Questions and Remarks to the Langlands Program

TL;DR

This paper surveys the classical Langlands correspondence, explores its generalization to higher-dimensional fields, and proposes a conjecture linking automorphic forms and L-functions, aiming to advance understanding of number theory and automorphic representations.

Contribution

It introduces a conjecture on automorphic induction in dimension 2, connecting Langlands correspondences across different dimensions and implying classical conjectures on L-functions.

Findings

Formulation of a direct image (automorphic induction) conjecture.

Linking Langlands correspondences in dimensions 2 and 1.

Implication of the Hasse-Weil conjecture for L-functions.

Abstract

A brief survey is given of the classical Langlands correspondence between n-dimensional representations of Galois groups of local and global fields of dimension 1 and irreducible representations of the groups GL(n). A generalization of the Langlands program to fields of dimension 2 is considered and the corresponding version for 1-dimensional representations is described. We formulate a conjecture on a direct image (=automorphic induction) of automorphic forms which links the Langlands correspondences in dimension 2 and 1. The direct image conjecture implies the classical Hasse-Weil conjecture on the analytical behaviour of the L-functions of curves defined over global fields of dimension 1.