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.