La curva de Fargues--Fontaine: Una motivación al estudio de la teoría de representaciones de Galois $p$-ádicas

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.