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[1] 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.
- ^ In this article, a local field is complete discrete valuation field whose residue field is perfect.