Euler Systems Karl Rubin
Euler systemsKarl RubinAuthor address:Department of MathematicsStanford UniversityStanford, CA 94305-2125USAE-mail address: rubin@math.stanford.edu
ContentsIntroductionviiChapter I. Galois cohomology of p-adic representations1. p-adic representations2. Galois cohomology3. Local cohomology groups4. Local duality5. Global cohomology groups6. Examples of Selmer groups7. Global duality11248111317Chapter II. Euler systems: definition and main results1. Euler systems2. Results over K3. Results over K 4. Twisting by characters of finite order2121232630Chapter III. Examples and Applications1. Preliminaries2. Cyclotomic units3. Elliptic units4. Stickelberger elements5. Elliptic curves6. Symmetric square of an elliptic curve33333339404754Chapter IV. Derived cohomology classes1. Setup2. The universal Euler system3. Properties of the universal Euler system4. Kolyvagin’s derivative construction5. Local properties of the derivative classes6. Local behavior at primes not dividing pr7. Local behavior at primes dividing r8. The congruence555558596267687477Chapter V. Bounding the Selmer group1. Preliminaries2. Bounding the order of the Selmer group797980iii
ivCONTENTS3. Bounding the exponent of the Selmer groupChapter VI. Twisting1. Twisting representations2. Twisting cohomology groups3. Twisting Euler systems4. Twisting theorems5. Examples and applications86898991929494Chapter VII. Iwasawa theory1. Outline2. Galois groups and the evaluation map3. The kernel and cokernel of the restriction map4. Proof of Theorem II.3.25. Galois equivariance of the evaluation maps6. Proof of Proposition 1.47. Proof of Proposition 1.69797102105110111115117Chapter VIII. Euler systems and p-adic L-functions1. The setting2. Perrin-Riou’s p-adic L-function and related conjectures3. Connection with Euler systems when d 14. Example: cyclotomic units5. Connection with Euler systems when d 1123123125127129131Chapter IX. Variants1. Rigidity2. Finite primes splitting completely in K /K3. Euler systems of finite depth4. Anticyclotomic Euler systems5. Adding additional local conditions6. Varying the Euler factors133133136136137140141Appendix A. Linear algebra1. Herbrand quotients2. p-adic representations145145146Appendix B. Continuous cohomology and inverse limits1. Preliminaries2. Continuous cohomology3. Inverse limits4. Induced modules5. Semilocal Galois cohomology151151151153155157Appendix C. Cohomology of p-adic analytic groups1. Irreducible actions of compact groups2. Application to Galois representations159159160
CONTENTSvAppendix D. p-adic calculations in cyclotomic fields1. Local units in cyclotomic fields2. Cyclotomic units163163167Bibliography171Notation Index175Subject Index177
viCONTENTS
IntroductionHistory. In 1986, Francisco Thaine [Th] discovered a remarkable method tobound ideal class groups of real abelian extensions of Q. Namely, if F is such afield, he used cyclotomic units in fields F (µ ), for a large class of rational primes ,to construct explicitly a large collection of principal ideals of F . His constructionproduced enough principal ideals to bound the exponent of the different Galoiseigencomponents of the ideal class group of F , in terms of the cyclotomic unitsof F . Thaine’s results were already known (as a Corollary of the proof by Mazurand Wiles [MW] of Iwasawa’s “Main Conjecture”) but Thaine’s proof was verymuch simpler. The author [Ru1] was able to apply Thaine’s method essentiallyunchanged to bound ideal class groups of abelian extensions of imaginary quadraticfields in terms of elliptic units, with important consequences for the arithmetic ofelliptic curves with complex multiplication.Shortly after this, Kolyvagin [Ko1] discovered independently a similar remarkable method, in his case to bound the Selmer group of an elliptic curve. SupposeE is a modular elliptic curve over Q, with sign 1 in the functional equation ofits L-function. Kolyvagin’s method used Heegner points on E over anticyclotomicextensions of prime conductor of an imaginary quadratic field K (in place of cyclotomic units in abelian extensions of Q) to construct cohomology classes over K(in place of principal ideals). He used these cohomology classes, along with dualitytheorems from Galois cohomology, to bound the exponent of the Selmer group ofE over Q. The overall structure of his proof was very similar to that of Thaine.Inspired by Thaine’s work and his own, Kolyvagin then made another fundamental advance. In his paper [Ko2] he introduced what he called “Euler systems.”In Thaine’s setting (the Euler system of cyclotomic units) Kolyvagin showed howto use cyclotomic units in fields F (µr ), for a large class of integers r (no longerjust primes), to bound the orders of the different Galois-eigencomponents of theideal class group of F , rather than just their exponents. Similarly, by using a largercollection of Heegner points in the situation described above, Kolyvagin was able togive a bound for the order of the Selmer group of E. Thanks to the theorem of Grossand Zagier [GZ], which links Heegner points with the L-function of E, Kolyvagin’sbound is closely related to the order predicted by the Birch and Swinnerton-Dyerconjecture.This book. This book describes a general theory of Euler systems for p-adicrepresentations. We start with a finite-dimensional p-adic representation T of theGalois group of a number field K. (Thaine’s situation is the case where T is lim µpntwisted by an even Dirichlet character, and Kolyvagin’s is the case where T is thevii
viiiINTRODUCTIONTate module of a modular elliptic curve.) We define an Euler system for T tobe a collection of cohomology classes in cF H 1 (F, T ), for a family of abelianextensions F of K, with properties relating cF 0 and cF when F F 0 . Our mainresults show how the existence of an Euler system leads to bounds on the orders ofSelmer groups attached to the Galois module Hom(T, µp ), bounds which dependonly on the given Euler system.The proofs of these theorems in this general setting parallel closely (with someadditional complications) Kolyvagin’s original proof. Results similar to ours haverecently been obtained independently by Kato [Ka2] and Perrin-Riou [PR5].What we do not do here is construct new Euler systems. This is the deepestand most difficult part of the theory. Since Kolyvagin’s introduction of the conceptof an Euler system there have been very few new Euler systems found, but each hasbeen extremely important. Kato [Ka3] has constructed a new Euler system for amodular elliptic curve over Q, very different from Kolyvagin’s system of Heegnerpoints (see Chapter III §5). Flach [Fl] has used a collection of cohomology classes(but not a complete Euler system in our sense) to bound the exponent but not theorder of the Selmer group of the symmetric square of a modular elliptic curve.One common feature of all the Euler systems mentioned above is that they areclosely related to special values of L-functions (and thereby to p-adic L-functions).An important benefit of this connection is that the bounds on Selmer groups thatcome out of the theory are then linked to L-values. Such bounds then provideevidence for the Bloch-Kato conjectures [BK], which predict the orders of theseSelmer groups in terms of L-values.Our definition of Euler system says nothing about L-values. If there is an Eulersystem for T then there is a whole family of them (for example, the collection ofEuler system cohomology classes is a Zp -module, as well as a Gal(K̄/K)-module). Ifone multiplies an Euler system by p, one gets a new Euler system but a worse boundon the associated Selmer groups. The philosophy underlying this book, althoughit is explicitly discussed only in Chapter VIII, is that under certain circumstances,not only should there exist an Euler system for T , but there should exist a “bestpossible” Euler system, which will be related to (and contain all the informationin) the p-adic L-function attached to T .A remark about generality. It is difficult to formulate the “most general” definition of an Euler system, and we do not attempt to do this here. The difficultyis partly due to the fact that the number of examples on which to base a generalization is quite small. In the end, we choose a definition which does not cover thecase of Kolyvagin’s Heegner points, because to use a more inclusive definition wouldintroduce too many difficulties. (In Chapter IX we discuss possible modificationsof our definition, including one which does include the case of Heegner points.)On the other hand, we do allow the base field K to be an arbitrary number field,instead of requiring K Q. Although this adds a layer of notation to all proofs,it does not significantly increase the difficulty. A reader wishing to restrict to thesimplest (and most interesting) case K Q should feel free to do so.
INTRODUCTIONixOrganization. In Chapter I we introduce the local and global cohomologygroups, and state the duality theorems, which will be required to state and proveour main results. Chapter II contains the definition of an Euler system, followed bythe statements of our main theorems bounding the Selmer group of Hom(T, µp )over the base field K (§2) and over Zdp -extensions K of K (§3).Chapter III contains sample applications of the theorems of Chapter II. Weapply those theorems to three different Euler systems: the first constructed fromcyclotomic units, to study ideal class groups of real abelian fields (§III.2); thesecond constructed from Stickelberger elements, to study the minus part of idealclass groups of abelian fields (§III.4); and the third constructed by Kato fromBeilinson elements in the K-theory of modular curves, to study the Selmer groupsof modular elliptic curves (§III.5).The proofs of the theorems of Chapter II are given in Chapters IV throughVII. In Chapter IV we give Kolyvagin’s “derivative” construction, taking the Eulersystem cohomology classes defined over abelian extensions of K and using them toproduce cohomology classes over K itself. We then analyze the localizations of thesederived classes, information which is crucial to the proofs of our main theorems.In Chapter V we bound the Selmer group over K by using the derived classes ofChapter IV and global duality. Bounding the Selmer group over K is similar butmore difficult; this is accomplished in Chapter VII after a digression in Chapter VIwhich is used to reduce the proof to a simpler setting.In Chapter VIII we discuss the conjectural connection between Euler systemsand p-adic L-functions. This connection relies heavily on conjectures of PerrinRiou [PR4]. Assuming a strong version of Perrin-Riou’s conjectures, and subjectto some hypotheses on the representation T , we show that there is an Euler systemfor T which is closely related to the p-adic L-function.Chapter IX discusses possible variants of our definition of Euler systems.Finally, there is some material which is used in the text, but which is outsideour main themes. Rather than interrupt the exposition with this material, weinclude it in four appendices.Notation. Equations are numbered consecutively within each chapter. Theorem 4.2 means the theorem numbered 4.2 in section 4 of the current chapter, whileLemma III.2.6 means Lemma 2.6 of Chapter III (and similarly for definitions, etc.).The chapters are numbered I through IX, and the appendices are A through D.If F is a field, F̄ will denote a fixed separable closure of F and GF Gal(F̄ /F ).(All fields we deal with will be perfect, so we may as well assume that F̄ is analgebraic closure of F .) Also F ab will denote the maximal abelian extension of F ,and if F is a local field F ur will denote the maximal unramified extension of F . IfF is a global field and Σ is a set of places of F , FΣ will be the maximal extension ofF which is unramified outside Σ. If K F is an extension of fields, we will writeK f F to indicate that [F : K] is finite.If F is a field and B is a GF -module, F (B) will denote the fixed field of thekernel of the map GF Aut(B), the smallest extension of F whose absolute Galoisgroup acts trivially on B.
xINTRODUCTIONIf O is a ring and B is an O-module then AnnO (B) O will denote theannihilator of B in O. If M O then BM will denote the kernel of multiplicationby M on B, and similarly if M is an ideal. If B is a free O-module and τ is anO-linear endomorphism of B, we will writeP (τ B; x) det(1 τ x B) O[x],the determinant of 1 τ x acting on B.The Galois module of n-th roots of unity will be denoted by µn .If p is a fixed rational prime and F is a field of characteristic different from p,the cyclotomic character εcyc : GF Z p is the character giving the action of GFon µp , and the Teichmüller character ω : GF (Z p )tors is the character givingthe action of GF on µp (if p is odd) or µ4 (if p 2). Hence ω has order at mostp 1 or 2, respectively (with equality if F Q) and hεi ω 1 εcyc takes values in1 pZp (resp. 1 4Z2 ).If B is an abelian group, Bdiv will denote the maximal divisible subgroup of B.If p is a fixed rational prime, we define the p-adic completion of B to be the doubledualBˆ Hom(Hom(B, Qp /Zp ), Qp /Zp )(where Hom always denotes continuous homomorphisms if the groups involvedcomes with topologies). For example, if B is a Zp -module then Bˆ B; if Bis a finitely generated abelian group then Bˆ B Z Zp . In general Bˆ is a Zpmodule and there is a canonical map from B to Bˆ. If τ is an endomorphism of Bthen we will often write B τ 0 for the kernel of τ , B τ 1 for the subgroup fixed byτ , etc.Most of these notations will be recalled when they first occur.Acknowledgments. This book is an outgrowth of the Hermann Weyl lecturesI gave at the Institute for Advanced Study in October, 1995. Some of the workand writing work was done while I was in residence at the Institute for AdvancedStudy and the Institut des Hautes Etudes Scientifiques. I would like to thank boththe IAS and the IHES for their hospitality and financial support, and the NSF foradditional financial support.I am indebted to many people for numerous helpful conversations, especiallyAvner Ash, Ralph Greenberg, Barry Mazur, Bernadette Perrin-Riou, Alice Silverberg, and Warren Sinnott. I would also like to thank Tom Weston and ChristopheCornut for their careful reading of the manuscript and their comments, and theaudiences of graduate courses I gave at Ohio State University and Stanford University for their patience as I was developing this material. Finally, special thanksgo to Victor Kolyvagin and Francisco Thaine for their pioneering work.
CHAPTER IGalois cohomology of p-adic representationsIn this chapter we introduce our basic objects of study: p-adic Galois representations, their cohomology groups, and especially Selmer groups.We begin by recalling basic facts about cohomology groups associated to padic representations, material which is mostly well-known but included here forcompleteness.A Selmer group is a subgroup of a global cohomology group determined by“local conditions”. In §3 we discuss these local conditions, special subgroups ofthe local cohomology groups. In §4 we state without proof the results we needconcerning the Tate pairing on local cohomology groups, and we study how ourspecial subgroups behave with respect to this pairing.In §5 and §6 we define the Selmer group and give the basic examples of idealclass groups and Selmer groups of elliptic curves and abelian varieties. Then in§7, using our local orthogonality results from §4 and Poitou-Tate duality of globalcohomology groups, we derive our main tool (Theorem 7.3) for bounding the sizeof Selmer groups.1. p-adic representationsDefinition 1.1. Suppose K is a field, p is a rational prime, and O is the ring ofintegers of a finite extension Φ of Qp . A p-adic representation of GK Gal(K̄/K),with coefficients in O, is a free O-module T of finite rank with a continuous, O-linearaction of GK .Let D denote the divisible module Φ/O. For a p-adic representation T , we alsodefineV T O Φ,W V /T T O D,WM M 1 T /T Wfor M O, M 6 0,so WM is the M -torsion in W . Note that T determines V and W , and W determinesT lim WM and V , but in general there may be different O-modules T giving riseto the same vector space V .Example 1.2. Suppose ρ : GK O is a character (continuous, but notnecessarily of finite order). Then we can take T Oρ , where Oρ is a free, rank-oneO-module on which GK acts via ρ. Clearly every one-dimensional representationarises in this way. When ρ is the trivial character we get T O, and when O Zp1
2I. GALOIS COHOMOLOGY OF p-ADIC REPRESENTATIONSand ρ is the cyclotomic character εcyc : GK Aut(µp ) Z pwe getT µpn , Zp (1) lim nµpn ,V Qp (1) Qp Zp lim nW (Qp /Zp )(1) µp .For general O we also write O(1) O Zp (1), Φ(1) Φ Qp (1), and D(1) D Zp (1).Definition 1.3. If T is a p-adic representation of GK then so is the dualrepresentationT HomO (T, O(1)).We will also writeV HomO (V, Φ(1)) HomO (T, Φ(1)) T O Φ,W V /T HomO (T, D(1)).Example 1.4. If ρ : GK O is a continuous character as in Example 1.2and T Oρ , then T Oρ 1 εcyc .Example 1.5. Suppose A is an abelian variety defined over K, and p is a primedifferent from the characteristic of K. Then we can take O to be Zp and T to bethe p-adic Tate module of A,Tp (A) limApn nnwhere Apn denotes the p -torsion in A(K̄), and we have rankZp T 2 dim(A). If Aand A0 are isogenous, the corresponding Tate modules T Tp (A) and T 0 Tp (A0 )need not be isomorphic (as GK -modules), but the corresponding vector spaces Vand V 0 are isomorphic.If the endomorphism algebra of A over K contains the ring of integers OF ofa number field F , and p is a prime of F above p, we can also take Φ Fp , thecompletion of F at p, andT Tp (A) limApn nwhich has rank 2 dim(A)/[F : Q] over the ring of integers O of Φ. If A is an ellipticcurve with complex multiplication by F K, this is another source of importantone-dimensional representations.2. Galois cohomologySuppose K is a field. If B is a commutative topological group with a continuousaction of GK , we have the continuous cohomology groupsH i (K, B) H i (GK , B),
2. GALOIS COHOMOLOGY3and if the action of GK factors through the Galois group Gal(K 0 /K) for someextension K 0 of K, we also writeH i (K 0 /K, B) H i (Gal(K 0 /K), B)See Appendix B for the basic facts which we will need about continuous cohomologygroups.Example 2.1. We haveH 1 (K, Qp /Zp ) Hom(GK , Qp /Zp ),H 1 (K, Zp ) Hom(GK , Zp ),and by Kummer theory and Proposition B.2.3, respectivelyH 1 (K, µp ) K (Qp /Zp ),nˆ p,H 1 (K, Zp (1)) limH 1 (K, µpn ) limK /(K )p K Z nnˆ denotes the (p-adically) completed tensor product.where Suppose T is a p-adic representation of GK with coefficients in O as in §1, andM O is nonzero. Recall that V T Φ and W V /T . We will frequently makeuse of the following three exact sequences.M0 WM W W 0M(1)M 10 T T WM 0, 1 yMy(2)0 T V W 0.Lemma 2.2. Suppose M O is nonzero.(i) The sequence (1) induces an exact sequence0 W GK /M W GK H 1 (K, WM ) H 1 (K, W )M 0.(ii) The bottom row of (2) induces an exact sequenceV GK W GK H 1 (K, T )tors 0.(iii) The kernel of the mapH 1 (K, T ) H 1 (K, W ) induced by T ³ T /M T WM , W isM H 1 (K, T ) H 1 (K, T )tors .Proof. Assertions (i) and (ii) are clear, once we show that the kernel of thenatural map H 1 (K, T ) H 1 (K, V ) is H 1 (K, T )tors . But this is immediate fromProposition B.2.4, which says that the map H 1 (K, T ) H 1 (K, V ) induces anisomorphism H 1 (K, V ) H 1 (K, T ) Qp .
4I. GALOIS COHOMOLOGY OF p-ADIC REPRESENTATIONSThe diagram (2) induces an exact commutative diagramMH 1 (K, T ) H 1 (K, T ) H 1 (K, WM ) φ y 1yφ2φ3H 1 (K, T ) H 1 (K, V ) H 1 (K, W )with φ1 induced by M 1 : T V . Sinceker(φ3 ) φ2 (H 1 (K, T )) φ1 (M H 1 (K, T )),we see thatker(φ3 φ1 ) M H 1 (K, T ) ker(φ1 ) M H 1 (K, T ) H 1 (K, T )torswhich proves (iii).3. Local cohomology groups3.1. Unramified local cohomology. Suppose for this section that K is afinite extension of Q for some rational prime . Let I denote the inertia subgroupof GK , let K ur K̄ I be the maximal unramified extension of K, and let Fr Gal(K ur /K) denote the Frobenius automorphism.Definition 3.1. Suppose B is a GK -module. We say that B is unramified ifI acts trivially on B. We define the subgroup of unramified cohomology classes1Hur(K, B) H 1 (K, B) by1Hur(K, B) ker(H 1 (K, B) H 1 (I, B)).Note that if T is as in §1,T is unramified V is unramified W is unramifiedand if the residue characteristic is different from p, then this is equivalent to T ,V , and/or W being unramified.Lemma 3.2. Suppose B is a GK module which is either a finitely generated Zp module, or a finite dimensional Qp -vector space, or a discrete torsion Zp -module.(i) H 1 (K, B) H 1 (K ur /K, B I ) B I /(Fr 1)B I .ur(ii) If the residue characteristic of K is different from p, then1H 1 (K, B)/Hur(K, B) H 1 (I, B)Fr 1 .Proof. The first isomorphism of (i) follows from the inflation-restriction exactsequence (Proposition B.2.5(i)). The second isomorphism of (i) (induced by the mapon cocycles c 7 c(Fr)) is Lemma B.2.8.The hypotheses on B guarantee (see Propositions B.2.5(ii) and B.2.7) that wehave a Hochschild-Serre spectral sequence0 H 1 (K ur /K, B I ) H 1 (K, B) H 1 (I, B)Fr 1 H 2 (K ur /K, B I ).Since Gal(K ur /K) has cohomological dimension one, H 2 (K ur /K, B I ) 0 so thisproves (ii).
3. LOCAL COHOMOLOGY GROUPS5Corollary 3.3. Suppose p 6 and V is a Qp [GK ]-module which has finitedimension as a Qp -vector space.1(i) dimQp (Hur(K, V )) dimQp (V GK ).1(ii) dimQp (H 1 (K, V )/Hur(K, V )) dimQp (H 2 (K, V )).Proof. Using Lemma 3.2(i) we have an exact sequenceFr 110 V GK V I V I Hur(K, V ) 0which proves (i).Since p 6 , I has a unique maximal p-divisible subgroup I 0 and I/I 0 Zpur(see [Fr] §8 Corollary 3). Thus both I and Gal(K /K) have p-cohomologicaldimension one. It follows thatH m (K ur /K, H n (I, V )) 0if m 1 or n 1. Therefore the Hochschild-Serre spectral sequence (PropositionsB.2.5(ii) and B.2.7) shows thatH 1 (K ur /K, H 1 (I, V )) H 2 (K, V ).On the other hand, Lemma 3.2 shows thatH 1 (K ur /K, H 1 (I, V )) H 1 (I, V )/(Fr 1)H 1 (I, V ),H 1 (K, V )/H 1 (K, V ) H 1 (I, V )Fr 1urso there is an exact sequenceFr 11(K, V ) H 1 (I, V ) H 1 (I, V ) H 2 (K, V ) 0.0 H 1 (K, V )/HurThis proves (ii).3.2. Special subgroups. Suppose now that K is a finite extension of someQ , but now we also allow , i.e., K R or C. Let T be a p-adic representation of GK , V T Φ and W V /T as in §1. Following many authors(for example Bloch and Kato [BK] §3, Fontaine and Perrin-Riou [FPR] §I.3.3,or Greenberg [Gr2]) we define special subgroups Hf1 (K, · ) of certain cohomologygroups H 1 (K, · ). We assume first that 6 p, , and discuss the other cases inRemarks 3.6 and 3.7 below.Definition 3.4. Suppose 6 p, 6 , and define the finite part of H 1 (K, V )by1Hf1 (K, V ) Hur(K, V ).Define Hf1 (K, T ) H 1 (K, T ) and Hf1 (K, W ) H 1 (K, W ) to be the inverse imageand image, respectively, of Hf1 (K, V ) under the natural mapsH 1 (K, T ) H 1 (K, V ) H 1 (K, W ).For every M O define Hf1 (K, WM ) H 1 (K, WM ) to be the inverse image ofHf1 (K, W ) under the map induced by the inclusion WM , W .Finally, for V , T , W , or WM define the singular quotient of H 1 (K, · ) byHs1 (K, · ) H 1 (K, · )/Hf1 (K, · )
6I. GALOIS COHOMOLOGY OF p-ADIC REPRESENTATIONSso there are exact sequences0 Hf1 (K, · ) H 1 (K, · ) Hs1 (K, · ) 0.Lemma 3.5. Suppose T is as above and 6 p, 6 . If A is a Zp -module letAdiv denote its maximal divisible subgroup.1(i) Hf1 (K, W ) Hur(K, W )div .11(ii) Hur (K, T ) Hf (K, T ) with finite index and Hs1 (K, T ) is torsion-free.(iii) Writing W W I /(W I )div , there are natural isomorphisms 1(K, W )/Hf1 (K, W ) W/(Fr 1)WHurand 1Hf1 (K, T )/Hur(K, T ) W Fr 1 .(iv) If T is unramified then1Hf1 (K, T ) Hur(K, T )and1Hf1 (K, W ) Hur(K, W ).Proof. It is immediate from the definitions that Hf1 (K, W ) is divisible andHs1 (K, T ) is torsion-free. The exact diagram10 Hur(K, T ) H 1 (K, T ) H 1 (I, T ) yy0 Hf1 (K, V ) H 1 (K, V ) H 1 (I, V ) yy10 Hur(K, W ) H 1 (K, W ) H 1 (I, W )11shows that Hf1 (K, W ) Hur(K, W ) and Hur(K, T ) Hf1 (K, T ). The rest ofassertions (i) and (ii) will follow once we prove (iii), since W I /(W I )div is finite.Note that the image of V I in W I is (W I )div . Taking I-cohomology and thenGal(K ur /K)-invariants of the exact sequence 0 T V W 0 gives an exactsequence0 (W I /(W I )div )Fr 1 H 1 (I, T )Fr 1 H 1 (I, V )Fr 1 .Therefore using Lemma 3.2 we have111(K, T ) H 1 (K, V )/Hur(K, V ))(K, T ) ker(H 1 (K, T )/HurHf1 (K, T )/Hur ker(H 1 (I, T )Fr 1 H 1 (I, V )Fr 1 ) (W I /(W I )div )Fr 1 ,111Hur(K, W )/Hf1 (K, W ) coker(Hur(K, V ) Hur(K, W )) coker(V I /(Fr 1)V I W I /(Fr 1)W I ) W I /((W I )div (Fr 1)W I ).This proves (iii).If T is unramified then W I W is divisible, so (iv) is immediate from (iii).
3. LOCAL COHOMOLOGY GROUPS7Remark 3.6. When the residue characteristic is equal to p, the choice of asubspace Hf1 (K, V ) is much more subtle. Fortunately, for the purpose of workingwith Euler systems it is not essential to make such a choice. However, to understandfully the arithmetic significance of the Selmer groups we will define in §5, and toget the most out of the applications of Euler systems in Chapter III, it is necessaryto choose a subspace Hf1 (K, V ) in the more difficult case p.In this case, Bloch and Kato define Hf1 (K, V ) using the ring Bcris defined byFontaine ([BK] §3). Namely, they define¡ Hf1 (K, V ) ker H 1 (K, V ) H 1 (K, V Bcris ) .For our purposes we will allow an arbitrary special subspace of H 1 (K, V ), which wewill still denote by Hf1 (K, V ). This notation is not as bad as it may seem: in ourapplications we will always choose a subspace Hf1 (K, V ) which is the same as theone defined by Bloch and Kato, but we need not (and will not) prove they are thesame. One could also choose, for example, Hf1 (K, V ) 0 or Hf1 (K, V ) H 1 (K, V ).Once Hf1 (K, V ) is chosen, we define Hf1 (K, T ), Hf1 (K, W ), and Hf1 (K, WM ) interms of Hf1 (K, V ) exactly as in Definition 3.4.Remark 3.7. If K R or C then H 1 (K, V ) 0, so Hf1 (K, V ) 0 andproceeding as above we are led to defineHf1 (K, W ) 0,Hf1 (K, T ) H 1 (K, T ),Hf1 (K, WM ) ker(H 1 (K, WM ) H 1 (K, W )) W GK /M W GK .Note that all of these groups are zero unless K R and p 2.Lemma 3.8. Suppose M O is nonzero.(i) Hf1 (K, WM ) is the image of Hf1 (K, T ) under the mapH 1 (K, T ) H 1 (K, WM )induced by T ³ M 1 T /T WM .1(ii) If 6 p, and T is unramified then Hf1 (K, WM ) Hur(K, WM ).Proof. The diagram (2) gives rise to a commutative diagram with exact rowsMH 1 (K, T ) H 1 (K, T ) H 1 (K, WM ) H 2 (K, T ) 1 yMy(3)H 1 (K, T ) H 1 (K, V ) H 1 (K, W ) H 2 (K, T ).It is immediate from this diagram and the definitions that the image of Hf1 (K, T )is contained in Hf1 (K, WM ).Suppose cWM Hf1 (K, WM ). Then the image of cWM in H 1 (K, W ) is theimage of some cV Hf1 (K, V ). Thus (3) shows that cWM is the image of somecT H 1 (K, T ), and the image of cT in H 1 (K, V ) differs from cV by an element c0of H 1 (K, T ). Therefore cT M c0 Hf1 (K, T ) and cT M c0 maps to cWM . Thisshows that Hf1 (K, WM ) is contained in the image of Hf1 (K, T ), and completes theproof of (i).
8I. GALOIS COHOMOLOGY OF p-ADIC REPRESENTATIONSIf 6 p and T is unramified then11Hf1 (K, WM ) image(Hf1 (K, T )) image(Hur(K, T )) Hur(K, WM )by (i) and Lemma 3.5(iv). Similarly if ιM is the map H 1 (K, WM ) H 1 (K, W )then Lemma 3.5(iv) shows that 1111Hf1 (K, WM ) ι 1M (Hf (K, W )) ιM (Hur (K, W )) Hur (K, WM )which proves (ii).Remark 3.9. We can view WM either as a subgroup of W or as a quotient ofT . Lemma 3.8(i) says that it makes no difference whether we define Hf1 (K, WM )as the inverse image of Hf1 (K, W ) (as we did) or as image of Hf1 (K, T ).Corollary 3.10. There are natural horizontal exact sequences and verticalisomorphisms0 Hf1 (K, W ) limHf1 (K, WM ) H 1 (K, W ) 0 limH (K, WM ) 0H 1 (K, T )Hs1 (K, T ) 0limHf1 (K, WM ) limH 1 (K, WM ) limHs1 (K, WM ) 0Hf1 (K, T ) 0limHs1 (K, WM ) 1M 0 M0Hs1 (K, W ) M M MMProof. The groups inside the inverse limits are finite (Proposition B.2.7(ii)),so the horizontal exact sequences are clear.1The isomorphism H 1 (K, W ) lim H (K, WM ) is a basic fact from Galois co1homology, and the isomorphism Hf1 (K, W ) lim Hf (K, WM ) follows immediately11from the definition of Hf (K, WM ). The isomorphism Hs1 (K, W ) lim Hs (K, WM )now follows.The second set of isomorphisms is similar, except that to handle the inverselimits we use Proposition B.2.3 for the center and Lemma 3.8(i) for the right.4. Local dualitySuppose that either K is a finite extension of Q for some rational prime orK R or C, and T is a p-adic representation of GK .Theorem 4.1 (Local duality). Suppose that either K is nonarchimedean andi 0, 1, 2, or K is archimedean and i 1. Then the cup product and the localinvariant map induce perfect pairingsH i (K, V )iH (K, WM )H i (K, T ) H 2 i (K, V ) H2 i (K, WM) H 2 i (K, W )H 2 (K, Φ(1)) ΦH (K, O(1)/M O(1)) O/M OH 2 (K, D(1)) D. 2
4. LOCAL DUALITY9Proof. See for example [Mi] Corollary I.2.3 or [Se2] §II.5.2 (and use Propositions B.2.3 and B.2.4).Without fe
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