dynamical Systems
math.FA — Functional Analysis
math.GM — General Mathematics
math.GN — General Topology
math.GR — Group Theory
math.GT — Geometric Topology
math.HO — History and Overview
math.IT — Information Theory
math.KT — K-Theory and Homology
math.LO — Logic
math.MG — Metric Geometry
math.NA — Numerical Analysis
math.NT — Number Theory
math.OA — Operator Algebras
math.OC — Optimization and Control
math.PR — Probability
math.QA — Quantum Algebra
math.RA — Rings and Algebras
math.RT — Representation Theory
math.SG — Symplectic Geometry
math.SP — Spectral Theory
math.ST — Statistics Theory
cs.AI — Artificial Intelligence
cs.CC — Computational Complexity
cs.CG — Computational Geometry
cs.DM — Discrete Mathematics
cs.DS — Data Structures and Algorithms
cs.FL — Formal Languages and Automata Theory
cs.IT — Information Theory
cs.LO — Logic in Computer Science
cs.MS — Mathematical Software
cs.NA — Numerical Analysis
cs.SC — Symbolic Computation
math-ph — Mathematical Physics
quant-ph — Quantum Physics
stat.TH — Statistics Theory