Some new directions in p-adic Hodge theory

TL;DR

This paper reviews foundational concepts in p-adic Hodge theory and discusses recent advances, especially the introduction of B-pairs, which extend the category of p-adic Galois representations and have implications for Galois cohomology.

Contribution

It introduces and explores the concept of B-pairs, expanding the framework of p-adic Galois representations and connecting to recent research on trianguline representations.

Findings

B-pairs provide a natural enlargement of p-adic Galois representations.

Galois cohomology formalism extends to B-pairs, including Tate local duality.

Recent results link B-pairs to trianguline representations.

Abstract

We recall some basic constructions from p-adic Hodge theory, then describe some recent results in the subject. We chiefly discuss the notion of B-pairs, introduced recently by Berger, which provides a natural enlargement of the category of p-adic Galois representations. (This enlargement, in a different form, figures in recent work of Colmez, Bellaiche, and Chenevier on trianguline representations.) We also discuss results of Liu that indicate that the formalism of Galois cohomology, including Tate local duality, extends to B-pairs.