La curva de Fargues--Fontaine: Una motivaci\'on al estudio de la teor\'ia de representaciones de Galois $p$-\'adicas

TL;DR

This paper reviews the Fargues-Fontaine curve's role in $p$-adic Hodge theory, highlighting its importance in classifying $p$-adic Galois representations and connecting arithmetic geometry with representation theory.

Contribution

It provides a detailed synthesis of the Fargues-Fontaine curve, its construction via Fontaine period rings, and its application to the classification of $p$-adic Galois representations.

Findings

Analysis of Fontaine period rings and their properties

Connection between the curve and admissible $p$-adic Galois representations

Integration of the curve with Harder-Narasimhan theory

Abstract

This article, written in Spanish, provides a comprehensive review of the Fargues-Fontaine curve, a cornerstone in $p$-adic Hodge theory, and its pivotal role in classifying $p$-adic Galois representations. We synthesize key developments surrounding this curve, emphasizing its connection between advanced concepts in arithmetic geometry and the practical theory of representations. We offer a detailed analysis of the Fontaine period rings ($B_{cris}, B_{st}, B_{dR}$), exploring their crucial algebraic and arithmetic properties and their contribution to the curve's construction and definition. Furthermore, we delve into the theory of admissible $p$-adic Galois representations, discussing how the curve, once defined, integrates with Harder-Narasimhan theory.