Some new directions in p-adic Hodge theory

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, Bellaïche, 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. 1. Setup and overview Throughout, K will denote a finite extension of the field Qp of p-adic numbers, and GK = Gal(Qp/K) will denote the absolute Galois group of K. We will write Cp for the completion of Qp; it is algebraically closed, and complete for a nondiscrete valuation. For any field F carrying a valuation (like K or Cp), we write oF for the valuation subring. One may think of p-adic Hodge theory as the p-adic analytic study of padic representations of GK , by which we mean finite dimensional Qp-vector spaces V equipped with continuous homomorphisms ρ : GK → GL(V ). (One might want to allow V to be a vector space over a finite extension of Qp; for ease of exposition, I will only retain the Qp-structure in this discussion.) A typical example of a p-adic representation is the (geometric) p-adic étale cohomology H i et(XQp ,Qp) of an algebraic variety X defined over K. Another typical example is the restriction to GK of a global Galois 286 Kiran S. Kedlaya representation GF → GL(V ), where F is a number field, K is identified with the completion of F at a place above p, and GK is identified with a subgroup of GF ; this agrees with the previous construction if the global representation itself arises as H i et(XF ,Qp) for a variety X over F . Examples of this sort may be thought of as having a “geometric origin”; it turns out that there are many p-adic representations without this property. For instance, there are several constructions that start with the p-adic representations associated to classical modular forms (which do have a geometric origin), and produce new p-adic representations by p-adic interpolation. These constructions include the p-adic families of Hida [11], and the eigencurve of Coleman and Mazur [5]. (Note that these are global representations, so one has to first restrict to a decomposition group to view them within our framework.) Our purpose here is to first recall the basic framework of p-adic Hodge theory, then describe some new results. One important area of application is Colmez’s work on the p-adic local Langlands correspondence for 2-dimensional representations of GQp [6, 7, 8, 9]. 2. The basic strategy The basic methodology of p-adic Hodge theory, as introduced by Fontaine, is to linearize the data of a p-adic representation V by tensoring with a suitable topological Qp-algebra B equipped with a continuous GK-action, then forming the space DB(V ) = (V ⊗Qp B)K of Galois invariants. One usually asks for B to be regular, which means that B is a domain, (FracB)GK = BGK (so the latter is a field), and any b ∈ B for which Qp · b is stable under GK satisfies b ∈ B×. It also forces the map (2.1) DB(V )⊗BGK B→ V ⊗Qp B to be an injection; one says that V is B-admissible if (2.1) is a bijection, or equivalently, if the inequality dimBGK DB(V ) ≤ dimQp V is an equality. In particular, Fontaine defines period rings Bcrys,Bst,BdR; we say V is crystalline, semistable, or de Rham if it is admissible for the corresponding period ring. We will define these rings shortly; for now, consider the following result, conjectured by Fontaine and Jannsen, and proved by FontaineMessing, Faltings, Tsuji, et al. Theorem 2.2. Let V = H i et(XQp ,Qp) for X a smooth proper scheme over K. (a) The representation V is de Rham, and there is a canonical isomorphism of filtered K-vector spaces DdR(V ) ∼= H i dR(X,K). p-adic Hodge theory 287 (b) If X extends to a smooth proper oK-scheme, then V is crystalline. (c) If X extends to a semistable oK-scheme, then V is semistable. In line with the previous statement, the following result was conjectured by Fontaine; its proof combines a result of Berger with a theorem concerning p-adic differential equations due to André, Mebkhout, and the author. Theorem 2.3. Let V be a de Rham representation of GK . Then V is potentially semistable; that is, there exists a finite extension L of K such that the restriction of V to GL is semistable.