Integral \(p\)-adic Hodge theory. (English) Zbl 1046.11085

Breuil, Christophe

Usui, Sampei (ed.) et al., Algebraic geometry 2000, Azumino. Proceedings of the symposium, Nagano, Japan, July 20–30, 2000. Tokyo: Mathematical Society of Japan (ISBN 4-931469-20-5/hbk). Adv. Stud. Pure Math. 36, 51-80 (2002).

This paper gives a survey of recent results on integral \(p\)-adic Hodge theory. In an initial section the author reviews the definition and basic properties of semi-stable \(p\)-adic representions of \(G_F\), the Galois group of \(\overline{\mathbb Q}_p\) over \(F\) where \(F\) is a finite extension of \({\mathbb Q}_p\), the \(p\)-adic numbers. He then introduces the idea of a strongly divisible lattice and conjectures that the category of these is equivalent to that of \({\mathbb Z}_p\) lattices in semi-stable \(p\)-adic representations of \(G_F\) under certain circumstances. In the third section of the paper he gives a proof of this conjecture for representations with Hodge-Tate weights between \(0\) and \(1\). Section 4 considers the higher weight case and gives a cohomological interpretation of strongly divisible lattices. The paper concludes with the computation mod \(p\) of Galois stable \({\mathbb Z}_p\) lattices in some semi-stable \(p\)-adic representations.
For the entire collection see [Zbl 1007.00031].

Reviewer: Keith Johnson (Halifax)

MSC:

11S20	Galois theory
14F40	de Rham cohomology and algebraic geometry
14F30	\(p\)-adic cohomology, crystalline cohomology
14F20	Étale and other Grothendieck topologies and (co)homologies
14L05	Formal groups, \(p\)-divisible groups

Keywords:

semi-stable representation; Galois lattice; strongly divisible lattice