Hodge Theory of $p$-adic varieties: a survey

TL;DR

This survey reviews the development and applications of $p$-adic Hodge Theory in arithmetic geometry, covering algebraic and analytic varieties and highlighting recent advances and number theory applications.

Contribution

It provides a comprehensive overview of $p$-adic Hodge Theory, including recent progress and its relevance to number theory, serving as an updated survey for researchers.

Findings

Summarizes key results in $p$-adic Hodge Theory for algebraic varieties.

Discusses recent advances in $p$-adic Hodge Theory for analytic varieties.

Highlights applications to number theory problems.

Abstract

$p$-adic Hodge Theory is one of the most powerful tools in modern Arithmetic Geometry. In this survey, we will review $p$-adic Hodge Theory for algebraic varieties, present current developments in $p$-adic Hodge Theory for analytic varieties, and discuss some of its applications to problems in Number Theory. This is an extended version of a talk at the Jubilee Congress for the 100th anniversary of the Polish Mathematical Society, Krak\'ow, 2019.