# Simple connexit\'e des fibres d'une application d'Abel-Jacobi et corps de classe local

## TL;DR

This paper provides a geometric Langlands proof of the local Langlands correspondence for GL_1, focusing on the Abel-Jacobi morphism and its properties over certain diamonds, advancing understanding in geometric representation theory.

## Contribution

It introduces a geometric proof of the local Langlands correspondence for GL_1 using Abel-Jacobi morphisms and studies their properties over punctured Banach-Colmez spaces.

## Key findings

- Proves the Abel-Jacobi morphism is a pro-étale locally trivial fibration in high degree.
- Studies the structure of absolute punctured Banach-Colmez spaces in detail.
- Establishes geometric properties relevant to the Langlands program.

## Abstract

We give a geometric Langlands type proof of the geometrization conjecture of the local Langlands correspondence introduced by the author for GL_1. For this we study an Abel-Jacobi morphism. We prove that this morphism is a pro-\'etale locally trivial fibration in simply connected diamonds in high degree. Those diamonds are absolute punctured Banach-Colmez spaces that we study in details.