In mathematics, p-adic Hodge theory is a theory that provides a way to classify and study p-adic Galois representations of characteristic 0 local fields with residual characteristic p (such as Qp). The theory has its beginnings in Jean-Pierre Serre and John Tate's study of Tate modules of abelian varieties and the notion of Hodge–Tate representation. Hodge–Tate representations are related to certain decompositions of p-adic cohomology theories analogous to the Hodge decomposition, hence the name p-adic Hodge theory. Further developments were inspired by properties of p-adic Galois representations arising from the étale cohomology of varieties. Jean-Marc Fontaine introduced many of the basic concepts of the field.
General classification of p-adic representations Let K {\displaystyle K} be a local field with residue field k {\displaystyle k} of characteristic p {\displaystyle p} . In this article, a p {\displaystyle p} -adic representation of K {\displaystyle K} (or of G K {\displaystyle G_{K}} , the absolute Galois group of K {\displaystyle K} ) will be a continuous representation ρ : G K → GL ( V ) {\displaystyle \rho :G_{K}\to {\text{GL}}(V)} , where V {\displaystyle V} is a finite-dimensional vector space over Q p {\displaystyle \mathbb {Q} _{p}} . The collection of all p {\displaystyle p} -adic representations of K {\displaystyle K} form an abelian category denoted R e p Q p ( K ) {\displaystyle \mathrm {Rep} _{\mathbb {Q} _{p}}(K)} in this article. p {\displaystyle p} -adic Hodge theory provides subcollections of p {\displaystyle p} -adic representations based on how nice they are, and also provides faithful functors to categories of linear algebraic objects that are easier to study. The basic classification is as follows:
Rep c r y s ( K ) ⊊ Rep s s ( K ) ⊊ Rep d R ( K ) ⊊ Rep H T ( K ) ⊊ Rep Q p ( K ) {\displaystyle \operatorname {Rep} _{\mathrm {crys} }(K)\subsetneq \operatorname {Rep} _{ss}(K)\subsetneq \operatorname {Rep} _{dR}(K)\subsetneq \operatorname {Rep} _{HT}(K)\subsetneq \operatorname {Rep} _{\mathbb {Q} _{p}}(K)}
where each collection is a full subcategory properly contained in the next. In order, these are the categories of crystalline representations, semistable representations, de Rham representations, Hodge–Tate representations, and all p-adic representations. In addition, two other categories of representations can be introduced, the potentially crystalline representations Rep p c r y s ( K ) {\displaystyle \operatorname {Rep} _{\mathrm {pcrys} }(K)} and the potentially semistable representations Rep p s s ( K ) {\displaystyle \operatorname {Rep} _{\mathrm {pss} }(K)} . The latter strictly contains the former which in turn generally strictly contains Rep c r y s ( K ) {\displaystyle \operatorname {Rep} _{\mathrm {crys} }(K)} ; additionally, Rep p s s ( K ) {\displaystyle \operatorname {Rep} _{\mathrm {pss} }(K)} generally strictly contains Rep s s ( K ) {\displaystyle \operatorname {Rep} _{\mathrm {ss} }(K)} , and is contained in Rep d R ( K ) {\displaystyle \operatorname {Rep} _{dR}(K)} (with equality when the residue field of K {\displaystyle K} is finite, a statement called the p-adic monodromy theorem).
… excerpt ends here. Continue reading the full article.
