A remark on the Langlands correspondence for tori

TL;DR

This paper refines the understanding of Langlands's correspondence for algebraic tori by analyzing its topological structure, showing the map's properties as a Lie group homomorphism and characterizing unramified characters.

Contribution

It provides a detailed topological analysis of Langlands's homomorphism for tori, establishing its structure as a complex Lie group homomorphism and characterizing unramified characters.

Findings

Langlands's map is a surjective, finite-to-one homomorphism of abelian complex Lie groups.

Unramified characters form the identity component of the character space.

The unramified characters correspond to Galois invariants of the dual torus.

Abstract

For an algebraic torus defined over a local (or global) field $F$, a celebrated result of R.P. Langlands establishes a natural homomorphism from the group of continuous cohomology classes of the Weil group, valued in the dual torus, onto the space of complex characters of the rational points of the torus (or automorphic characters in the global case). We expand on this result by detailing its topological aspects. We show that if we topologize the relevant spaces of continuous homomorphisms and continuous cochains using the compact-open topology, Langlands's map becomes a (surjective, finite-to-one) homomorphism of abelian complex Lie groups. Moreover, we demonstrate that, in both the local and global settings, the subset of unramified characters is the identity component of the relevant space of characters. Finally, we compare the group of unramified characters with the Galois (co)invariants of the dual torus.