Abstract

$p$-adic Hodge theory originated in the 1960s and has since become a fundamental tool in arithmetic geometry, with applications ranging from the Langlands program to homotopy theory and birational geometry. In recent years, its basic structure has been reorganized---and significantly better understood---in the language of algebraic stacks. This geometrization brings tools from algebraic geometry to bear on a questions of an arithmetic nature, e.g., Tate's duality for the cohomology of Galois modules (which includes local class field theory) is recast as Serre's duality for coherent sheaves. These lectures give an overview of this story, highlighting key examples where the algebro-geometric perspective yields new insights.