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This book is, on the one hand, a pedagogical introduction to the formalism of slopes, of semi-stability and of related concepts in the simplest possible context. It is therefore accessible to any graduate student with a basic knowledge in algebraic geometry and algebraic groups. On the other hand, the book also provides a thorough introduction to the basics of period domains, as they appear in the geometric approach to local Langlands correspondences and in the recent conjectural p-adic local Langlands program. The authors provide numerous worked examples and establish many connections to topics in the general area of algebraic groups over finite and local fields. In addition, the end of each section includes remarks on open questions, historical context and references to the literature.

Geometry, Algebraic. --- Finite fields (Algebra) --- p-adic fields. --- Algebraic fields --- p-adic numbers --- Modular fields (Algebra) --- Algebra, Abstract --- Galois theory --- Modules (Algebra) --- Algebraic geometry --- Geometry --- Symmetric domains. --- Domains, Symmetric --- Functions of several complex variables