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Kiran S. KEDLAYA Some new directions in p-adic Hodge theory Tome 21, n o 2 (2009), p Université Bordeaux 1, 2009, tous droits réservés. L

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Kiran S. KEDLAYA Some new directions in p-adic Hodge theory Tome 21, n o 2 (2009), p http://jtnb.cedram.org/item?id=jtnb_ _2_285_0 Université Bordeaux 1, 2009, tous droits réservés. L accès aux articles de la revue «Journal de Théorie des Nombres de Bordeaux» (http://jtnb.cedram.org/), implique l accord avec les conditions générales d utilisation (http://jtnb.cedram. org/legal/). Toute reproduction en tout ou partie cet article sous quelque forme que ce soit pour tout usage autre que l utilisation à fin strictement personnelle du copiste est constitutive d une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques Journal de Théorie des Nombres de Bordeaux 21 (2009), Some new directions in p-adic Hodge theory par Kiran S. KEDLAYA Résumé. Nous rappelons quelques constructions fondamentales de la théorie de Hodge p-adique, et décrivons ensuite quelques résultats nouveaux dans ce domaine. Nous traitons principalement la notion de B-paire, introduite récemment par Berger, qui fournit une extension naturelle de la catégorie des représentations Galoisiennes p-adiques. (Sous une autre forme, cette extension figure dans les travaux récents de Colmez, Bellaïche et Chenevier sur les représentations triangulables.) Nous discutons aussi quelques résultats de Liu qui indiquent que le formalisme de la cohomologie Galoisienne, y compris la dualité locale de Tate, se prolonge aux B-paires. 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, 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 Q p of p-adic numbers, and G K = Gal(Q p /K) will denote the absolute Galois group of K. We will write C p for the completion of Q p ; it is algebraically closed, and complete for a nondiscrete valuation. For any field F carrying a valuation (like K or C p ), we write o F for the valuation subring. One may think of p-adic Hodge theory as the p-adic analytic study of p- adic representations of G K, by which we mean finite dimensional Q p -vector spaces V equipped with continuous homomorphisms ρ : G K GL(V ). (One might want to allow V to be a vector space over a finite extension of Q p ; for ease of exposition, I will only retain the Q p -structure in this discussion.) A typical example of a p-adic representation is the (geometric) p-adic étale cohomology H i et(x Qp, Q p ) of an algebraic variety X defined over K. Another typical example is the restriction to G K of a global Galois 286 Kiran S. Kedlaya representation G F GL(V ), where F is a number field, K is identified with the completion of F at a place above p, and G K is identified with a subgroup of G F ; this agrees with the previous construction if the global representation itself arises as Het(X i F, Q p ) 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 G Qp [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 Q p -algebra B equipped with a continuous G K -action, then forming the space D B (V ) = (V Qp B) G K of Galois invariants. One usually asks for B to be regular, which means that B is a domain, (Frac B) G K = B G K (so the latter is a field), and any b B for which Q p b is stable under G K satisfies b B. It also forces the map (2.1) D B (V ) B G K 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 dim B G K D B (V ) dim Qp V is an equality. In particular, Fontaine defines period rings B crys, B st, B dr ; 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 Fontaine- Messing, Faltings, Tsuji, et al. Theorem 2.2. Let V = H i et(x Qp, Q p ) 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 D dr (V ) = H i dr(x, K). p-adic Hodge theory 287 (b) If X extends to a smooth proper o K -scheme, then V is crystalline. (c) If X extends to a semistable o K -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 G K. Then V is potentially semistable; that is, there exists a finite extension L of K such that the restriction of V to G L is semistable. 3. Period rings We now describe some of the key rings in Fontaine s theory. (We recommend [3] for a more detailed introduction.) Keep in mind that everything we write down will carry an action of G Qp, so we won t say this explicitly each time. Write o for o Cp. The ring o/po admits a p-power Frobenius map; let Ẽ+ be the inverse limit of o/po under Frobenius. That is, an element of Ẽ+ is a sequence x = (x n ) n=0 of elements of o/po with x n = x p n+1. By construction, Frobenius is a bĳection on Ẽ+. If x Ẽ+ is nonzero, then p n v Cp (x n ) is constant for those n for which x n 0. The resulting function v E (x) = lim n pn v Cp (x n ) is a valuation, and Ẽ+ is a valuation ring for this valuation; in particular, Ẽ + is an integral domain (even though o/po is nonreduced). It will be convenient to fix the choice of an element ɛ Ẽ+ such that ɛ n is a primitive p n -th root of unity; then Ẽ+ is the valuation ring in a completed algebraic closure of F p ((ɛ 1)). Given x Ẽ+, let x n o be any lift of x n. The elementary fact that a b (mod p m ) = a p b p (mod p m+1 ) implies first that the sequence (x pn n ) n=0 is p-adically convergent, and second that the limit is independent of the choice of the x n. We call this limit θ(x); the resulting function θ : Ẽ+ o is not a homomorphism, but it is multiplicative. In particular, θ(ɛ) = 1. Let Ã+ = W (Ẽ+ ) be the ring of p-typical Witt vectors with coefficients in Ẽ+. Although this ring is non-noetherian (because the valuation on Ẽ+ is not discrete), one should still think of it as a two-dimensional local ring, since Ã+ /pã+ = Ẽ+. By properties of Witt vectors, the multiplicative map θ lifts to a ring homomorphism θ : Ã+ o taking a Teichmüller lift [x] to θ(x). (The Teichmüller lift [x] is the unique lift of x which has p n -th roots for all n 0.) Also, there is a Frobenius map φ : Ã+ Ã+ lifting 288 Kiran S. Kedlaya the usual Frobenius on Ẽ+. Put B + = Ã+ [ 1 p ]; then θ extends to a map θ : B + C p, and φ also extends. With this, we are ready to make Fontaine s rings. It can be shown that in B +, ker(θ) is principal and generated by ([ɛ] 1)/([ɛ] 1/p 1). Also, Ã + is complete for the ker(θ)-adic topology. However, B + is not (despite being p-adically complete); denote its completion by B + dr. This ring does not admit an extension of φ, because φ is not continuous for the ker(θ)- adic topology. Instead, choose p Ẽ+ with θ( p) = p, form the p-adically completed polynomial ring B + x, and let B + max B + dr be the image of the map B + x B + dr under x [ p]/p (that makes sense because [ p]/p 1 ker(θ)). Then φ does extend to B + max, and we can define B + rig = n 0 φ n (B + max). (Note for experts: we will substitute B + max for B + crys, and this is okay because they give the same notion of admissibility.) Put B + st = B+ max[log[ p]]; this logarithm is only a formal symbol, but there is a logical way to extend φ to it, namely φ(log[ p]) = p log[ p]. To embed B + st into B+ dr, one must choose a branch of the p-adic logarithm (i.e., a value of log(p)), and then map (1 [ p]/p) i log[ p] log(p). i i=1 One defines a monodromy operator N = d/d(log[ p]) on B + st, satisfying Nφ = pφn. Finally, the non-plus variants B max, B st, B dr are obtained from their plus counterparts by adjoining 1/t for t = log([ɛ]) = i=1 (1 [ɛ])i i ker(θ). Note that B dr is a complete discrete valuation field with uniformizer t; it is natural to equip it with the decreasing filtration Fil i B dr = t i B + dr. Also, the ring B e = B φ=1 max sits in the so-called fundamental exact sequence: 0 Q p B φ=1 max B dr /B + dr 0. This is loosely analogous to the exact sequence for u t 1 k t \ k t. 0 k k[u] k((t))/k t 0 p-adic Hodge theory Permuting the steps We note in passing that it is possible to permute the steps of the above constructions, with the aim of postponing the choice of the prime p. This might help one present the theory so that the infinite place becomes a valid choice of prime, under which one should recover ordinary Hodge theory. So far, this is more speculation than reality; we report here only the first steps, leaving future discussion to another occasion. For starters, Ã+ can be constructed as the inverse limit of W (o) under the Witt vector Frobenius map, and θ can be recovered as the composition Ã + W (o) o of the first projection of the inverse limit with the residue map on Witt vectors. (This observation was made by Lars Hesselholt and verified by Chris Davis.) This still involves the use of p in constructing o = o Cp ; that can be postponed as follows. Let Z be the ring of algebraic integers. Let R be the inverse limit of W (Z) under the Witt vector Frobenius; then we get a map θ : R W (Z) Z by composing as above. Choose a prime p of Z above p (thus determining a map Q Q p, up to an automorphism of Q p ), and put ( p) = Z pz p ; in other words, ( p) are the algebraic integers whose p-adic valuation is at least that of p itself. Then Ã+ is the completion of R for the θ 1 ( p)-adic topology. We are still using p via the definition of W using p-typical Witt vectors. One can postpone the choice of p further by using the big Witt vectors, taking the inverse limit under all of the Frobenius maps, instead of the p-typical Witt vectors. In this case, the completion for the θ 1 ( p)-adic topology splits into copies of Ã+ indexed by positive integers coprime to p, which are shifted around by the prime-to-p Frobenius maps. 5. B-pairs as p-adic Hodge structures One of the principal points of departure for ordinary Hodge theory is the comparison isomorphism H Betti(X, Z) Z C = H dr(x, C). One then defines a Hodge structure as a C-vector space carrying the extra structures brought to the comparison isomorphism by the extra structures on both sides: the integral structure on the left side, and the Hodge decomposition on the right side. The notion of a B-pair, recently introduced by Berger [4], performs an analogous function for the comparison isomorphism H i et(x Qp, Q p ) Qp B dr = H i dr (X, K) K B dr, 290 Kiran S. Kedlaya where the extra structures are a Hodge filtration on the right side, and the compatible Galois actions on both sides. A B-pair over K is a pair (W e, W dr + ), where W e is a finite free B e - module equipped with a continuous semilinear G K -action, and W dr + is a finite free B + dr -submodule of W dr = W e Be B dr stable under G K such that W dr + B + B dr W dr is an isomorphism. (Note that B e is a Bézout domain, i.e., an integral domain in which finitely generated ideals are dr principal [4, Proposition 1.1.9].) The rank of a B-pair W is the common quantity rank Be W e = rank B + W dr +. dr Lemma 5.1. Every B-pair W of rank 1 over K can be uniquely written as (B e (δ), t i B + dr (δ)) for some character δ : G K Q p and some i Z. Proof. See [4, Lemme 2.1.4]. We refer to the integer i as the degree of W. If W has rank d 1, we define the degree of W as the degree of the determinant det W = d W. We refer to the quotient µ(w ) = (deg W )/(rank W ) as the average slope of W. (This is meant to bring to mind the theory of vector bundles on curves; see the next section for an extension of this analogy.) There is a functor V W (V ) = (V Qp B e, V Qp B + dr ), from p-adic representations to B-pairs; it has the one-sided inverse W V (W ) = W e W + dr, and so is fully faithful. However, not every B-pair corresponds to a representation; for instance, in Lemma 5.1, only those B-pairs with i = 0 correspond to representations. We will return to the structure of a general B-pair in the next two sections. One can generalize many definitions and results from p-adic representations to B-pairs. For instance, we set for {max, st, dr}, the map D max (W ) = (W e Be B max ) G K D st (W ) = (W e Be B st ) G K D dr (W ) = (W e Be B dr ) G K ; (5.2) D (W ) G B K B V Qp B is injective, and we say that W is -admissible if (5.2) is a bĳection. In these terms, we have the following extension of Theorem 2.3, again due to Berger [2]. (The closest analogue of this result in ordinary Hodge theory is a theorem of Borel [18, Lemma 4.5], that any polarized variation of rational Hodge structures is quasi-unipotent.) p-adic Hodge theory 291 Theorem 5.3. Let W be a de Rham B-pair over K. Then there exists a finite extension L of K such that the restriction of W to L is semistable. Moreover, the B-pairs which are already crystalline over K can be described more explicitly; they form a category equivalent to the category of filtered φ-modules over K. (Such an object is a finite dimensional K 0 - vector space V, for K 0 the maximal unramified extension of K, equipped with a semilinear φ-action and an exhaustive decreasing filtration Fil i V K of V K = V K0 K.) 6. Sloped representations For h a positive integer and a Z coprime to h, we define an (a/h)- representation as a finite-dimensional Q p h-vector space V a,h equipped with a semilinear G K -action and a semilinear Frobenius action φ commuting with the G K -action, satisfying φ h = p a. For instance, we may view a p-adic representation as a 0-representation by taking φ = id. We say that the B-pair W is isoclinic of slope a/h if it occurs in the essential image of W a,h ; we say that W is étale if it is isoclinic of slope 0. Theorem 6.1. The functor V a,h W a,h (V a,h ) = ((B max Qp h V a,h) φ=1, B + dr Q p h V a,h) from (a/h)-representations to B-pairs of slope a/h is fully faithful. (Here Q p h denotes the unramified extension of Q p with residue field F p h.) Proof. See [4, Théorème 3.2.3] for the construction of a one-sided inverse. Using the equivalence of categories between B-pairs and (φ, Γ)-modules (Theorem 9.1), one deduces the following properties of isoclinic B-pairs. Lemma 6.2. (a) If 0 W 1 W W 2 0 is exact and any two of W 1, W 2, W are isoclinic of slope s, then so is the third. (b) Let W 1, W 2 be isoclinic B-pairs of slopes s 1 s 2. Then Hom(W 1, W 2 ) = 0. (c) Let W 1, W 2 be isoclinic B-pairs of slopes s 1, s 2. Then W 1 W 2 is isoclinic of slope s 1 + s 2. Proof. See Theorem 1.6.6, Corollary 1.6.9, and Corollary 1.6.4, respectively, of [12]. The following is a form of the author s slope filtration theorem for Frobenius modules over the Robba ring. Theorem 6.3. Let W be a B-pair. Then there is a unique filtration 0 = W 0 W l = W of W by B-pairs with the following properties. 292 Kiran S. Kedlaya (a) For i = 1,..., l, the quotient W i /W i 1 is a B-pair which is isoclinic of slope s i. (b) We have s 1 s l. Proof. See [12, Theorem 1.7.1]. A corollary is that a B-pair W is semistable (in the sense of vector bundles, up to a reversal of the sign convention), meaning that if and only if W is isoclinic. 0 W 1 W = µ(w 1 ) µ(w ), Corollary 6.4. The property of a B-pair having all slopes s (resp. s) is stable under extensions. 7. Trianguline B-pairs One might reasonably ask why the study of B-pairs should be relevant to problems only involving p-adic representations. One answer is that there are many p-adic representations which become more decomposable when viewed as B-pairs. Specifically, following Colmez [6], we say that a B-pair W is trianguline (or triangulable) if W admits a filtration 0 = W 0 W l = W in which each quotient W i /W i 1 is a B-pair of rank 1. For instance, by a theorem of Kisin [13], if V is the two-dimensional representation corresponding to a classical or overconvergent modular form of finite slope, then W (V ) is trianguline. As one might expect, the extra structure of a triangulation makes trianguline B-pairs easier to classify. For example, Colmez has classified all two-dimensional trianguline B-pairs over Q p, by calculating Ext(W 1, W 2 ) whenever W 1, W 2 are B-pairs of rank 1. He has also shown that their L- invariants (in the sense of Fontaine-Mazur) can be read off from the triangulation. A study of the general theory of trianguline B-pairs has been initiated by Bellaïche and Chenevier [1], with the aim of applying this theory to the study of Selmer groups associated to Galois representations of dimension greater than 2 (e.g., those arising from unitary groups). It is also hoped that this study will give insight into questions like properness of the Coleman-Mazur eigencurve. One feature apparent in the work of Bellaïche and Chenevier, connected to the results of the next section, is that the trianguline property of a representation is reflected by the structure of the corresponding deformation ring. (A related notion is Pottharst s definition of a triangulordinary representation [17], which generalizes the notion of an ordinary representation in a manner useful when considering duality of Selmer groups.) p-adic Hodge theory Cohomology of B-quotients In this section, we describe a theorem of Liu [15] generalizing Tate s fundamental results on the Galois cohomology of p-adic representations, and also the Ext group calculations of Colmez mentioned above. However, to do this properly, we must work with a slightly larger category than the B-pairs, because this category is not abelian: it contains kernels but not cokernels. One can construct a minimal abelian category containing the B-pairs as follows. Define a B-quotient as an inclusion (W 1 W 2 ) of B-pairs. We put these in a category in which the morphisms from (W 1 W 2 ) to (W 1 W 2 ) consist of subobjects X of W 2 W 2 containing W 1 W 1, such that the composition X W 2 W 2 W 2 is surjective and the inverse image of W 1 is W 1 W 1. (One must also define addition and composition of morphisms, but the reader should have no trouble reconstructing them.) It can be shown that this yields an abelian category, into which the B-pairs embed by mapping W to 0 W. For W = (W 1 W 2 ) a B-quotient, we define the rank of W as rank(w 2 ) rank(w 1 ), and we say W is torsion if rank(w

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