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Fontaine's rings and p-adic L-functions Tung, Shen-Ning 2015

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Fontaine’s rings and p-adic L-functionsbyShen-Ning TungA THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinTHE FACULTY OF GRADUATE AND POSTDOCTORIAL STUDIES(Mathematics)The University of British Columbia(Vancouver)February 2015c© Shen-Ning Tung, 2015AbstractIn the first part, we introduce theory of p-adic analysis for one variable p-adic functions and then usethem to construct Kubota-Leopoldt p-adic L-functions.In the second part, we give a description of the Iwasawa modules attached to p-adic Galois represen-tations of the absolute Galois group of K in terms of the theory of (ϕ,Γ)-modules of Fontaine. Whenthe representation is de Rham when K be finite extension of Qp. This gives a natural construction of theexponential map of Perrin-Riou which is used in the construction and the study of p-adic L-functions.In the third part, we give formulas for Bloch-Kato’s exponential map and its dual for an alsolutelycrystalline p-adic representation V . As a corollary of these computation, we can give a improved descrip-tion of Perrin-Riou’s exponential map, which interpolates Bloch-Kato’s exponentials for the twists of V.Finally we use this map to reconstruct Kubota-Leopoldt p-adic L-functions.iiPrefaceThis thesis is a reorganisation of classical materials on Perrin-Riou’s big reciprocity law maps and theoryof (ϕ,Γ) module. The author is benefited from papers of Cherbonnier, Colmez, Berger, Loeffler andZerbes. The topic of this thesis was suggested by my supervisor, Professor Sujatha Ramdorai.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 One variable p-adic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1 Functions on Zp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Distributions on Zp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Operations on the distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Kubota-Leopoldt p-adic L-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.1 The Riemann zeta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2 Kummer congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3 p-adic Mellin transform and Leopoldt’s Γ-transform . . . . . . . . . . . . . . . . . . . . . 103.4 Construction of the Kubota-Leopoldt zeta function . . . . . . . . . . . . . . . . . . . . . 133.5 The residue at s = 1 and the p-adic zeta function . . . . . . . . . . . . . . . . . . . . . . 143.6 Dirichlet L-function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.7 p-adic L-function attached to Dirichlet character . . . . . . . . . . . . . . . . . . . . . . . 153.8 Behavior at s = 1 of Dirichlet L-function . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.9 Twist by a character of conductor power of p . . . . . . . . . . . . . . . . . . . . . . . . 174 (ϕ ,Γ)-modules and p-adic representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.1 The field E˜ and its subrings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20iv4.2 The field B˜ and its subrings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.3 (ϕ,Γ)-module and Galois representations . . . . . . . . . . . . . . . . . . . . . . . . . . 225 (ϕ,Γ)-modules and Galois cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.1 The complex Cϕ,γ(K,V ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.2 The operator ψ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.3 The compactness of D(V )ψ=1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.4 The module D(V )ψ−1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 Iwasawa theory and p-adic representations . . . . . . . . . . . . . . . . . . . . . . . . . . . 326.1 Iwasawa cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326.2 Corestriction and (ϕ,Γ)-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336.3 Interpretation of D(V )ψ=1 and D(V )ψ−1 in Iwasawa theory . . . . . . . . . . . . . . . . . . . 357 De Rham representations and overconvergent representations . . . . . . . . . . . . . . . . 377.1 De Rham representations and crystalline representations . . . . . . . . . . . . . . . . . . 377.2 Overconvergent elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387.3 Overconvergent representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 Explicit reciprocity laws and de Rham Representation . . . . . . . . . . . . . . . . . . . . . 418.1 The Bloch-Kato exponential map and its dual . . . . . . . . . . . . . . . . . . . . . . . . 418.2 Explicit reciprocity law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438.3 Connection with the Perrin-Riou’s logarithm . . . . . . . . . . . . . . . . . . . . . . . . . 459 The Qp(1) representation and Coleman’s power series . . . . . . . . . . . . . . . . . . . . . 479.1 The module D(Zp(1))ψ=1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479.2 Kummer theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479.3 Multiplicative representatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489.4 Generalized Coleman’s power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499.5 The map LogQp(1) and Exp∗Qp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509.6 Cyclotomic units and Coates-Wiles homomorphisms . . . . . . . . . . . . . . . . . . . . 5210 (ϕ ,Γ)-modules and differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5510.1 The rings Bmax and B˜+rig . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5510.2 The structure of D(T )ψ=1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5610.3 p-adic representations and differential equations . . . . . . . . . . . . . . . . . . . . . . . 5710.4 The Fontaine isomorphism revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5810.5 Iwasawa algebra and power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6010.6 Iwasawa algebras and differential equations . . . . . . . . . . . . . . . . . . . . . . . . . 61v10.7 Bloch-Kato’s exponential maps: Three explicit reciprocity formulas . . . . . . . . . . . . 6310.8 The Bloch-Kato’s exponential map and its dual revisited . . . . . . . . . . . . . . . . . . 6310.9 Perrin-Riou’s big exponential map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6710.10The explicit reciprocity formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7011 Perrin-Riou’s big regulator map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7211.1 Perrin-Riou’s big logarithm map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7211.2 Cyclotomic units and Kubota-Leopoldt p-adic L-functions . . . . . . . . . . . . . . . . . 75Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78viAcknowledgmentsI would like to thank my supervisors Sujatha Ramdorai for her continuous support and guidance, forsuggesting me this topic and giving a seminar on this subject and for the innumerable things that I havelearned from them throughout graduate school.I owe an enormous debt of gratitude to Ming-Lun Hsieh, from whom I have learned so much andhelping me finishing this thesis. Ming-Lun has always been unfailingly patient and willing to share hisknowledge and expertise. I have benefitted tremendously from his intellectual generosity, attentiveness todetail, and time, and his feedback on multiple drafts of this thesis has led to significant improvements inclarity and exposition. As an undergraduate at National Taiwan University, I was fortunate to have terrificmentors who encouraged me to develop my passions and mathematical foundation.It is a pleasure and privilege to thank Subhajit Juna and Li Zheng for many helpful discussions and ex-changes related to this project. We have shared conversations and experiences that have been an invaluablepart of this journey. I also thank Justin Scarfy for pointing out several errors in my previous draft.viiChapter 1Introduction1.1 OverviewWe fix an algebraic closure Qp of Qp. If K is a finite extension of Qp and n ∈ N, we put Kn = K(µp)with µp the p-th root of unity and we denote by K∞ the union of Kn. We denote by GK the Galois groupGal(Qp/K) and HK the kernel of the restriction of χ on GK where χ : GQp → Z∗p is the cyclotomic units.Thus HK = Gal(Qp/K∞) and we put ΓK = GK/HK = Gal(K∞/K).The theory of (ϕ,Γ)-modules, introduced by Fontaine, associate a p-adic representation V of HK toa module D(V ) over a dimension 2 local field (ring of Laurent series in one variable with coefficient in afinite unmarried extension of Qp with a Frobenius action ϕ and a left invere ψ of ϕ . If V is a representationof GK , then the module D(V ) is endowed with an residue action ΓK commuting with ϕ . The reason thistheory is interested is because we can reconstruct V via D(V ), which is a more amenable object. Indeed,since ΓK is procyclic, the structure of D(V ) is totally decided by the action of two operators (ϕ and thegenerator of ΓK) verifying the commutative relation. Thus, to study the p-adic Galois representations ofGK is equivalent to (ϕ,Γ)-modules. Two main results we will prove are1. the construction of the isomorphism Exp∗V ∗(1) from the Iwasawa module H1K,V =Qp⊗(lim←−H1(Kn,T ))to D(V )ψ=1 where T is a Galois fixed lattice in V.2. an explicit reciprocity law of Exp∗V ∗(1) where V is a de Rham representation in terms of the theBloch-Kato’s exponential maps.The first one using the result of Herr, which gives a description of H1Iw(K,V ) in terms of D(V ). Inthe case V = Qp(1) and K is unramified extension over Qp, the map Exp∗V ∗(1) can be described in termsof derivatives of classical Coleman’s power series (See [8]). We therefore obtain a generalized Colemanpower series without any restriction of K. To prove the explicit reciprocity law, we use the fact that everyrepresentation of GK is overconvergent. This is an ingredient permits us to link D(V ) and the of DdR(V ).1When K is unramified extension of Qp and V is crystalline representation of GK , Perrin-Riou hasconstructed in [18] a period map ΩV,h which interpolates the expK,V (k) as k runs over the positive integers.It is a generalization of Coleman’s map. Using the inverse of Perrin-Riou’s map, one can then associate toan Euler system a p-adic L-function. If one starts with V = Qp(1), then Perrin-Riou’s map is the inverseof the Coleman isomorphism and one recovers Kubota-Leopoldt’s p-adic L-functions.1.2 Structure of the thesisIn chapter 2-3, following [11] and [12], we introduce the theory of one variable p-adic functions and useit to construct Kubota-Leopoldt p-adic L-functions which interpolate the value of Dirichlet L-functions atnegative integers.In chapter 4-6, we begin with background of (ϕ,Γ)-modules and its relation with p-adic Galois repre-sentations. Following [14], we can use Herr complexes to compute Galois cohomology groups of p-adicGalois representations. Finally, we prove the Fontaine’s isomorphism in [6] which relates Iwasawa coho-mology groups of p-adic Galois representations with its associated (ϕ,Γ)-modules.In chapter 7-8, we introduce Fontaine’s period rings and overconvergent representations. Following[5], we can compare DdR and D†. We then introduce Bloch-Kato’s exponential map and its dual in [4] anduse overconvergent representation to deduce Colmez’s explicit reciprocity law in [6].In chapter 9, we recall the definition of Coleman’s power series in [8]. Following [6], we study itsrelation with the (ϕ,Γ)-modules associated to Zp(1). Combining with Colmez’s explicit reciprocity laws,we reinterpret classical Coates-Wiles homomorphisms.In chapter 10-11, we first introduce the Robba ring B†rig, which can be used to reinterpret Fontaine’sisomorphism. By [2], we can get three explicit reciprocity formulas for Bloch-Kato’s exponential mapand its dual and use these explicit reciprocity formulas to deduce Perrin-Riou’s big exponential map in[18].In chapter 12, we follows the variance in appendix of [17], which allows us to add a finite order char-acter in the inverse of Perrin-Riou’s big exponential map constructed in chapter 10. Via this map, we canreconstruct Kubota-Leopoldt p-adic L-functions via cyclotomic units.2Chapter 2One variable p-adic functions2.1 Functions on ZpLet C 0(Zp,L) be the space of continuous functions from Zp to L. Since Zp is compact, every contin-uous function on Zp is bounded. This allows us to define a valuation vC 0 on C0(Zp,L) by vC 0(φ) =infx∈Zp(φ(x)), which makes C 0(Zp,L) an L-Banach space.If n ∈ N, let(xn)be the polynomial defined by(xn)=1 if n = 0x(x−1)···(x−n+1)n! if n≥ 1.Theorem 2.1.1. (Mahler) {(xn),n∈ N} forms a Banach basis of C 0(Zp,L).If h ∈ N, let LAh(Zp,L) be the space of functions from Zp to L which is analytic on a+ phZp for alla ∈ Zp. i.e. if φ ∈ LAh(Zp,L), x0 ∈ Zp, then φ can be written as the formφ(x) =∞∑k=0ak(φ ,x0)(x− x0)k ∀x ∈ x0 + phZp,where ak(x0,φ) is a sequence in L such that νp(ak(φ ,x0))+ kh tends to +∞ as k tends to +∞. We endowLAh(Zp,L) a valuation vLAh defined byvLAh(φ) = infx0∈Zpinfk∈Nνp(ak(φ ,x0))+ kh,which makes LAh(Zp,L) an L-Banach space. One can show that vLAh(φ) = infa∈S infk∈Nνp(φ ,ak(a)) + khwhere S is a representative of Zp/phZp. (See [12, remark I.4.4.])We denote LA(Zp,L) the space of locally analytic functions on Zp. Since Zp is compact, it is aninductive limit of LAh(Zp,L), h ∈ N, and we endow it with the inductive limit topology.3Theorem 2.1.2. (Amice) {[ nph ]!(xn),n ∈ N} forms a Banach basis of LAh(Zp,L), where [ ] is the Gausssymbol and ! is the factorial.Theorem 2.1.3. The function φ =∑+∞n=0 an(φ)(xn)∈C 0(Zp,L) is in LA(Zp,L) if and only if liminfn7→∞ 1nνp(an(φ))>0.A function φ : Zp→ L is differentiable at x0 ∈ Zp if limh→0 φ(x0+h)−φ(x0)h exists. The limit is denotedby φ ′(x0). A function is said to be differentiable of order 1 if it is differentiable at all x0 ∈ Zp. Inductively,we say that a function is differentiable of order k if its differentiation is of order k−1.If r ≥ 0, we say that φ : Zp→ L is of class C r if there exist functions φ ( j) : Zp→ L for 0 ≤ j ≤ [r],such that, if we define εφ ,r : Zp×Zp→ L and Cφ ,r : N→ R∪{+∞} byεφ ,r(x,y) = φ(x+ y)−[r]∑j=0φ ( j)(x)yjj!and Cφ ,r(h) = infx∈Zp,y∈phZpνp(εφ ,r(x,y))− rh,then Cφ ,r(h) tends to +∞ as h tends to +∞.We denote C r(Zp,L) the set of functions φ : Zp→ L of class C r. We endow C r(Zp,L) the valuationvC r defined byvC r(φ) = inf(inf0≤ j≤[r],x∈Zpνp(φ ( j)(x)j!), infx,y∈Zpνp(εφ ,r(x,y)− rνp(y))),which makes it an L-Banach space.Proposition 2.1.4. If h ∈ N, and if r ≥ 0, then LAh(Zp,L) ⊂ C r(Zp,L). Moreover, if φ ∈ LAh(Zp,L),thenvC r(φ)≥ vLAh(φ)− rh.Proof. See [12, proposition I.5.7].If i ∈ N, we denote l(i) the least integer n such that pn > i, then we havel(0) = 0 and l(i) = [log ilog p]+1, if i≥ 1.Theorem 2.1.5. (Mahler) The function φ = ∑+∞n=0 an(φ)(xn)∈ C 0(Zp,L) is in C r(Zp,L), r ≥ 0 if and onlyif νp(an(φ))− rl(n)→+∞ as n→+∞. Moreover, the valuation v′C r defined on C r(Zp,L) by the formulav′C r(φ) = infn∈N(νp(an(φ))− rl(n))is equivalent to the valuation vC r .Proof. See [12, proposition I.5.18].4Corollary 2.1.6. p[rl(n)](xn), n ∈ N forms a Banach basis of C r(Zp,L).2.2 Distributions on ZpA continuous distribution on Zp is a continuous linear function on LA(Zp,L), that is, a linear functionon LA(Zp,L) whose restriction to LAh(Zp,L) is continuous. We denote D(Zp,L) the set of continuousdistributions on Zp with values in L and endow D(Zp,L) with the Fre´chet topology defined by the familyof valuations vLAh , h ∈ N.Given a continuous distribution µ , we associate it with the formal series:Aµ(T ) =∫Zp(1+T )xµ =+∞∑n=0T n∫Zp(xn)µ,which is called the Amice transform of µ . Hence we define a map µ 7→Aµ from continuous distributionsto formal power series.Lemma 2.2.1. If µ ∈D(Zp,L) and if νp(x) > 0, then∫Zp(1+ z)xµ(x) =Aµ(z).Let R+ be the ring of power series f = ∑∞n=0 anTn with coefficients in L, which is convergent ifνp(T ) > 0.We say that an element f = ∑∞n=0 anTn ∈R+ is of order r if νp(an)+ rl(n) is bounded. We denoteby R+h the subset of R+ of elements of order r, and we endow R+r the valuation vr defined by vr( f ) =infn∈N νp(an)+ rl(n), which makes it a L-Banach space. We endow R+ the Fre´chet topology defined bythe family of valuations vr.Theorem 2.2.2. The map µ 7→Au is an isomorphism of Fre´chet space from D(Zp,L) to R+.Proof. See [12, Theorem II.2.2].If r ≥ 0, we say a continuous distribution µ on Zp is of order r if it can be extended by continuity toC r. We denoteDr(Zp,L) the set of distributions of order r, which is equipped with a valuation vDr definedbyvDr(µ) = inff∈C r(Zp,L)−{0}(νp(∫Zpfµ)− vC r( f )),which gives Dr(Zp,L) the dual topology of C r(Zp,L).A distribution is said to be tempered if there exist r ∈ R+ such that it is of order r. We denoteDtemp(Zp,L) the space of tempered distributions.Proposition 2.2.3. The map µ 7→Aµ induces an isometry from Dr(Zp,L) equipped with valuation vDr toR+r equipped with valuation vr.5A distribution of order 0 is called measure. By definition, D0(Zp,L) is the topological dual of thespace of continuous functions. By proposition 2.2.3, we have a one-one correspondence from a measureto a power series of bounded coefficients.To sum up, we haveC 0 ⊃ C r ⊃ LA⊃ LArD0 ⊂Dr ⊂D ⊂ LA∗r2.3 Operations on the distributions1. Haar measure: µ(Zp) = 1 and µ is invariant by translation. We must have µ(i+ pnZp) = 1pn which isnot bounded. Hence there exists no Harr measure on Zp.2. Dirac measure: For a ∈ Zp, we define δa the Dirac measure associated to f (a). The Amice transformof δa is Aδa(T ) = (1+T )a.3. Multiplication by a function: If µ is a distribution on Zp and f is a locally analytic function on Zp, wedefine the distribution fµ by∫Zp φ( fµ) =∫Zp( fφ)µ .• Multiplication by x: We have x ·(xn)= ((x−n)+n)(xn)= (n+1)( xn+1)+n(xn), and hence we haveAxµ(T ) = ∂Aµ where∂ = (1+T )ddT.• Multiplication by zx when νp(z−1)> 0: By lemma 2.2.1, if νp(y−1)> 0, and if λ is a continuousdistribution on Zp, then∫Zp yxλ (x)=Aλ (y−1). Applying this to λ = zxµ , we obtainAλ (y−1)=Aµ(yz−1). We have the formulaAzxµ(T ) =Aµ((1+T )z−1).4. Restriciton to compact open subset: If X is a compact open subset of Zp, then the characteristic function1X is continuous on Zp. If µ is a distribution on Zp, the measure 1Xµ is the restriction of µ to X and isdenoted by ResX(µ). In particular for n∈N and a∈Zp, we have 1a+pnZp(x) = p−n∑zpn=1 z−azx, henceAResa+pnZp (µ)(T ) = p−n ∑zpn=1z−aAµ((1+T )z−1).5. Derivation of distribution: If µ ∈D(Zp,L), we define dµ by∫Zpφ(x)dµ =∫Zpφ ′(x)µ, and therefore Adµ(T ) = log(1+T ) ·Aµ(T ).66. Actions of Z∗p, ϕ and ψ:• If a ∈ Z∗p, and if µ ∈D(Zp,L), we define σa(µ) ∈D(Zp,L) by∫Zpφ(x)σa(µ) =∫Zpφ(ax)µ, and thereforeAσa(µ)(T ) =Aµ((1+T )a−1).• ϕ acts on distribution µ by∫Zpφ(x)ϕ(µ) =∫Zpφ(px)µ, and thereforeAϕ(µ)(T ) =Aµ((1+T )p−1).• If µ is a distribution on Zp, we denote ψ(µ) the distribution on Zp defined by∫Zpφ(x)ψ(µ) =∫pZpφ(p−1x)µ and therefore Aψ(µ) = ψ(Aµ),where ψ :R+→R+ is defined by ψ(F)((1+T )p−1) = 1p ∑ζ p=1 F((1+T )ζ −1).The action of Z∗p, ϕ and ψ satisfy the relations:(a) ψ ◦φ =id.(b) ψ ◦σa = σa ◦ψ and ϕ ◦σa = σa ◦ϕ if a ∈ Z∗p.(c) ψ(Aµ) = 0 if and only if µ has support on Z∗p, and AResZ∗p (µ) = (1−ϕψ)Aµ .7. Convolution of distribution: If λ and µ are two distributions on Zp, we define the convolution λ ∗µ by∫Zpφ ·λ ∗µ =∫Zp(∫Zpφ(x+ y)µ(x))λ (y).Let φ(x) be the function x 7→ zx, where νp(z− 1) > 0, then we have Aλ∗µ(z) = Aλ (z)Aµ(z). Hencewe deduce Aλ∗µ =Aλ ·Aµ .7Chapter 3Kubota-Leopoldt p-adic L-functions3.1 The Riemann zeta functionLet ζ (s) = ∑+∞n=1 n−s =∏p:Prime(1− p−s)−1 be the Riemann zeta function. Let Γ(s) =∫ +∞t=0 e−tts dtt be theGamma function, which is holomorphic on Re(s) > 0 and satisfies the functional equation Γ(s+ 1) =sΓ(s), and thus it can be extended to a meromorphic function on C.Recall we have:Lemma 3.1.1. If Re(s) > 1, thenζ (s) = 1Γ(s)∫ +∞01et −1tstdt.Proposition 3.1.2. If f is a C ∞ function on R+ which decreases rapidly at infinity, then the functionL( f ,s) =1Γ(s)∫ +∞0f (t)tsdttdefined on Re(s) > 0 admits a holomorphic extension to C and if n ∈ N, then L( f ,−n) = (−1)n f (n)(0).Apply proposition 3.1.2 to f0(t) = tet−1 . Let ∑+∞n=0 Bntnn! be the Taylor expension of f0 at 0, where Bn isthe Bernoulli number. We have, in particularB0 = 1, B1 =−12, B2 =16, B4 =−130· · · .Since f0(t)− f0(−t) =−t, we have B2k+1 = 0 if k ≥ 1.Theorem 3.1.3.i) The function ζ has a meromorphic continuation to C, which has a simple pole at s = 1 with residue1.8ii) If n ∈ N, then ζ (−n) = (−1)n Bn+1n+1 , and in particular ζ (−n) ∈Q.3.2 Kummer congruencesIf a ∈ R∗+, by applying proposition 3.1.2 to the function fa(t) =1et−1 −aeat−1 , which is C∞ (the pole att = 0 canceling out) on R+ and decreases rapidly at infinity, we haveCorollary 3.2.1. If a ∈R∗+, the function (1−a1−s)ζ (s) = L( fa,s) has an analytic continuation on C, andif n ∈ N, then (1−a1+n)ζ (−n) = (−1)n f (n)a (0). In particular, if a ∈Q, then (1−a1+n)ζ (−n) ∈Q.Given a continuous distribution µ , we associate it with the formal series:Lµ(t) =+∞∑n=0∫Zpetxµ =+∞∑n=0tnn!∫Zpxnµ,which is called the Laplace transform of µ .We have Lµ(t) =Aµ(et −1).Proposition 3.2.2. If a ∈ Z∗p, there exists a measure µa whose Laplace transform is fa(t). MoreovervD0(µa)≥ 0 and if n ∈ N, then∫Zp xnµa = (−1)n(1−a1+n)ζ (−n).Proof. To show the existance of µa, it suffices to prove that the coefficients of series obtained by replaceet by 1+T (Amice transform of µa) is bounded by proposition 2.2.3. Since (1+T )a− 1 is of the formaT (1+T g(T )) where g(T ) = ∑+∞n=21a(an)T n−2 ∈ Zp[[T ]], we have1T−a(1+T )a−1=+∞∑n=1(−T )n−1gn ∈ Zp[[T ]].Since the coefficients are in Zp, we have vD0(µa)≥ 0. Moreover, we have∫Zp xnµa =L (n)µa (0) = f(n)a (0).Corollary 3.2.3. (Kummer congruences) If a ∈ Z∗p and k ≥ 1, if n1 and n2 are two integers ≥ k such thatn1 ≡ n2 mod (p−1)pk−1, thenνp((1−a1+n1)ζ (−n1)− (1−a1+n2)ζ (−n2))≥ k.Proof. Since by assumption n1 ≥ k and n2 ≥ k, we have νp(xn1)≥ k and νp(xn2)≥ k if x ∈ pZp. On theother hand, since the order of (Z/pkZ)∗ is (p−1)pk−1, and by assumption n1 ≡ n2 mod (p−1)pk−1, wehave xn1−xn2 ∈ pkZp if x∈Z∗p. To sum up, we have νp(xn1−xn2)≥ k if x∈Zp and hence vC 0(xn1−xn2)≥k. Since vD0(µa)≥ 0, which impliesνp((1−a1+n1)ζ (−n1)− (1−a1+n2)ζ (−n2)) = νp(∫Zp(xn1− xn2)µa(x))≥ k.9Proposition 3.2.4. If a ∈ Z∗p, theni) ψ(µa) = µa.ii) ResZ∗p(µa) = (1−ϕ)µaiii)∫Z∗pxnµa = (1− pn)∫Zp xnµa for all n ∈ N.Proof. Let F(T ) = ψ( 1T ). By definition, we haveF((1+T )p−1) =1p ∑ζ p=11(1+T )ζ −1 =−1p ∑ζ p=1+∞∑n=0((1+T )ζ )n=−+∞∑n=0(1+T )pm =1(1+T )p−1.Thus, we have ψ( 1T ) =1T . On the other hand, since the Amice transform of µa is1T −a(1+T )a−1 =1T −aσa( 1T ), the action of ψ commutes with σa, and by ψ(Aµ) =Aψ(µ) if µ is a distribution, we deduce i).ii) follows from i) since we have ResZ∗p(µ) = (1−ϕψ)µ if µ is a distribution. iii) follows from ii)and∫Zp xnϕ(µ) =∫Zp(px)nµ .Corollary 3.2.5. Let a ∈ N−{1} be prime to p. Let k ≥ 1. If n1 and n2 are two integers ≥ k such thatn1 ≡ n2 mod (p−1)pk−1, thenνp((1−a1+n1)(1− pn1)ζ (−n1)− (1−a1+n2)(1− pn2)ζ (−n2))≥ k.By corollary 3.2.5, the function n 7→ (1− pn)ζ (−n) is continuous under the p-adic topology. To havea uniform formula, we put q = 4 if p = 2 and q = p if p 6= 2. We denote by φ the Euler φ− function, andthus φ(q) = 2 if q = 4 and φ(q) = p−1 if p 6= 2.Theorem 3.2.6. If i ∈ Z/φ(q)Z, there exist an unique function ζp,i continuous on Zp (resp. Zp−{1}) ifi 6= 1 (resp. i = 1) such that the function (s−1)ζp,i is analytic on Zp (resp. i+2Zp if p = 2) and one hasζp,i(−n) = (1− pn)ζ (−n) if n ∈ N satisfying −n≡ i mod p−1.Remark 3.2.7. ζp,i is called the i-th branch of Kubota-Leopoldt zeta function. If i is even, then ζp,i isidentically zero since ζ (−n) = 0 if n≥ 2 is even.3.3 p-adic Mellin transform and Leopoldt’s Γ-transformWe denote ∆ the group of roots of unity of Q∗p. Therefore ∆ is a cyclic group of order φ(q) and Z∗p isdisjoint union of ε + qZp with ε ∈ ∆. We denote ω : Zp→ ∆∪{0} the function defined by ω(x) = 0 ifx ∈ pZp, and x−ω(x) ∈ qZp, if x ∈ Z∗p. If x ∈ Z∗p, we define 〈x〉 ∈ 1+qZp by 〈x〉= xω(x)−1.10Proposition 3.3.1. If i ∈ Z/φ(q)Z, the function x 7→ ω(x)i〈x〉s is a locally analytic function on Zp. More-over, we havei) ω(x)i〈x〉n = xn if n≡ i mod φ(q) and if x ∈ Z∗pii) ω(x)i〈x〉s = limn→sn≡imodφ(q)xn for x ∈ Zp.Proof. Note that we have ω(x)i〈x〉s = 0 on pZp andω(x)i〈x〉s = ε i( xε )s =+∞∑n=0(sn)ε i−n(x− ε)n,if x ∈ ε+qZp and ε ∈ ∆, thus the function is locally analytic.Since the order of ∆ is φ(q), we have ω(x)n = ω(x)i if n≡ i mod φ(q), i) and ii) follows.If i ∈ Z/φ(q)Z, we defined the i-th branch of the Mellin transform of a continuous distribution µ bythe formulaMeli,µ(s) =∫Zpω(x)i〈x〉sµ(x) =∫Z∗pω(x)i〈x〉sµ(x)the second equality is because ω(x) = 0 if x ∈ pZp. On the other hand, we have Meli,µ(n) =∫Z∗pxnµ ifn≡ i mod ψ(q).Let u be a topological generator of multiplicative group of 1+ qZp, and let θ : 1+ qZp → Zp thehomomorphism which sends x to logxlogu . This homomorphism is analytic and also its inverse be. If f is alocally analytic function (resp. continuous) function on 1+ qZp, the function θ ∗ f defined by θ ∗φ(x) =φ(θ(x)) is locally analytic (resp. continuous) on Zp.If µ is a distribution support on 1+qZp, we define a distribution θ∗µ on Zp by the formula∫Zpφ(θ∗µ) =∫1+qZp(θ ∗φ)µ.In particular, θ∗ sends measure to measure.Lemma 3.3.2. If X is a open compact subset of Zp, If α ∈ Z∗p, and if µ is a continuous distribution onZp, thenResX(σα(µ)) = σα(Resα−1X(µ))11Proof. Since we have 1X(αx) = 1α−1X(x) if X ⊂ Zp, we deduce the formula∫Zpφ(x)ResX(σα(µ)) =∫Zp1X(x)φ(x)σαµ=∫Zp1X(αx)φ(αx)µ(x)=∫Zpφ(αx)(1α−1X(x)µ(x))=∫Zpφ(αx)Resα−1X(µ)=∫Zpφ(x)σα(Resα−1X(µ)),which proves the lemma.Definition 3.3.3. If µ is a distribution on Z∗p and if i ∈ Z/φ(q)Z, we define Γ(i)µ the i-th branch of theΓ-transform of µ byΓ(i)µ = θ∗Res1+qZp(∑ε∈∆ε−iσε(µ)) = θ∗(∑ε∈∆ε−iσε(Resε−1+qZp(µ))),where the second equality follows from the above lemma. Moreover, it is clear that if µ is a measure onZ∗p, then Γ(i)µ is a measure on Zp, and we have vD0(Γ(i)µ )≥ vD0(µ).Proposition 3.3.4. If µ is a continuous distribution on Z∗p and i ∈ Z/φ(q)Z, thenMeli,µ(s) =∫Z∗pω(x)i〈x〉sµ(x) =∫ZpusyΓ(i)µ (y) =AΓ(i)µ(us−1)Proof. The first equality is by the definition of Mellin transform and the third equality is by the definitionof Amice transform. If y = θ(x) = logxlogu , we have usy = exp(s logx) = 〈x〉s and∫ZpusyΓ(i)µ (y) =∫1+qZp〈x〉s ∑ε∈∆ε−iσε(Resε−1+qZp(µ)).Using the fact that ω(x) = ε−1 if x ∈ ε−1 +qZp and 〈εx〉= 〈x〉, we obtain∫ZpusyΓ(i)µ (y) = ∑ε∈∆∫ε−1+qZpω(x)i〈x〉sµ(x),and the proposition follows from that Z∗p is the disjoint union of ε+qZp for ε ∈ ∆.Corollary If µ is a continuous distribution and i ∈ Z/φ(q)Z, the function Meli,µ(s) is an analytic function of sand even us−1.ii) If µ is a measure verifying vD0(µ)≥ 0, and if i ∈ Z/φ(q)Z, then there exists gi,µ ∈ OL[[T ]] such thatMeli,µ(s) = gi,µ(us−1).3.4 Construction of the Kubota-Leopoldt zeta functionIf i ∈ Z/φ(q)Z and a ∈ Z∗p such that 〈a〉 6= 1, we define the function ga,i on Zp by the formulaga,i(s) =11−ω(a)1−i〈a〉1−s Mel−i,µa(−s) =11−ω(a)1−i〈a〉1−s∫Z∗pω(a)−i〈a〉−sµa.By corollary 3.3.5, Mel−i,µa(−s) is an analytic function of s. On the other hand, if ω(a)1−i 6= 1, thefunction s 7→ 1−ω(a)1−i〈a〉1−s is a nonzero analytic function on Zp since 〈a〉s ∈ 1+qZp and ω(a)1−i ∈∆−{1}, therefore ω(a)1−i 6∈ 1+qZp and if ω(a)1−i = 1, the function 1−〈a〉1−s vanishes only at s = 1.We deduce that ga,i is a function continuous on Zp−{1} and even continuous on Zp if ω(a)1−i 6= 1.Moreover, if −n≡ i mod φ(q), we have ω(a)1−i = ω(a)1+n and ω(x)−i = ω(x)n if x ∈ Z∗p. Thereforega,i(−n) =11−ω(a)1+n〈a〉1+n∫Z∗pω(x)n〈a〉nµa(x) =11−a1+n∫Z∗pxnµa(x) = (−1)n(1− pn)ζ (−n)does not depend on the choice of a. If a and a′ two elements of Z∗p, the function ga,i− ga′,i is a quotientof analytic functions on Zp vanishing at infinitely many points, which implies it identical zero and thefunction ga,i is independent of choice of a. Thus we set ζp,i = ga,i for any a satisfying 〈a〉 6= 1 andω(a)1−i 6= 1 if i 6= 1 to construct Kubota-Leopoldt zeta function.Let Fn = Qp(εpn) and F∞ = ∪Fn. The norm NFn+1/Fn induces a homomorphism from µpn+1 to µpn ,where µpn be the set of pn-th roots of unity in Fn. We denote the projective limit of µpn with respect toNFn+1/Fn by µp∞ (the Tate module), which is a compact Zp-module.The following theorem is due to Mazur and Wiles:Theorem 3.4.1. If i∈ (Z/(p−1)Z)∗ is odd and if s∈Zp, then the following two conditions are equivalent:i) ζp,i(s) = 0;ii) There exists an element u ∈ µp∞ which is not killed by a power of p such that σ ∈ Gal(F∞/Qp) actsby the formulaσ(u) = ω(χcycl(σ))i〈χcycl(σ)〉s ·u.133.5 The residue at s = 1 and the p-adic zeta functionThe formal power series log(1+T )T converges on open unit disk, thus it is an Amice transform of a uniquedistribution µKL. The Laplace transform of µKL is tet−1 = f0(t) and∫ZpxnµKL = (−1)n−1nζ (1−n)Lemma 3.5.1.∫a+pnZp µKL =1pnProof. Since∫a+pnZp µKL =1pn ∑ε pn=1 ε−aAµKL(ε−1) and since logε = 0 if ε is a root of unity of order apower of p, all terms of the sum are zero except for the term corresponding to ε = 1, we get the result.Proposition 3.5.2. We havei) ψ(µKL) = p−1µKLii) ResZ∗p(µKL) = (1− p−1ϕ)µKLiii)∫Z∗pµKL = (−1)n−1n(1− pn−1)ζ (1−n) if n ∈ N.Proof. i) follows from the formula ψ( 1T ) =1T (c.f. proposition 3.2.4) and ϕ(log(1+T )) = p log(1+T )and ψ(ϕ(a)b) = aψ(b). The rest can be deduced from proposition 3.2.4.Theorem 3.5.3. The p-adic zeta function ζp,1 has a simple pole at s = 1 with residue 1− 1p .Proof. According to the above, we can define the function ζp,i, if i ∈ Z/φ(q)Z by the formulaζp,i(s) =(−1)i−1s−1Mel1−i,µKL(1− s) =(−1)i−1s−1∫Z∗pω1−i〈x〉1−sµKL(x).Indeed, the function is analytic on Zp−{1} by the above formula, and takes the same value ζp,i(−n) =(1− pn)ζ (−n) if n ∈ N satisfies −n≡ i mod p−1. Moreover,lims→1(s−1)ζp,i(s) =∫Z∗pω(x)1−iµKL(x)= ∑α∈∆ω(α)1−i∫α+pZpµKL(x) =1− 1p if i = 10 otherwise.143.6 Dirichlet L-functionFor χ a Dirichlet character of conductor D > 1 and if n ∈ Z, we define the Gauss sum G(χ) byG(χ) = ∑amodDχ(a)e2piiaD .LetL(χ,s) =+∞∑n=1χ(n)ns= ∏p:prime(1−χ(p)p−s)−1, for Re(s)≥ 1,be the Dirichlet L-function attached to χ . By the formulaχ(n) = 1G(χ−1) ∑bmodDχ−1(b)e2piinbD ,we obtainL(χ,s) = 1G(χ−1) ∑bmodDχ−1(b)e2pii nbDns.Using the formula∫ +∞0 e−ntts dtt =Γ(s)ns with εD := e2piiD , we obtainL(χ,s) = 1G(χ−1)1Γ(s) ∑bmodDχ−1(b)∫ +∞0+∞∑n=1εnbD e−nttsdtt=1G(χ−1)1Γ(s)∫ +∞0∑bmodDχ−1(b)ε−bD et −1tsdtt.In particular, proposition 3.1.2 implies that L(χ,s) can be extended to a holomorphic function on C.Moreover, L(χ,−n) = ( ddt )nLχ(t) |t=0 whereLχ(t) =−1G(χ−1) ∑bmodDχ−1(b)εbDet −1.3.7 p-adic L-function attached to Dirichlet characterLet χ be a Dirichlet character of conductor D > 1 prime to p. If χ−1(b) 6= 0, then εbD is a root of unity oforder prime to p and distinct from 1, this implies νp(εbD−1) = 0. We deduce that the power seriesFχ(T ) =−1G(χ−1) ∑bmodDχ−1(b)(1+T )εbD−1=1G(χ)−1 ∑bmodDχ−1(b)+∞∑n=0εnbD(εbD−1)n+1T nis of bounded coefficients (since νp(G(χ)G(χ−1)) = νp(D) = 0) and hence there exists an Amice trans-form of a measure µχ on Zp whose Laplace transform Fχ(et−1) =Lχ(t). We have∫Zp xnµχ =L (n)χ (0) =15L(χ,−n) and vD0(µχ)≥ 0.Definition 3.7.1. We define the p-adic L-function associated to χ as the Mellin transform of µχ , that is,the function β 7→ Lp(χ⊗β ) defined byLp(χ⊗β ) =∫Z∗pβ (x)µχ(x).where β is a locally analytic character on Z∗p. On the other hand, if i ∈ Z/φ(q)Z, we putLp,i(χ,s) = Lp(χ⊗ (ω−i(x)〈x〉−s)) =∫Z∗pω−i〈x〉−sµχ(x).Proposition 3.7.2. If i∈Z/φ(q)Z, the function Lp,i(χ,s) is an analytic function on Zp and thus Lp,i(χ,−n)=(1−χ(p)pn)L(χ,−n) if n ∈ N satisfying −n≡ i mod φ(q).Proof. The fact that Lp,i(χ,s) is an analytic function on Zp follows from corollary 3.3.5. On the otherhand, we have∑η p=11(1+T )εDη−1= p1(1+T )pε pbD −1,thus we deduce the Amice transform of µχ restriction to Z∗p is−1G(χ) ∑bmodDχ−1(b)(1+T )εbD−1−χ−1(b)(1+T )pε pbD −1,which can be written as Aµχ (T )−χ(p)Aµχ ((1+T )p−1). Hence we deduce the formulaLResZ∗p (µχ )(t) =Lµχ (t)−χ(p)Lµχ (pt) and∫Z∗pxnµχ = (1−χ(p))L(χ,−n),and the proposition follows.3.8 Behavior at s = 1 of Dirichlet L-functionBy section 3.6, we haveL(χ,1) = 1G(χ−1) ∑bmodDχ−1(b)+∞∑n=0εnbDn=−1G(χ−1) ∑bmodDχ−1(b) log(1− εbD).We will establish the p-adic analogue of this formula by calculating∫Z∗px−1µχ . To do this, we willcalculate the Amice transform of x−1µχ and then restrict it to Z∗p.16Proposition 3.8.1. The Amice transform of x−1µχ isAx−1µχ (T ) =−1G(χ)−1 ∑bmodDχ−1(b) log((1+T )εbD−1).Proof. If µ is a distribution, the relation of Amice transform of µ and x−1µ is given by(1+T )ddTAx−1µ(T ) =Aµ(T ).Applying the operator (1+T ) ddT on the right hand side of the equality in the proposition we obtain−1G(χ−1) ∑bmodDχ−1(b) (1+T )εbD(1+T )εbD−1=−1G(χ−1) ∑bmodDχ−1(b)(1(1+T )εbD−1+1)which is equal to Aµχ since ∑bmodD χ−1(b) = 0. We deduce that the two elements have the same imageby (1+T ) ddT and therefore differ by a locally constant function. To conclude, we must verify that theright hand side is given by a series which converges on the open unit disk. Sincelog((1+T )εbD−1) = log(εbD−1)+ log(1+εbDTεbD−1) = log(εbD−1)++∞∑n=1(−1)n−1n( εbDTεbD−1)nand we suppose (D, p) = 1, we have νp(εbD−1) = 0, and hence the series converges on open unit disk.Lemma 3.8.2. The Amice transform of the restriction of x−1µχ to Z∗p is defined byAResZ∗p (x−1µχ )(T ) =−1G(χ−1) ∑bmodDχ−1(b)(log((1+T )εbD−1)−1plog((1+T )pε pbD −1))=Ax−1µχ (T )−χ(p)pAx−1µχ ((1+T )p−1).Proof. Use the formula for Amice transform of ResZ∗p .By taking T = 0 in the above formula, we obtainLp,1(χ,1) = Lp(χ⊗ x−1) =∫Z∗px−1µχ =−1G(χ−1)(1−χ(p)p) ∑bmodDχ−1(b) log(εbD−1).which differs from the complex L-function case by an Euler factor.3.9 Twist by a character of conductor power of pLet χ be a Dirichlet character of conductor D prime to p and β be a Dirichlet character of conductorpk. We denote χ⊗β to be the Dirichlet character of conductor Dpk defined by (χ⊗β )(a) = χ(a)β (a),17where χ and β are viewed as characters mod Dpk via the projections from (Z/DpkZ)∗ to (Z/DZ)∗ and(Z/pkZ)∗.Lemma 3.9.1. Let k≥ 1, β a Dirichlet character of conductor pk and µ a continuous distribution on Zp,then we have∫Zpβ (x)(1+T )xµ(x) = 1G(β )−1 ∑c mod pkβ−1(c)Aµ((1+T )εcpk −1).Proof. We have∫Zpβ (x)(1+T )xµ(x) = ∑amod pkβ (a)∫a+pkZp(1+T )xµ= ∑amod pkβ (a)(1pk ∑η pk=1η−aAµ((1+T )η−1))= ∑η pk=1Aµ((1+T )η−1)(1pk ∑amod pkβ (a)η−a),and the lemma follows from the identity1pkβ−1(−c)G(β ) = β−1(c)G(β−1) .Proposition 3.9.2. If µ is a measure on Zp with Amice transform of the formAµ(T ) =1G(χ−1) ∑b mod Dχ−1(b)F((1+T )εbD−1)and if β is a Dirichlet character of conductor pk with k ≥ 1, then∫Zpβ (x)(1+T )xµ(x) = 1G((χ⊗β )−1) ∑a mod Dpk(χ⊗β )−1(a)F((1+T )εbDεcpk −1).Proof. By the preceding lemma we have∫Zpβ (x)(1+T )xµ(x) = −1G(χ−1)G(β−1) ∑b mod D∑c mod pkχ−1(b)β−1(c)F((1+T )εbDεcpk −1).Using the fact that every element of Z/DpnZ can be written uniquely of the form Dc+ pkb, where b ∈18Z/DZ and c ∈ Z/pkZ, we have the following formulasεaDpn =εbDεcpk(χ⊗β )−1(a) =χ−1(pk)β−1(D)χ−1(b)β−1(c)G((χ⊗β )−1) = ∑a mod Dpk(χ⊗β )−1(a)εaDpk=χ−1(pk)β−1(D)(∑b mod Dχ−1(b)εbD)(∑c mod pkβ−1(c)εcpk)=χ−1(pk)β−1(D)G(χ−1)G(β−1)and the conclusion follows.Proposition 3.9.3. If β is a non-trivial Dirichlet character of conductor a power of p and if n ∈ N, thenLp(χ⊗ (xnβ )) = L(χ⊗β ,−n)Proof. By proposition 3.9.2 and the formula for the Amice transform of µχ , we have the Amice transformof βµχ is−1G((χ⊗β )−1) ∑x mod Dpn(χ⊗β )−1(x)(1+T )εxDpn−1and thus its Laplace transform is the function Lχ⊗β (t).19Chapter 4(ϕ ,Γ)-modules and p-adic representationsThroughout this article, k will denote a finite field of characteristic p > 0, so if W (k) denotes the ring ofWitt vectors over k, then F = W (k)[ 1p ] is a finite unramified extension of Qp. Let Qp be the algebraicclosure Qp, let K be a totally ramified extension of F , and let GK = Gal(Qp/K) be the absolute Galoisgroup of K. Let µpn be the group of pn-th roots of unity; for every n, we will choose a generator ε(n) of µpnwith the additional requirement that (ε(n))p = ε(n−1), This makes lim←−ε(n) into a generator lim←−µpn 'Zp(1).We set Kn = K(µpn) and K∞ =⋃n≥0Kn. Recall that the cyclotomic character χ : GK → Z∗p is defined by therelation: g(ε(n)) = (ε(n))χ(g) for all g ∈GK . The kernel of the cyclotomic character is HK = Gal(Qp/K∞),and χ therefore identifies ΓK = GK/HK with an open subgroup of Z∗p.4.1 The field E˜ and its subrings.Let Cp be the completion of Qp for the p-adic topology and letE˜ = lim←−Cp = {(x(0),x(1), ...) | (x(n+1))p = x(n)and let E˜+ be the set of x ∈ E˜ such that x(0) ∈ OCp . If x = (x(i)) and y = (y(i)) are two elements of E˜, wedefine the sum x+ y and their product xy by(x+ y)(i) = limj→+∞(x(i+ j)+ y(i+ j))pjand (xy)(i) = x(i)y(i),which makes E˜ an algebraically closed field of characteristic p. (c.f. [13] proposition 4.8) If x=(x(n))∈ E˜,let νE(x) = νp(x(0)). This is a valuation on E˜ and E˜ is complete for this valuation; the ring of integers ofE˜ is E˜+. If a is an ideal satisfying p ∈ a ⊂ m, where m is the maximal ideal of OCp , the E˜+ is identifiedwith the projective limit of An, where if n ∈ N, we put An = OCp/a and the transition amp from An+1 toAn is given by x 7→ xp.Let ε = (1,ε(1), ...,ε(n), ...) be an element of E˜ such that ε(1) 6= 1, this implies that ε(n) is a primitive20pn-th root of unity if n ≥ 1. Let pi = ε−1, we have νE(pi) = pp−1 and let EQp be the subfield Fp((pi)) ofE˜. We denote by E the separable closure of EQp in E˜ and E+ (resp. mE) the ring of integers (resp. themaximal ideal of E+).By ramification theory, if K is a finite extension of Qp, then for all η > 0, there exists nη ∈ N suchthat if n ≥ nη , and if τ ∈ ΓKn , then νp(τ(x)− x) ≥ 1p −η . In particular if a is an ideal of OCp defined bya = {x ∈ OCp | νp(x) ≥ 1p}, then NKn+1/Kn(x)− xp ∈ a if n is large enough and x ∈ OKn+1 . This allows usto construct a map ιK from the projective limit lim←−OKn of OKn with respect to norm map to E˜+ (field ofnorms), such that u = (u(n))n∈N is associated to ιK(u) = (x(n))n∈N, where x(n) is the image of u(n) in OCp/aif n large enough. Hence we have the following proposition:Proposition 4.1.1. If K is a finite extension of Qp, then ιK induces a bijection from lim←−OKn to the ring ofintegers E+K of EK = EHK .By this proposition, one can show that EK is a finite separable extension of EQp of degree [HQp : HK ] =[K∞ : Qp(µp∞)] and that one can identify Gal(E/EK) with HK .Remark 4.1.2.i) If F is a finite unramified extension of Qp with residue field kF , the field EF is the composition of kFand EQp , that is, kF((pi)).ii) If K is a finite extension of Qp and F = K ∩Qnrp it maximal unramified subfield, then EK is anextension of EF of degree [K∞ : F∞] which is equal to [Kn : Fn] for n large enough.4.2 The field B˜ and its subringsLet A˜ = W (E˜) (resp. A˜+ = W (E˜+)) the Witt vectors with coefficients in E˜ (resp. E˜+). By construction,we have A˜/pA˜ = E˜ (resp. A˜+/pA˜+ = E˜+). LetB˜ = A˜[1/p] = { ∑k−∞pk[xk] | xk ∈ E˜} (resp. B˜+ = A˜+[1/p] = { ∑k−∞pk[xk] | xk ∈ E˜+}),where [x] ∈ A˜ is the Teichm¨uller lift of x ∈ E˜ (resp. E˜+).We endow A˜ (resp. A˜+) with the topology by taking the collection of open sets {[pi]kA˜++ pnA˜}k,n≥0(resp. {([pi]k + pn)A˜+}k,n≥0) as family of neighborhoods of 0 and endow B˜ = ∪n∈N p−nA˜ (resp. B˜+)the inductive limit topology. The action of GQp on E˜ can be extended by continuity to A˜ and B˜ whichcommutes with the Frobenius action ϕ .For F a finite unramified extension over Qp, let pi = [ε]−1, we define AF the closure of OF [[pi,pi−1]]in A˜ by the above topology, thusAF = {∑k∈Zakpin | an ∈ OF , limk→−∞νp(ak) = +∞}21which is a complete discrete valuation ring with residue field EF and the Galois action and Frobeniusaction is defined byϕ(pi) = (1+pi)p−1 and g(pi) = (1+pi)χ(g)−1 g ∈ GF ,and its fraction field BF = AF [ 1p ] is stable by actions of ϕ and GF .Let B be the completion for the p-adic topology of the maximal unramified extension of BF in B˜and A = B∩ A˜. We have B = A[ 1p ] and A are complete discrete valuation ring with fraction field B andresidual field E. We then define B+ = B∩ B˜+ and A+ = A∩ A˜+. These rings are endowed with an actionof Galois and a Frobenius induced from those on E˜.If K is a finite extension of Qp, we put AK = AHK and BK = AK [1/p], this makes AK a completediscrete valuation ring with residue field EK and fraction field BK = AK [1/p]. On the other hand, whenK = F , the definitions of AF and BF coincide with previous definitions. We put A+F = (A+)HF andB+F = (B+)HF then by using fields of norm above, we can show that A+F = OF [[pi]] and B+F = F [[pi]].If L is a finite extension of K, BL is an unramified extension of BK of degree [L∞ : K∞]. If L/K is aGalois extension, then the extension B˜L/B˜K (resp. BL/BK) is Galois with Galois group Gal(B˜L/B˜K) =Gal(BL/BK) = Gal(EL/EK) = Gal(L∞/K∞) = HK/HL.Remark 4.2.1.i) If piK is a uniformizer of EK , let piK be any lifting of piK in AK and F ′ is the maximal unramifiedextension of F contained in K∞. Then,AK = {∑k∈ZakpikK | ak ∈ OF ′ , limk→−∞νp(ak) = +∞}.ii) In the above construction, the correspondence R−→ R˜ is obtained by making ϕ bijective and then dothe completion with respect to the given topology on R, where R = {EK ,E,AK ,A,BK ,B}.4.3 (ϕ,Γ)-module and Galois representationsA p-adic representation V is a finite dimensional Qp-vector space with a continuous linear action of GK .It is easy to see that there is always a Zp-lattice of V which is stable by the action of GK , and such latticeswill be denoted by T (called a Zp-representation). The main strategy due to Fontaine for studying p-adicrepresentations of a Galois group G is to construct topological Qp-algebras B (period rings), endowedwith an action of G and some additional structures so that if V is a p-adic representation, thenDB(V ) := (B⊗Qp V )G22is a BG-module which inherits these structures, and so that the functor V 7→ DB(V ) gives interestinginvariants of V . We say that a p-adic representation V of G is B-admissible if we have B⊗Qp V ' Bd asB[G]-modules, where d = dimV .Definition 4.3.1. If K is a finite extension of Qp, we sayi) A (ϕ,Γ)-module over AK (resp. BK) is an AK-module of finite type (resp. a finite dimensionalBK-vector space) equipped with a ΓK-action and a Frobenius action ϕ which commutes with ΓK .ii) A (ϕ,Γ)-module D over AK is e´tale if ϕ(D) generates D as an AK-module. A (ϕ,Γ)-module D overBK is e´tale if it has an AK-lattice which is e´tale, equivalently, there exists a basis {e1, ...,ed} overBK , such that the matrix of ϕ in terms of the basis is in GLd(AK).If K is a finite extension of Qp and V is a Zp-representation (resp. p-adic representation) of GK , weputD(V ) = (A⊗Zp V )HK (resp. D(V ) = (B⊗Qp V )HK )Since the action of ϕ commutes with GK , D(V ) is a equipped with a Frobenius action ϕ which commuteswith the residual action GK/HK = ΓK . This make D(V ) a (ϕ,Γ)-module over AK (resp. BK).On the other hand, if V is a Zp-representation (resp. a p-adic representation) of GK , then (A⊗AKD(V ))ϕ=1 (resp. (B⊗AK D(V ))ϕ=1) is canonically isomorphic to V as a representation of GK . In otherwords, V is determined by the (ϕ,Γ)-module D(V ).Theorem 4.3.2. (Fontaine) The correspondenceV 7−→ D(V ) = (A⊗Zp V )HKis an equivalence of ⊗ categories from the category of Zp-representations (resp. p-adic representation) ofGK to the category of e´tale (ϕ,Γ)-module over AK (resp. BK), and its inverse functor isD 7−→V (D) = (A⊗AK D)ϕ=1.23Chapter 5(ϕ,Γ)-modules and Galois cohomology5.1 The complex Cϕ,γ(K,V )Let K be an finite extension of Qp such that ΓK is isomorphic to Zp (i.e. contains Qp(µp) if p≥ 3 or threequadratic ramified extensions of Q2 if p = 2). Let γ be a generator of ΓK . If V is a Zp-representation orp-adic representation of GK and f : D(V )→ D(V ) is a Zp-linear map which commutes with action of Γ,we denote by C f ,γ(K,V ) the complex0 // D(V ) // D(V )⊕D(V ) // D(V ) // 0where the maps D(V ) to D(V )⊕D(V ) and D(V )⊕D(V ) to D(V ) are respectively given byx 7→ (( f −1)x,(γ−1)x) and (a,b) 7→ (γ−1)a− ( f −1)bwe denote Zi(C f ,γ(K,V )) (resp. Bi(C f ,γ(K,V )), resp. H i(C f ,γ(K,V )) =Zi(C f ,γ (K,V ))Bi(C f ,γ (K,V ))) the i-th cocycles(resp. coboundaries, resp. cohomology) of the complex C f ,γ(K,V ).The cohomology groups of the complex C f ,γ(K,V ) can be canonically and functorially identified withthe Galois cohomology group H i(K,V ) (c.f. [14]). The following proposition gives the case of H1.Let ΛK = Zp[[ΓK ]] the complete group algebra of ΓK . Since ΓK acts continuously on D(V ), we canview D(V ) as a ΛK-module. On the other hand, ΓK is pro-cyclic, if γ is a generator of ΓK and γ ′ isany element of ΓK , then the elementγ ′−1γ−1 of Frac(ΛK) is indeed in ΛK . Moreover, the GK action factorsthrough ΓK on D(V ), so the expression σ−1γ−1 y makes sense if y ∈D(V ), σ ∈GK and γ is a generator of ΓK .Proposition 5.1.1.i) If (x,y) ∈ Z1(Cϕ,γ(K,V )) and b ∈A⊗Zp V is a solution of (ϕ−1)b = x, then σ 7→ cx,y(σ) = σ−1γ−1 y−(σ −1)b is a cocycle of GK with values in V .24ii) The map which sends (x,y) ∈ Z1(Cϕ,γ(K,V )) to the class of cx,y in H1(K,V ) induces an isomorphismιϕ,γ of H1(Cϕ,γ(K,V )) to H1(K,V ).Proof. It clear that σ 7→ cx,y(σ) is a cocycle by definition. On the other hand, we have(ϕ−1)(cx,y(σ)) =σ −1γ−1 ((ϕ−1)y)− (σ −1)x = 0since (γ−1)x = (ϕ−1)y. Hence cx,y(σ) ∈ (A⊗Zp V )ϕ=1 =V . This proves (i).To prove (ii), suppose the image of cx,y in H1(K,V ) is zero, there exist z ∈V such thatσ −1γ−1 y− (σ −1)(b+ z) = 0 ∀σ ∈ GK .We deduce that b+z is stable by HK and therefore belongs to D(V ). Take σ = γ , we have y=(γ−1)(b+z)and hence x = (ϕ−1)(b+ z), which implies (x,y) ∈ B1(Cϕ,γ(K,V )) and the injectivity of ιϕ,γ follows.To prove the surjectivity, let c ∈H1(K,V ) and V ′ an extension of Zp by V corresponding to c. That is,an exact sequence0 // V // V ′ // Zp // 0such that e ∈ V ′ sends to 1 ∈ Zp and σ(e) = e+ cσ , where σ 7→ cσ is the cocycle of GK represents c.Applying functor D, we get0 // D(V ) // D(V ′) // D(Zp) // 0,let e˜ ∈ D(V ′) be an element maps to 1 ∈ Zp = D(Zp) and let x,y be elements of D(V ) defined by x =(ϕ−1)e˜ and y = (γ−1)e˜. Since γ and ϕ commute, (x,y) is belongs to Z1(Cϕ,γ(K,V )). On the other hand,b = e˜− e ∈ A⊗Zp V satisfies (ϕ−1)b = x, so we havecx,y(σ) =σ −1γ−1 y− (σ −1)b = (σ −1)(e˜−b) = (σ −1)e = cσ .From this, we deduce the surjectivity of ιϕ,γ .If γ ′ is another generator of ΓK , then γ−1γ ′−1 ∈ Frac(ΓK) is indeed a unit in ΓK and the diagramCϕ,γ(K,V ) : 0 // D(V ) //γ−1γ ′−1D(V )⊕D(V ) //γ−1γ ′−1⊕idD(V ) //id0Cϕ,γ ′(K,V ) : 0 // D(V ) // D(V )⊕D(V ) // D(V ) // 0is commutative. It hence induces via cohomology an isomorphism ιγ,γ ′ from H1(Cϕ,γ(K,V )) to H1(Cϕ,γ ′(K,V )).Since we assume ΓK is torsion free, we have χ(γ) ∈ 1 + pZp for γ ∈ ΓK , then there exists k ≥ 125such that logp(χ(Γ)) ∈ pkZ∗p and we’ll write log0p(γ) = logp(χ(γ))/pk. The following lemma shows thatlog0p(γ)ιϕ,γ does not depend on the choice of generator γ of ΓK .Lemma 5.1.2. If γ and γ ′ are two generators of ΓK , then the isomorphisms log0p(γ)ιϕ,γ and log0p(γ ′)ιϕ,γ ′ ◦ιγ,γ ′ from H1(Cϕ,γ(K,V )) to H1(K,V ) are equal.Proof. If (x,y) ∈ Z1(Cϕ,γ(K,V )). Let b (resp. b′) be element of A⊗Zp V verifies (ϕ − 1)b = x (resp.(ϕ−1)b′ = γ−1γ ′−1 x). Sincelog0p(γ)γ−1 −log0p(γ ′)γ ′−1 ∈ Zp[[ΓK ]], we can write the cocycle associates to log0p(γ ′)ιϕ,γ ◦ιγ,γ ′(x,y)− log0p(γ)ιϕ,γ(x,y) as σ 7→ (σ −1)c, wherec = (log0p(γ ′)γ ′−1 −log0p(γ)γ−1 )y− (log0p(γ ′)b′− log0p(γ)b)and the relation (ϕ − 1)y = (γ − 1)x implies (ϕ − 1)c = 0, hence c ∈ V and the cocycle is indeed acoboundary, which leads to the conclusion.5.2 The operator ψTo calculate H1(Cϕ,γ(K,V )) we have to understand the group D(V )ϕ=1 andD(V )ϕ−1 . The problem is that thegroup D(V )ϕ−1 is too complicated to write down. To solve this difficulty, we introduce the left inverse of ϕ .The field B is an extension of degree p of ϕ(B), which allows up to define the operator ψ : B→ B bythe formula ψ(x) = 1pϕ−1(TrB/ϕ(B)(x)). More explicitly, one can verify that {1, [ε], ..., [ε]p−1} is a basisof A over ϕ(A) (hence B over ϕ(B)) so we haveψ(p−1∑i=0[ε]iϕ(xi)) = x0 xi ∈ B and ψ(ϕ(x)) = x x ∈ B.The operator ψ commute with the action of GK and ψ(A)⊂ A.Since ψ commutes with the action of GK , if V is a Zp-representation or a p-adic representation of GK ,the module D(V ) inherit the action of ψ and commute with ΓK . That is, the unique map ψ : D(V )→D(V )withψ(ϕ(a)x) = aψ(x), ψ(aϕ(a)) = ψ(a)xif a ∈ AK , x ∈ D(V ).Proposition 5.2.1. If V is a Zp-representation or a p-adic representation of GK , then γ − 1 is invertibleon D(V )ψ=0.Proof. See [14].26Lemma 5.2.2. We have a commutative diagram of complexesCϕ,γ(K,V ) : 0 // D(V ) //idD(V )⊕D(V ) //(−ψ,id)D(V ) //−ψ0Cψ,γ(K,V ) : 0 // D(V ) // D(V )⊕D(V ) // D(V ) // 0which induces an isomorphism ι from H1(Cϕ,γ(K,V )) to H1(Cψ,γ(K,V )).Proof. The commutativity of the diagram follows from definition. Since ψ is surjective, the cokernelcomplex is 0. The kernel complex is0 // 0 // D(V )ψ=0γ−1 // D(V )ψ=0 // 0,which has no cohomology by proposition 5.2.1.Notation 5.2.3. We denote ιψ,γ the isomorphism from H1(Cψ,γ(K,V )) to H1(K,V ) obtained by compositeιϕ,γ (see proposition 5.1.1) and ι−1 (see lemma 5.2.2).Remark 5.2.4. The same proof as lemma 5.1.2 shows that log0p(γ)ιψ,γ does not depend on the generatorγ of ΓK .Lemma 5.2.5. The map which sends (x,y) ∈ Z1(Cϕ,γ(K,V )) to the image of x inD(V )ψ−1 induces an exactsequence0 // D(V )ψ=1ΓK// H1(Cψ,γ(K,V )) //(D(V )ψ−1)ΓK // 0Proof. x ∈ D(V )ψ−1 is fixed by ΓK if and only if there exists (x,y) ∈ Z1(Cϕ,γ(K,V )) whose image inD(V )ψ−1 isequal to x. The kernel of the map is the sum of B1(Cψ,γ(K,V )) and the set X of elements of the form (0,y)where y ∈ D(V )ψ=1. One observes that X ∩B1(Cϕ,γ(K,V )) is constituted by couples of the form (0,y)where y ∈ (γ−1)D(V )ψ=1.Remark 5.2.6. By [14], one can show that the Herr complex Cψ,γ(K,V ) indeed computes the Galoiscohomology groups H i(K,V ), hence we have• H0(K,V )' D(V )ψ=1,γ=1 ' D(V )ϕ=1,γ=1.• H2(K,V )' D(V )(ψ−1,γ−1) .• H i(K,V ) = 0 if i≥ 2.Similar to the case of ϕ , the modules D(V )ψ=1 and D(V )ψ−1 can be interpreted naturally as Iwasawaalgebra. Moreover, the module D(V )ψ−1 is ”small” compared toD(V )ϕ−1 , thus we can write H1(K,V ) mainlyas the submodule D(V )ψ=1. More precisely, we have the following proposition which is proved in thesubsequent two subsections.27Proposition 5.2.7. If V is a Zp-representation (resp. a p-adic representation) of GK , theni) D(V )ψ=1 is compact (resp. locally compact) and generates the AK-module (BK-vector space) D(V ).ii) D(V )ψ−1 is a free Zp-module of finite rank (resp. a finite dimensional Qp-vector space).Remark 5.2.8. Since the p-adic representation case can be deduced from the Zp-representation case bytensoring with Qp, we only need to treat the Zp-representation case.5.3 The compactness of D(V )ψ=1The goal of this paragraph is to prove the following lemma. In particular, when n = 0 and N = +∞ isequivalent to the compactness of D(V )ψ=1.Lemma 5.3.1. If V is a Zp-representation of GK , x ∈ D(V ) and N ∈ N∪ {+∞}, the set of solutionsy ∈ D(V )/pN+1D(V ) of the equation (ψ−1)y = x is compact.Let A+Qp be the subring Zp[[pi]] of AQp , and let A = A+Qp[[ ppi p−1 ]], then A is a compact subring of AQpsuch that elements of A can be written as x = ∑n∈Z xnpin where (xn)n∈Z is a sequence in Zp such that wehave νp(xn)≥− np−1 if n≤ 0.If x ∈ AQp , let wn(x) ∈ N be the smallest integer k such that x belongs to pi−kA+ pn+1AQp . If x isfixed, the sequence {wn(x)}n∈N is increasing and we havewn(x+ y)≤ sup(wn(x),wn(y))wn(xy)≤ supi+ j=n(wi(x)+w j(y))≤ wn(x)+wn(y)wn(ϕ(x))≤ pwn(x)the first two inequalities follow from the fact that A is a ring and the third one holds because ϕ(pi)pi p is anunit in A (This is the reason for working with A instead of A+Qp by defining the map wn) and such thatx ∈ pi−kA+ pn+1AQp implies ϕ(x) ∈ ϕ(pi)−kA+ pn+1AQp = pi−pkA+ pn+1AQp .Lemma 5.3.2.i) If k ∈ N, then ψ(pik) ∈ A+Qp and ψ(pi−k) ∈ pi−kA+Qpii) ψ(A)⊂ A.Proof. ii) follows from i) and the definition of A. Since ϕ(pi) = (1+pi)p− 1 is a monic polynomial ofdegree p in pi and [ε]i = (1+pi)i is a monic polynomial of degree i in pi , hence {[ε]iϕ(pi) j}0≤i≤p−1, j∈N28forms a basis of polynomials in pi . Moreover, ψ([ε]iϕ(pi) j) =0 i 6= 0pi j i = 0, we thus deduce that ψ(pik) ∈A+Qp if k ≥ 0. If k ≥ 1, thenTrAQp/ϕ(AQp )(pi−k) = ∑ζ p=1((1+pi)ζ −1)−k,which can be written in the form P(ϕ(pi))ϕ(pi)k , where P is a polynomial with coefficient in Zp. Thus theconclusion follows.Corollary 5.3.3. If x ∈ AQp and n ∈ N, then wn(ψ(x))≤ 1+[wn(x)p ]≤ 1+wn(x)p .Proof. Since ϕ(pi)pi p is an unit in A and ψ(xϕ(pi)k ) =ψ(x)pik , we haveψ(pi−kpA+ pn+1AQp) = ψ(ϕ(pi)−kA+ pn+1AQp)⊂ pi−kA+ pn+1AQp ,the conclusion follows.If U = (ai, j)1≤i, j≤d ∈Md(AQp) and n ∈ N, we define wn(U) by wn(U) = supi, j wn(ai, j). Similarly ifV is a Zp-representation of GK and if e1, ...,ed is a basis of D(V ) over AQp , we put wn(a) = supi wn(ai) ifa = ∑di=1 aiei ∈ D(V ). Note that wn depends on the choice of basis e1, ...,ed .Lemma 5.3.4. Let V be a Zp-representation of GK , and let e1, ...,ed be a basis of D(V ) over AQp andΦ = (ai, j) be the matrix defined by e j = ∑di=1 ai, jϕ(ei). If x,y ∈ AQp satisfy the equation (ψ − 1)y = x,then wn(y)≤ sup(wn(x),pp−1(wn(Φ)+1))for all n ∈ N.Proof. Since ϕ(e1), ...,ϕ(ed) is a basis of D(V ) over φ(D(V )), we can write x = ∑di=1 xiϕ(ei) and y =∑di=1 yiϕ(ei). We have ψ(y) = ∑di=1ψ(yi)ei and the equation ψ(y)− y = x translates to a system of equa-tionsyi =−xi +d∑j=1ai, jψ(y j) 1≤ j ≤ d.One gets the inequalitieswn(yi)≤ sup(wn(xi), sup1≤ j≤d(wn(ai, j)+wn(ψ(y j))))≤ sup(wn(x),wn(Φ)+wn(y)p+1)for 1≤ i≤ d, which gives us the inequalitywn(y)≤ sup(wn(x),wn(Φ)+wn(y)p+1)and the conclusion follows.29We now deduce lemma 5.3.1. If n∈N∪{+∞}, let Xn be the set of solutions of the equation (ψ−1)y =x in D(V )/pn+1D(V ). We want to show that Xn is compact. If n∈N, let rn = sup(wn(x),pp−1(wn(Φ)+1)).The set Xn is closed (since ψ−1 is continuous). By the previous lemma, the image of (pi−rnA)d is compactsince A is. If N is finite, it suffices to take n = N to conclude. If N = +∞, the map from x ∈ X+∞ to thesequence of its images modulo pn+1 allows us to identify X+∞ with the closed subset of compact set∏n∈N Xn, and the conclusion follows.5.4 The module D(V )ψ−1Lemma 5.4.1. Let V be a Zp-representation of GK . Then the moduleD(V )ψ−1 has no nonzero p-divisibleelement.Proof. Let x be a p-divisible element of D(V )ψ−1 . For each n∈N, there exist elements yn,zn of D(V ) such thatx = pnyn +(ψ−1)zn. If we fix m ∈ N and if n≥ m+1, then zn is a solution of the equation ψ(z)− z = xmod pm+1. Since the set of solutions is compact due to lemma 5.3.1, there exists a subsequence of {zn}n∈Nwhich converges modulo pm for all m and we have a limit Z in D(V ). By passing to the limit, we obtainx = (ψ−1)z and hence x = 0 in D(V )ψ−1 .Lemma 5.4.2. If V is a Fp-representation of GK and x ∈ mE⊗V , where mE is the maximal ideal of E,then the series ∑+∞n=0ϕn(x) and ∑+∞n=1ϕn(x) converges in mE⊗V and we have(ψ−1)( +∞∑n=0ϕn(x))= ψ(x) and (ψ−1)( +∞∑n=1ϕn(x))= x.Proof. If e1, ...,ed is a basis of V over Fp and x = x1e1 + ...+ xned ∈mE⊗V , there exists r ≥ 0 such thatif νE(xi) ≥ r for 1 ≤ i ≤ d implies that νE(ϕn(xi)) ≥ pnr tends to +∞ and hence we have ϕn(x) tends to0 as n tends to +∞. We thus deduce the convergence of the series. These formulas are the consequencesof the fact that ψ is a left inverse of ϕ .Lemma 5.4.3.i) Let V be a Fp-representation of GK , thenD(V )ψ−1 is a finite dimensional Fp-vector space.ii) There exists a open subgroup of ΓK which acts trivially onD(V )ψ−1 .Proof. Let M = (mE⊗V )HK , which is a lattice of D(V ) fixed by ϕ . If x ∈ M, the sereis ∑+∞n=1ϕn(x)converges in M, and by the previous lemma, we have x = (ψ − 1)(∑+∞n=1ϕn(x)), which proves that (ψ −1)D(V ) contains M.Since ψ is continuous, there exists c ∈ N such that ψ(M) ⊂ pi−cM and since ψ(pi−pkx) = pi−kx, wehave ψ(pi−pkM) ⊂ pi−k−cM. We deduce that if n ≥ b = [ pcp−1 ] + 1, then ψ = 0 inpi−n+1Mpi−nM and ψ − 1 is30bijective on pi−n+1Mpi−nM . Since D(V ) =⋃n∈Npi−nM, which implies the natural map from pi−bMψ−1 toD(V )ψ−1 is anisomorphism.To prove i), it suffices to note that (ψ−1)M contained in M, which implies that D(V )ψ−1 is a quotient ofpi−bMψ−1 . To prove ii), we note that ΓK fixes M and hence pikM for all k ∈Z and the action of ΓK is continuouson D(V ) and M is closed in D(V ), there exists an open subgroup of ΓK acts trivially on pi−bMψ−1 since themodule is endowed with discrete topology.Corollary 5.4.4. If V is a Zp-representation of GK , thenD(V )ψ−1 is a Zp-module of finite type.Proof. D(V )ψ−1 /pD(V )ψ−1 =D(V )(p,ψ−1) =D(V/p)ψ−1 is a Fp-vector space of finite type by the preceding lemma, to-gether with lemma 5.4.1, we get the conclusion.Hence we deduce ii) of proposition 5.2.7 and it remains to prove that D(V )ψ=1 generate D(V ). Wewill need the following lemma.Lemma 5.4.5. Let V be a Fp-representation of GK and X be a sub-Fp-vector space of D(V )ψ=1 of finitecodimension. Then X contains a basis of D(V ) over EK .Proof. Let M = (mE⊗V )HK as above. Note that by lemma 5.4.2, if x ∈Mψ=0, then the series ∑+∞n=0ϕn(x)converges in D(V ) to an element of D(V )ψ=1. We denote it by eul(x). Let e1, ...,ed be a basis of M overE+K . Let r the codimension of X in D(V )ψ=1. If 1 ≤ i ≤ d and j ≥ 1, let zi, j = eul(εϕ(pi jei)). If i andn≥ 1 are fixed, the {zi, j}n≤ j≤n+r form a set of r+1 elements in D(V )ψ=1 and since X is of codimensionr in D(V )ψ=1, we can find elements {a(n)i, j }0≤ j≤r of Fp such that fi,n = ∑rj=0 a(n)i, j zi, j+n belongs to X . Letβi,n = pin∑rj=0 a(n)i, j pi j. We have limn→+∞(εϕ(βi,n))−1 fi,n = ϕ(ei), which implies that the determinant off1,n, ..., fd,n in the basis ϕ(e1), ...,ϕ(ed) is nonzero if n 0 and we have f1,n, ..., fd,n form a basis of D(V )over EK if n is large enough. The lemma follows.Corollary 5.4.6. If V is a Zp-representation of GK , then D(V )ψ=1 generates the AK-module D(V ).Proof. The snake lemma shows that the cokernel of the injective map D(V )ψ=1/pD(V )ψ=1 to D(V/p)ψ=1is identified with the p-torsion part of D(V )/(ψ − 1). In particular, it is of finite dimension over Fp. Bythe preceeding lemma, we have D(V )ψ=1/pD(V )ψ=1 contains a basis of D(V/p) over EK , which lifts toa basis in D(V )ψ=1 that generates D(V ) over AK .31Chapter 6Iwasawa theory and p-adic representations6.1 Iwasawa cohomologyRecall that if n ∈ N, we denote by Kn the field K(ε(n)) = K(µpn). On the other hand, if n≥ 1 (resp. n≥ 2if p = 2), the group ΓKn is isomorphic to Zp. We choose a generator γ1 of ΓK1 and put γn = γ[KN :K1]1 if n≥ 1(if p = 2, we can start from n = 2), this makes γn a generator of ΓKn .Let V be a p-adic representation of GK . The Iwasawa cohomology groups H iIw(K,V ) are defined byH iIw(K,V ) = Qp⊗Zp HiIw(K,T ) where T is any GK-stable lattice of V and whereH iIw(K,T ) = lim←−corKn+1/KnH i(Kn,T )Each of the cohomology group H i(K,T ) is a Zp[ΓK/ΓKn ]-module, and HiIw(K,T ) is then endowed withthe structure of a ΛK-module. Roughly speaking, theses cohomology groups are where Euler systems live(at least locally).If V is a Zp-representation or a p-adic representation of GK , we endow ΛK ⊗Zp V with the naturaldiagonal action of GK . If we consider ΛK ⊗Zp V as the space of measures of ΓK with values in V (seesection ??), the measure σ(µ) is the map sends a continuous map f : ΓK 7→V to the element∫ΓKf (x)σ(µ) = σ(∫ΓKf (σx)µ) ∈VIf V is a Zp-representation or a p-adic representation of GK and k ∈ Z, we denote by V (k) the twist ofV by the k-th power of the cyclotomic character and if x ∈V , we denote x(k) its image in V (k).If µ ∈ Hm(K,ΛK ⊗Zp V ) and if τ 7→ µτ1,...,τm is a continuous m-cocycle represents µ , then τ 7→(∫ΓKnχ(x)kµτ1,...,τm)(k) is a m-cocycle of GK with values in V (k) whose class (∫ΓKnχ(x)kµ)(k) in Hm(Kn,V (k))does not depend on the choice of cocycle representing µ .Shapiro’s lemma allows us to replace the projective limit in the definition of HmIw(K,V ) by a group32cohomology.Proposition 6.1.1. Let V be a Zp-representation or a p-adic representation of GK .If m ∈N and k ∈ Z, themap which sends µ to (...,∫Kn χ(x)kµ(k), ...) is an isomorphism from H i(K,ΓK ⊗Zp V ) to H iIw(K,V (k)).In particular, if k ∈ Z, the cohomology groups HmIw(K,V ) and HmIw(K,V (k)) are isomorphic.Proof. The case of Qp follows from the case of Zp by tensoring withQp. If M is a GKn-module, wedenote IndKKnM the set of continuous maps from GK to M satisfying a(hx) = ha(x) if h ∈GKn . The moduleIndKKnM is provided with a continuous action of GK , the image ga of a by g ∈ GK , is given by the formula(ga)(x) = a(xg). If M is a GK module, and a ∈ IndKKnM, the map sends x ∈ GK to x−1(a(x)) is constantmodulo GKn , and the map IndKKnM→Zp[Gal(Kn/K)]⊗M a 7→∑x∈Gal(Kn/K) x−1(ax)δx−1 is an isomorphismof GK-modules. By Shapiro’s lemma, we have a canonical isomorphism from H i(K,Zp[Gal(Kn/K)]⊗M)to H i(Kn,M). On the other hand, the corestriction map from H i(Kn+1,M) to H i(Kn,M) is derived fromthe previous isomorphosm and the natural map from Zp[Gal(Kn+1/K)] to Zp[Gal(Kn/K)]. we thus obtaina natural mapH i(K,ΛK⊗M)→ lim←−Hi(K,(Λ/ωn)⊗M)' lim←−Hi(Kn,M).It remains to show that this map is an isomorphism.Surjectivity is a obvious. To prove injectivity, it suffices to verify that the map from H i(K,ΛK⊗M) toH i(K,Λ/(ωn, pn)⊗M) is injective. SinceΛK = lim←−ΛK/(ωn, pn), it suffices to show that H i(K,(Λ/ωn, pn)⊗M) satifies the Mittag-Leffler condition (c.f. [15] ), which is obvious since the group is finite.By lemma 5.2.5, the map ιψ,γn identifiesD(V )ψ=1γn−1 with a subgroup of H1(Kn,V ) if ΓKn is torsionfree, we thus obtain a map h1Kn,V : D(V )ψ=1 → H1(Kn,V ). Explicitly, if y ∈ D(V )ψ=1, then (ϕ − 1)y ∈D(V )ψ=0 and since γn− 1 is invertible on D(V )ψ=0, there exists xn ∈ D(V )ψ=0 satisfying (γn− 1)xn =(ϕ − 1)y (i.e. (xn,y) ∈ Z1ϕ,γn(Kn,V )). On the other hand, lemma 5.1.2 implies that the image ιψ,n(y) andlog0p(γn)ιϕ,γn(xn,y) in H1(Kn,V ) does not depend on the choice of γ .By lemma 6.2.1 below, we have corKn+1/Kn ◦ h1Kn+1,V = h1Kn,V . On the other hand, if ΓKn is no longertorsion free, we define h1Kn,V by the relation corKn+1/Kn ◦h1Kn+1,V = h1Kn,V . Thus we associate every elementin D(V )ψ=1 to a collection of Galois cohomology classes h1Kn,V (y) ∈H1(Kn,V ) for n≥ 1. The main resultof this section is:Theorem 6.1.2. (Fontaine) Let V be a Zp-representation or a p-adic representation of GK .i) If y ∈ D(V )ψ=1, then (...,h1Kn,V (y), ...) ∈ H1Iw(K,V ).ii) The map Log∗V ∗(1) : D(V )ψ=1→ H1Iw(K,V ) defined by previous paragraph is an isomorphism.6.2 Corestriction and (ϕ,Γ)-modulesi) of theorem 6.1.2 is a consequence of the following lemma.33Lemma 6.2.1. If n≥ 1, letTγ,n : H1(Cϕ,γn(Kn,V ))→ H1(Cϕ,γn−1(Kn−1,V ))be the map induced by (x,y) ∈ Z1(Cϕ,γn(Kn,V )) to (γn−1γn−1−1 x,y) ∈ Z1(Cϕ,γn−1(Kn−1,V )). Then the diagramH1(Cϕ,γn(Kn,V ))Tγ,n //ιϕ,γnH1(Cϕ,γn−1(Kn−1,V ))ιϕ,γn−1H1(Kn,V )corKn/Kn−1 // H1(Kn−1,V )is commutative.Proof. Recall that if G is a group, M is a G-module and H a subgroup of finite index of G, the core-striction map cor : H1(H,N)→ H1(G,M) can be written in the following way: let X ⊂ G is a system ofrepresentatives of G/H and, if g ∈ G, let τg is the permutation of X defined by τg(x)H = gxH if x ∈ X . Ifc ∈ H1(H,M) and h 7→ ch is a cocycle which represents c, theng→ ∑x∈Xτg(x)(cτg(x)−1gx)is a cocycle of G with values in M whose class in H1(G,M) does not depend on the choice of X and isequal to cor(c).If N is a G-submodule of M such that the image of c in H1(H,N) is trivial (i.e. there exists b ∈ N suchthat we have ch = (h−1)b for all h ∈ H), then cor(c) is the class of the cocycle g 7→ (g−1)(∑x∈X xb).In particular, we put G = GKn−1 , H = GKn and, if γ˜n−1 is a lift of γn−1 in GKn−1 , we take X ={1, γ˜n−1, ..., γ˜ p−1n−1 }. Take N = Frac(Zp[[GKn−1 ]])⊗Zp[[GKn−1 ]] (A⊗Zp V ). If (x,y) ∈ Z1(Cϕ,γ(Kn,V )) andif b ∈ A⊗ T , the cocycle cx,y is given by the formula cx,y(τ) = (τ − 1)c, where c = yγ˜n−1 − b ∈ N. Itfollows that corKn/Kn−1(ιϕ,γn(x,y)) is represented by the cocycleτ → (σ −1)(p−1∑i=0γ˜ in−1c) = (σ −1)(yγ˜n−1−1−p−1∑i=0γ˜ in−1b)and since(ϕ−1)(p−1∑i=0γ˜ in−1b) =p−1∑i=0γ˜ in−1((ϕ−1)b) =γ˜ pn−1−1γ˜n−1−1x =γn−1γn−1−1x,we see that this cocycle is just ιϕ,γn−1(Tγ,n(x,y)), and the conclusion follows.34Remark 6.2.2. One can also hide the explicit calculation by noting that, if n≥ 1, the diagramCϕ,γn(Kn,V ) : 0 // D(V ) //γn−1γn−1−1D(V )⊕D(V ) //( γn−1γn−1−1,id)D(V ) //id0Cϕ,γn−1(Kn,V ) : 0 // D(V ) // D(V )⊕D(V ) // D(V ) // 0is commutative and functorial on V and induces a homomorphism of cohomology group from H∗(Kn, ·)to H∗(Kn−1, ·) which coincides with with the corestriction map at ∗= 0 and hence is corestriction map.6.3 Interpretation of D(V )ψ=1 and D(V )ψ−1 in Iwasawa theoryWe now turn to the proof of ii) of theorem 6.1.2. Lemma 6.2.1 implies that the map (ιψ,γn)n∈N induces anisomorphism from the projective limit of H1(Cψ,γn(Kn,V )) with respect to the map Tγ,n to H1Iw(K,V ). Onthe other hand, lemma 5.2.5, implies by passing to the projective limit, that we have an exact sequence:0 // lim←−D(V )ψ=1γn−1// lim←−H1(Cψ,γn(Kn,V )) // lim←−(D(V )ψ−1 )ΓKnThe projective limit of D(V )ψ=1γn−1 is take with respect to the natural maps induced by the identity on D(V )ψ−1and that of (D(V )ψ−1 )γn=1 with respect to the mapγn+1−1γn−1: (D(V )ψ−1)γn=1→ (D(V )ψ−1)γn−1=1.Hence ii) of theorem 6.1.2 holds by the following proposition:Proposition 6.3.1. If V is a Zp-representation of GK , theni) The natural map from D(V )ψ=1 to lim←−D(V )ψ=1γn−1 is an isomorphism.ii) lim←−(D(V )ψ−1 )γn=1 = 0Proof. i) Let (xn)n∈N ∈ lim←−D(V )ψ=1γn−1 . The compactness of D(V )ψ=1 [c.f. proposition 5.2.7 i)] impliesthat the sequence xn admits an accumulation point x ∈ D(V )ψ=1 and the image of x in lim←−D(V )ψ=1γn−1 is byconstruction (xn)n∈N. The natural map from D(V )ψ=1 to lim←−D(V )ψ=1γn−1 is hence surjective.By the compactness of D(V )ψ=1 and the fact that if x∈D(V ), then (γn−1)x tends to 0 when n tends to+∞ implies that if U is open in D(V ) fixed by Γ, then there exist nU ∈ N such that (γn−1)D(V )ψ=1 ⊂Uif n≥ nV . This implies that⋂n∈N(γn−1)(D(V )ψ=1) = {0} and we prove the injectivity.ii) D(V )ψ−1 is a free Zp-module of finite rank [c.f. proposition 5.2.7 ii)], the sequence (D(V )ψ−1 )γn=1 isstationary since it is increasing. One can deduce the fact that there exists n0 ∈ N such thatγn−1γn−1−1 is35multiplication by p on (D(V )ψ−1 )γn=1 if n ≥ n0, which proves the statement sinceD(V )ψ−1 has no p-divisibleelement [c.f. lemma 5.4.1].Remark 6.3.2. We have H2(Kn,V ) ∼= H2(Cψ,γn(Kn,V )) =D(V )(ψ−1,γn−1) . We deduce that if V is a Zp-representation, then H2Iw(K,V ) is the projective limit ofD(V )(ψ−1,γn−1) sinceD(V )ψ−1 is a Zp-module of finitetype on which ΓK acts continuously by ii) of lemma 5.4.3, the natural map fromD(V )ψ−1 to the projectivelimit of D(V )(ψ−1,γn−1) is an isomorphism, this proves thatD(V )ψ−1 is identified with H2Iw(K,V ).By the above proposition, one can summarize the above results as follows:Corollary 6.3.3. The complex of Qp⊗Zp ΛK-modules0 // D(V )1−ψ // D(V ) // 0computes the Iwasawa cohomology of V . i.e. H1Iw(K,V ) = D(V )1−ψ and H2Iw(K,V ) =D(V )ψ−1There is a natural projection map prKn,V : HiIw(K,V )→ Hi(Kn,V ) and when i = 1 it is of course equalto the composition of:H1Iw(K,V ) // D(V )ψ=1 h1Kn ,V // H1(Kn,V ).The H iIw(K,V ) have been studied in detail by Perrin-Riou, who proved the followingProposition 6.3.4. If V is a p-adic representation of GK , theni) The torsion submodule of H1Iw(K,V ) is a Qp⊗Zp ΛK-module isomorphic to VHK and H1Iw(K,V )/VHKis a free Qp⊗Zp ΛK-module whose rank is [K : Qp]dimQpV .ii) H2Iw(K,V ) is isomorphic to V (−1)HK as Qp⊗Zp ΛK-module. In particular, it is torsion.iii) H iIw(K,V ) = 0 when i 6= 1,2.Proof. See [18, 3.2.1].36Chapter 7De Rham representations andoverconvergent representations7.1 De Rham representations and crystalline representationsRecall A˜+ = W (E˜+), the ring of Witt vectors with coefficients in E˜+. We define the homomorphismθ : A˜+→ OCp byθ(∑k≥0pk[xk]) = ∑k≥0pkx(0)kOne can show that this is a surjective map and ker(θ : A˜+→ OCp) is generated by ω = pi/ϕ−1(pi).We can extend θ to a homomorphism from B˜+ = A˜+[ 1p ] to Cp, and we denote by B+dR the ringlim←− B˜+/(kerθ)n, thus θ can be extended by continuity to a homomorphism from B+dR to Cp. This makesB+dR a discrete valuation ring with maximal ideal kerθ and residue field Cp. The action of GQp on A˜+extends by continuity to an action of GQp on B+dR. The series log[ε] = ∑+∞n=1(−1)n−1n pin converges in B+dR toan element which we denote by t, which is a generator of kerθ with a GQp action defined by σ(t) = χ(σ)twhere σ ∈ GQp . This element can be viewed as a p-adic analogue of 2pii.We put BdR =B+dR[t−1], this makes BdR a field with filtration defined by FiliBdR = t iB+dR. This filtrationis stable by the action of GK .Let K be a finite extension of Qp and V be a p-adic representation of GK . We say that V is de Rham ifit is BdR-admissible, which is equivalent to the assertion that K-vector space DdR(V ) = (BdR⊗Qp V )GK isof dimension d = dimQp(V ). On the other hand, DdR(V ) is endowed with a filtration induced by BdR. Wehave FiliDdR(V ) = DdR(V ) if i 0 and FiliDdR(V ) = {0} if i 0.The ring B+cris is defined byB+cris = {∑n≥0anωnn!| an ∈ B˜+ is a sequence converging to 0},37and Bcris = B+cris[1t ]. The ring Bcris is a subring of BdR stable under GQp containing t and the action of ϕon B˜+ is extended by continuity to an action of B+cris. In particular, we have ϕ(t) = pt.We say V is crystalline if it is Bcris-admissible, which is equivalent to the assertion that F = K∩Qurp -vector space Dcris(V ) = (Bcris⊗V )GK is of dimension d = dimQp(V ). The action of ϕ on Bcris commuteswith the action of GQp , which endows Dcris(V ) a natural semi-linear action of ϕ . i.e ϕ( f d) = ϕ( f )ϕ(d)where f ∈ F and d ∈ Dcris(V ).We have (BdR⊗Qp V )GK = DdR(V ) = K⊗F Dcris(V ), thus the crystalline representation is de Rhamand K⊗F Dcris(V ) is a filtered K-vector space. Hence if V is de Rham (resp. crystalline) and k ∈ Z, so isV (k), and we have DdR(V (k)) = t−kDdR(V ) (resp. Dcris(V (k)) = t−kDcris(V )).7.2 Overconvergent elementsEvery element x of B˜ can be written uniquely in the form ∑k−∞ pk[xk], where xk is an element of E˜ andthe series converges in B+dR if and only if the series ∑k−∞ pkx(0)k converges in Cp, which is equivalentto k+ νE(xk) tends to +∞ as k tends to +∞. More generally, if n ∈ N, ϕ−n(x) converges if and only ifk+ p−nνE(xk) tends to +∞ as k tends to +∞.For r ≥ 0, we setB˜†,r = {x ∈ B˜ | limk→+∞νE(xk)+prp−1k =+∞}.This makes B˜†,r into an intermediate ring between B˜+ and B˜. We denote B˜† =∪r≥0B˜†,r, which is a subfieldof B˜ with action of GK and ϕ . On the other hand, we have a well-defined injective map ϕ−n : B˜†,rn →B+dR,where rn = pn−1(p−1).We denote A˜†,r = B˜†,r∩A˜, that is, the subring of elements x=∑+∞k=0 pk[xk] of A˜ such that νE(xk)+ prp−1 ktends to +∞ as k tends to +∞. We have B˜†,n = A˜†,n[ 1p ].By putting B† = B∩ B˜†, A†,r = A∩ A˜†,r and B†,r = B∩ B˜†,r, we define a subring B† of B fixed byϕ and GQp , and if r ∈ R, subrings A†,r and B†,r of B are fixed by GQp . By construction, ϕ−n(B†,rn) isnaturally identified with a subring of B+dR. Finally, if K is a finite extension of Qp, we set B†K = (B†)HK ,A†,rK = (A†,r)HK and B†,rK = (B†,r)HK .Let eK be the ramification index of K∞ over F∞ and F ′ ⊂ K∞ be the maximal unramified extension ofQp contained in K∞. Let piK be a uniformizer of EK = kF ′((piK)) and PK ∈ EF ′ be a minimal polynomialof piK and δ = νE(P′(piK)). Choose PK ∈ AF ′ such that its image modulo p is PK . By Hensel’s lemma,there exists a unique piK ∈ AK such that PK(piK) = 0 and piK = piK modulo p. In particular, if K = F ′, onecan take piK = pi .The terminology ”overconvergent” can be explained by the following proposition:Proposition 7.2.1. If r≥ r(K), then the map f 7→ f (piK) from BeKrF ′ to B†,rK is an isomorphism, where BαF ′is the set of power series f (T ) = ∑k∈Z akTk such that ak is a bounded sequence of elements of F ′, andsuch that f (T ) is holomorphic on the p-adic annulus {p−1/α ≤ ‖T‖< 1}.38Proof. See lemma II.2.2 [5].Proposition 7.2.2. If K is a finite extension of Qp, then B†K is an extension of B†Qpof degree [BK : BQp ] =[K∞ : Qp(µp∞)] and there exists a(K) ∈ N such that if n ≥ a(K), then ϕ−n(B†,rnK ) ⊂ Kn[[t]], where rn =pn−1(p−1).Proof. In the case K is unramified over Qp, one can follow proposition 7.2.1 i) using the fact that Kn[[t]]is closed in B+dR and the formulaϕ−n(pi) = ϕ−n([ε]−1) = [ε p−n ]−1 = ε(n) exp(t/pn)−1 ∈ Kn[[t]].For the general case, by remark 4.1.2, there exists ω = (ω(n))n∈N ∈ lim←−OKn such that ω(n) is a uniformizerof OKn if n is large enough and then piK = ιK(ω) is a uniformizer of EK such that it is totally ramified ofdegree eK over EF ′ . Let P(X) = XeK + aeK−1XeK−1 + ...+ a0 ∈ EF ′ [X ] be the minimal polynimial of piKover EF ′ and let δ = νE(P′(piK)). If 0 ≤ i ≤ eK −1, let ai ∈ OF [[pi]] ⊂ AF whose reduction modulo p isai and let P(X) = XeK +aeK−1XeK−1 + ...+a0 ∈ AF [X ]. By Hensel’s lemma, the equation P(X) = 0 has aunique solution piK in AK whose reduction modulo p is piK and we can write it in the formpiK = [piK ]++∞∑i=1pi[αi], (7.1)where αi are elements of E˜ verifying νE(αi)≥−iδ . In particular, piK ∈ A†,rK ifpp−1 r ≥ δ , hence we haveA†,rK = A†,rF [piK ] ifpp−1 r ≥ δ . Thus it suffices to prove it when n large enough, then piK,n = ϕ−n(piK) ∈Kn[[t]].Let Pn (resp. Qn) be the polynomial obtained by the map θ ◦ϕ−n (resp. ϕ−n) applied on the coefficientsof P, which is a polynomial with coefficients in OFn (resp. Fn[[t]]) with θ(piK,n) (resp. piK,n) as a root. Onthe other hand, by definition of ιK (c.f. 4.1.2), we have νp(ω(n)−pi(n)K )≥ 1p if n large enough and formula(7.1) shows that νp(θ(piK,n)−pi(n)K )≥ (1− δpn ). Then we have νp(Pn(ω(n)))≥ 1p if n large enough andνp(P′n(ω(n)) =1pnνE(P′(piK)) =δpn<12pif n large enough. By Hensel’s lemma, the equation Pn(X) = 0 has a unique solution in Cp close to ω(n)and hence belongs to OKn since ω(n) and the coefficients of Pn do. We deduce that θ(piK,n) belongs to Kn.By using Hensel’s lemma again, one can show that Qn has a unique solution in B+dR whose image by θ isθ(piK,n) and thus belongs to Kn[[t]].We endow BQp with the differential operator ∂ defined by continuity and the derivation ∂pi = 1+pi .We therefore have ∂ = [ε] ddpi =ddt (Note that t 6∈ BQp). The derivation can be extended uniquely to amaximal unramified extension of BQp in B˜, hence by continuity to a derivation ∂ from B to B.39Lemma 7.2.3. If K is a finite extension of Qp, there exists m(K) ∈ Z such that, if n≥m(K) and x in B†,rnK ,theni) ∂x ∈ B†,rnK .ii) ϕ−n(∂x) = pn∂ (ϕ−n(x)).Proof. If K = Qp, explicit calculation using proposition 7.2.1 i), shows that we can take m(K) = 1. Forthe general case, let α be a generator of B†K over B†Qpand P be its minimal polynomial. The identity,0 = ∂ (P(α)) = P′(α)∂α+∂P(α),where ∂P is the polynomial obtained by applying ∂ on the coefficients of P, shows that ∂α = − ∂P(α)P′(α) ∈B†K . It is then possible to take m(K) any integer such that B†,m(K)K contains ∂α and α .For ii), it suffices to note that ϕ−n ◦∂ is pn∂ ◦ϕ−n are two derivations of B†,rnK coincides on B†,rnQpbyϕ−n ◦∂ ([ε]) = ϕ−n([ε]) = ε(n) exp(p−nt)pn∂ ◦ϕ−n([ε]) = pn ddt(ε(n) exp(p−nt)) = ε(n) exp(p−nt).7.3 Overconvergent representationsDefinition 7.3.1. If V is a p-adic representation of GK , we setD†(V ) = (B†⊗Qp V )HK and D†,r(V ) = (B†,r⊗Qp V )HKWe have dimB†KD†(V )≤ dimQpV and we say that V is overconvergent if equality holds, which is equivalentto the assertion that D(V ) has a basis over BK made up of elements of D†(V ).Proposition 7.3.2.i) Every p-adic representation of GK is overconvergent.ii) There exists r(V ) such that D(V )ψ=1 ⊂ D†,r(V )(V ).iii) If V is overconvergent and n∈N, then γn−1 admits a an continuous inverse on D†(V )ψ=0. Moreover,there exists n2(V ) such that if n≥ n2(V ), then(γn−1)−1(D†,rn(V )ψ=0)⊂ D†,rn+1(V )ψ=0Proof. i), iii) see [5]. ii) follows from lemma 8Explicit reciprocity laws and de RhamRepresentation8.1 The Bloch-Kato exponential map and its dualLet K be a finite extension of Qp and V a p-adic representation of GK . We have the fundamental exactsequence0 // Qp // Bϕ=1cris// BdR/B+dR // 0(c.f. [9, proposition III 3.5]). Tensoring this exact sequence with V and taking the invariants under theaction of GK , we obtain:0 // V GK // Dcris(V )ϕ=1 // ((BdR/B+dR)⊗V )GK // H1e (K,V ) // 0where we denote H1e (K,V ) the kernel of the natural map from H1(K,V ) to H1(K,Bϕ=1cris ⊗V ). We call theisomorphism induced by the connecting homomorphismexpK,V :DdR(V )Fil0DdR(V )+Dcris(V )ϕ=1−→ H1e (K,V )⊂ H1(K,V )the Bloch-Kato exponential of V over K and we denote its inverse bylogK,V : H1e (K,V )−→DdR(V )Fil0DdR(V )+Dcris(V )ϕ=1the Bloch-Kato logarithm of V over K. Moreover, if V is de Rham and k  0, then expK,V (k) is anisomorphism from DdR(V (k)) to H1(K,V (k)).The choice of t gives an isomorphism from DdR(Qp(1)) = t−1K to K. If V is a p-representation of41GK , the couple [, ]DdR(V ) is defined by the composition of mapsDdR(V )⊗DdR(V ∗(1))∼= DdR(V ⊗V ∗(1)) // DdR(Qp(1))∼= KTrK/Qp // Qp .This composition is non-degenerate, hence DdR(V ∗(1)) can be naturally identified with the dual of DdR(V ).Similarly, via the cup productH1(K,V )×H1(K,V ∗(1))→ H2(K,Qp(1)) = Qp,H1(K,V ∗(1)) is naturally identified with the dual of H1(K,V ). This allows us to view the map exp∗K,V ∗(1)as the transpose of the map expK,V ∗(1) : DdR(V∗(1))→ H1(K,V ∗(1)) as a map from H1(K,V ) to DdR(V ),whose image is contained in Fil0(DdR(V )). If V is de Rham and k  0, the map exp∗K,V ∗(1+k) is anisomorphism from H1(K,V (−k)) to DdR(V (−k)).If x ∈ K∞ and n ∈ N, then 1pm TrKm/Kn(x) does not depend on the choice of integer m≥ n+1 such thatx is belongs to Km. We denote Tn the above Qp-linear map from K∞ to Kn. If n ≥ 1 and x ∈ Kn, thenTn(x) = p−nx. We haveTm = TrKn/Km ◦Tn if n≥ m.We also denote by Tn the map from K∞((t)) to Kn((t)) defined by Tn(∑+∞k=0 aktk) = ∑+∞k=0 Tn(ak)tk.Proposition 8.1.1.i) K∞((t)) is dense in BHKdR and Tn can be extended to a Qp-linear map from BHKdR to Kn((t)).ii) If F ∈ BHKdR , then limn→+∞ pnTn(F) = F.Proof. See [9], proposition V.4.5.Let V be a de Rham representation of GK , we have BdR⊗Qp V ∼=BdR⊗K DdR(V ) and H1(K,BdR⊗V )=H1(K,BdR⊗DdR(V )) = H1(K,BdR)⊗DdR(V ). Since K ∼= H1(K,BdR) via x 7→ x∪ logχ . We thus get anisomorphismDdR(V )→ H1(K,BdR⊗V ); x 7→ x∪ logχProposition 8.1.2. If V is a de Rham representation, the map x ∈ DdR(V ) to a cocycle τ 7→ x logχ(τ) ∈DdR(V )⊂ BdR⊗V induces an isomorphism from DdR(V ) to H1(K,BdR⊗V ) and the map exp∗V ∗(1) is thecomposition of the inverse of the above isomorphism and the natural map from H1(K,V ) to H1(K,BdR⊗V ).Proof. See [16] proposition 1.4. of chapter II.We define the map prKn : BHKdR → Kn((t)) by the formula prKn(x) =1[Km:Kn]TrKm/Kn(x) if x ∈ K∞ andm ≥ n such that x ∈ Km and there exists a′(K) ≥ 1 such that one has pnTn = prKn if n ≥ a′(K). From (ii)of proposition 8.1.1, we can show that limn→+∞ prKnx = x if x ∈ BHKdR42If V is a de Rham representation, the natural map from BHKdR ⊗K DdR(V ) to (BdR⊗Qp V )HK is an isomor-phism and we can extend the map Tn and prKn for n∈N by linearity to BHKdR ⊗K DdR(V ). On the other hand,if F ∈ K∞((t))⊗DdR(V ), we can write F uniquely as the form ∑k−∞ tkdk, where dk ∈ K∞⊗DdR(V ). Wedenote ∂V (−k)(F) the element tkdk of K∞⊗DdR(V (−k)).Proposition 8.1.3. Let V be a p-adic representation of GK and n,m∈N be two integers. If c∈H1(Km,V (−k)),there exists a cocycle τ 7→ cτ on ΓKm with values in (BdR⊗V (−k))HK which has the same image as c inH1(Km,BdR⊗V (−k)). Moreover, if V is de Rham. thenexp∗V ∗(1+k)(c) = ∂V (−k) ◦prKm(1logp(χ(γ))cγ)for all γ ∈ ΓKm such that logp(χ(γ)) 6= 0Proof. Since H1(K∞,BdR⊗V ) is zero (c.f. [9] theorem IV.3.1), the inflation map from H1(ΓKm ,(BdR⊗V )HK ) to H1(Km,BdR⊗V ) is an isomorphism, hence we have the existance of coycle τ 7→ cτ . On the otherhand, if V is de Rham, the map τ 7→ ∂V (−k) ◦ prKm(cτ) is a cocycle on ΓKm with values in DdR(V (−k))which ΓKm acts trivially. It is of the form τ 7→ d logp χ(τ), where d ∈ DdR(V (−k)) and if c is zero, whichimplies τ 7→ cτ is a coboundary, hence d = 0. One cae deduce that ∂V (−k)◦prKm(1logp χ(γ)cγ)∈DdR(V (−k))does not depend on γ ∈ ΓKm such that χ(γ) 6= 0 and the choice of cocycle τ 7→ cτ representing c, whichprovides us a natural map from H1(K,V (−k)) to DdR(V (−k)) coincides with exp∗V ∗(1+k) by proposition8. Explicit reciprocity lawLet V be a de Rham representation of GK and let n(V )≥ n1(V ) be the smallest integer satisfies rn(V ) ≥ rV(c.f. proposition 7.3.2). If µ ∈ H1Iw(K,V ), then Exp∗V ∗(1)(µ) ∈ D(V )ψ=1. On the other hand, D(V )ψ=1 ⊂D†,rV (V ). If n≥ n(V ), we can view ϕ−n(Exp∗V ∗(1)(µ)) as an element in BdR⊗V . Since ϕ−n(Exp∗V ∗(1)(µ))is an element of BdR⊗V fixed by HK , we can consider its image under Tm.Theorem 8.2.1. Let V be a de Rham representation and m ∈ N.i) If n ≥ sup(m,n(V )) and µ ∈ H1Iw(K,V ), then Tm(ϕ−n(Exp∗V ∗(1)(µ))) is an element in Km((t))⊗KDdR(V ) independent of n, we denote it by Exp∗V ∗(1),Km(µ).ii) If µ ∈ H1Iw(K,V ), thenExp∗V ∗(1),Km(µ) = ∑k∈Zexp∗V ∗(1+k)(∫ΓKmχ(x)−kµ).iii) There exists m(V )≥ n(V ) such that if m≥ m(V ) and µ ∈ H1Iw(K,V ), thenExp∗V ∗(1),Km(µ) = p−mϕ−m(Exp∗V ∗(1)(µ)).43Remark 8.2.2.i) The image of H1(Km,V (−k)) by exp∗V ∗(1+k) is contained in Fil0DdR(V (−k)) = Fil0(tKDdR(V )) whichis zero if k 0. Hence the series in ii) converges in BdR⊗DdR(V ).ii) We have a map µ ∈ H1Iw(K,V ) 7→∫ΓKnχkµ ∈ H1(GKn ,V (k)), thus exp∗V (1+k)(∫ΓKnχ−kµ) ∈ tkKn⊗KDdR(V ).iii) For n≥ n(V ), we have ϕ−n(D†,rn(V ))⊂ Kn((t))⊗DdR(V ).Proof. Given that Tr = TrKm/Kr ◦Tm if r≤m and if L1⊂ L2 are two finite extension of K, then the diagramH1(L2,V )exp∗V∗(1)//corL2/L1L2⊗DdR(V )TrL2/L1⊗idH1(L1,V )exp∗V∗(1)// L1⊗DdR(V )is commutative. Thus, to prove i) and ii), it suffices to prove them for m large enough. We can thereforesuppose that m≥ n(V )+1, prKm = pmTm and log0p(γm) =logp(χ(γm))pm .Denote y the element Exp∗V ∗(1)(µ) in D(V )ψ=1 and if i∈Z, denote y(i) the image of y in D(V (i))ψ=1 =D(V )ψ=1 (same as set but differerent as Galois module by twist χ i). By construction of Exp∗V ∗(1) (indeedits inverse),∫ΓKnχ(x)−kµ is represented by the cocycleσ 7→ c′σ = log0p(γm)( σ −1γm−1y(−k)− (σ −1)b),where b ∈ A⊗V is a solution of the equation (ϕ−1)b = (γm−1)−1((ϕ−1)y)(−k)).By definition of n(V ), we have y ∈ D†,rn(V )(V ) ⊂ D†,rm−1(V ) and (ϕ − 1)y ∈ D†,rm(V ), which impliesthat (γm− 1)−1(ϕ − 1)y(−k) ∈ D†,rm+1(V ) by proposition 7.3.2, and the same argument as lemma 5.3.4implies that b ∈A†,rm⊗V . Since we suppose that n≥ sup(m,n(V )), we have ϕ−n(b) and ϕ−n(y) are bothin B+dR⊗V and c′σ = ϕ−n(c′σ ) is a cocycle with values in BdR⊗V which differs from the coycleσ 7→ cσ =logp χ(γm)pmσ −1γm−1ϕ−n(y(−k))by a coboundary σ 7→ logp χ(γm)pm (σ −1)ϕ−n(b). Since y is fixed by HK , the cocycle σ 7→ cσ has values in(BdR⊗V )HK which allows us to use proposition 8.1.3 to calculate it and we obtainexp∗V ∗(1+k)(∫ΓKmχ(x)−kµ)=1logp χ(γm)∂V (−k)(prKm(cγm)) =1pm∂V (−k)(prKm(ϕ−n(y)))and since 1pm prKm = Tm and Tm(x) = ∑k∈Z ∂V (−k)(Tm(x)) if x ∈ (BdR⊗V )HK , we deduce i) and ii).44To prove iii), it suffices to show that if m is large enough, then ϕ−m(Exp∗V ∗(1)(µ)) ∈ Km((t))⊗KDdR(V ). We need the following lemma:Lemma 8.2.3. Let d be an integer ≥ 1. If U ∈ GLd(BHKdR ) and there exists n ∈ N such that U−1γ(U) ∈GLd(Kn((t))), then there exists m ∈ N such that U ∈ GLd(Km((t))).Proof. Let A = U−1γ(U). If m≥ n, let Um = prKm(U). Using the fact that prKm is Kn((t))-linear if m≥ n,we obtain, by applying prKm to the identity UA = γ(U), the relation UmA = γ(Um). On the other hand,since limm→+∞Um = U , there exists m ≥ n such that Um is invertible. Subtract A by the above identity,We have UU−1m is fixed by γ and therefore belongs to GLd(K). We hence deduce that U belongs toGLd(Km((t))).Let e1, ...,ed be a basis of D†,rn(V )(V ) over B†,rn(V )K which contains D(V )ψ=1 and f1, ..., fd a basis ofDdR(V ) over K. Let A = (ai, j) ∈ GLd(B†K), B = (bi, j) ∈ GLd(B†K) and if m ≥ n(V ), C(m) = (c(m)i, j ) ∈GLd(BHKdR ) the matrices defined byγ(ei) =d∑j=1ai, je j, ϕ(ei) =d∑j=1bi, je j and ϕ−m(ei) =d∑j=1c(m)i, j f j.The relation γ ◦ϕ−m = ϕ−m ◦ γ and ϕ−m = ϕ−(m+1) ◦ϕ is translated toγ(C(m)) =C(m)ϕ−m(A) and C(m) =C(m+1)ϕ−(m+1)(C−1)since f1, ..., fd is fixed by γ . There exists n0 ≥ n(V ) such that A and B belongs to GLd(B†,rn0K ). Since thereexists m0 ∈N such that ϕ−m(B†,rmK ) ∈ Km[[t]], if m≥m0. By above relations and lemma 8.2.3, there existsm(V )≥ sup(n0,m0) = m1 such that C(m1) ∈ GLd(Km(V )((t))), which implies that C(m) ∈ GLd(Km(V )((t)))for m ≥ m(V ) by second relation. Since x ∈ D(V )ψ=1 is of the form ∑di=1 xiei where x ∈ B†,rn(V ) andϕ−m(B†,rn(V )) ⊂ Km[[t]] if m ≥ m(V ) by the choice of m(V ), we have the inclusion ϕ−m(D(V )ψ=1) ⊂Km((t))⊗K DdR(V ) if m≥ m(V ). This proves iii).8.3 Connection with the Perrin-Riou’s logarithmOur Goal in this paragraph is to compare Exp∗V ∗(1) and Perrin-Riou’s logarithm constructed in [9]. Let usrecall the construction of logarithm map.Proposition 8.3.1. Let V be a de Rham representation. Let W be the finite dimensional Qp-vector space∪n∈N(Bϕ=1cris ⊗V )GKn . Let µ ∈ H1Iw(Kn,V ) such that∫ΓKnµ ∈ H1e (K,V ) for all n ∈ N and τ → µτ a contin-uous cocycle representing µ . Finally, if n 0, let cn be the unique element of (Bϕ=1cris ⊗V )/W verifying(1− τ)cn =∫Kn µτ for all τ ∈ GKn .45i) The sequence pncn converges in (Bϕ=1cris ⊗V )/W to an element of (Bϕ=1cris ⊗V )GK/W denoted byLogV (µ).ii) If n ∈ N, thentddtTn(LogV (µ)) = ∑k∈Nexp∗V ∗(1+k)(∫ΓKnχ(x)−kµ).Proof. See [9, Theorem VI.3.1 and Theorem VII.1.1].Remark 8.3.2.i) There exists k0 ∈N such that the condition∫ΓKnµ ∈H1e (K,V ) for all n ∈N holds automatically if wereplace V by V (k) for k ≥ k0.ii) The operator ddt annihilates K∞⊗DdR(V ) and hence W , which explains why we don’t need to pass toquotient W in formula (ii).The connection between LogV and Exp∗V ∗(1) in the case V is de Rham is given by:Theorem 8.3.3. Let V be a de Rham representation of GK . There exists m(V ) ≥ n(V ) such that if m ≥m(V ) and µ ∈ H1Iw(K,V ) such that∫ΓKnµ ∈ H1e (Kn,V ) for all n ∈ N, thenp−mϕ−m(Exp∗V ∗(1)(µ)) = tddt(Tm(LogV (µ)))Proof. Given ii) of proposition 8.3.1, it follows immediately by theorem 8.2.1.Remark 8.3.4. It is possible that the theorem is empty, that is there exists no nonzero element in H1Iw(K,V )satisfying the assumptions in proposition 8.3.1, but as we noted above, if we replace V by V (k) for k 0,then the assumptions of the theorem is verified for all elements of H1Iw(K,V ).46Chapter 9The Qp(1) representation and Coleman’spower series9.1 The module D(Zp(1))ψ=1The module Zp(1) is just Zp with the action of GQp defined by g ∈ GQp , x ∈ Zp(1), g(x) = χ(g)x. Weshall study the exponential mapExp∗Qp : H1Iw(Qp,Zp(1))→ D(Zp(1))ψ=1.Note that D(Zp(1)) = (A⊗Zp(1))HQp = AQp(1), with usual actions of ϕ and ψ , and for γ ∈ Γ, γ( f (pi)) =χ(γ) f ((1+pi)χ(γ)−1), for all f (pi) ∈ AQp(1).Proposition 9.1.1. (AQp)ψ=1 = Zp · 1pi ⊕ (A+Qp)ψ=1.Proof. Note that we have ψ(A+Qp)⊂A+Qp, ψ( 1pi ) =1pi and νE(ψ(x)≥ [νE (x)p ] if x ∈ E+Qp. These facts implythat ψ − 1 is bijective on EQp/pi−1E+Qp and hence it is also bijective on AQp/pi−1A+Qp . Thus ψ(x) = ximplies x ∈ pi−1A+Qp .9.2 Kummer theoryWe define the Kummer map κ : K∗→ H1(K,Qp(1)) as follows: For a ∈ K∗, we choose x any element inE˜ satisfying x(0) = a, then τ 7→ (1− τ)( log[x]t (1)) is a 1-cocycle on GK with values in Qp(1) whose imagein H1(K,Qp(1)) is defined to be κ(a).Recall that ε = (1,ε(1), · · ·) ∈ E+Qp , ε(1) 6= 1. Let Fn = Qp(ε(n)) and κn : F∗n → H1(Fn,Qp(1)) be theKummer maps defined above. Since corFn+1/Fn ◦κn+1 = κn ◦NFn+1/Fn , which induces a mapκ : lim←−F∗n → H1Iw(Qp,Qp(1)).47We haveH1Iw(Qp,Zp(1)) = Zp ·κ(pi)⊕κ(lim←−O∗Fn).9.3 Multiplicative representativesRecall that B is a purely inseparable extension of degree p of ϕ(B) (totally ramified since residualextension is purely inseparable). Define the multiplicative map N : B → B by the formula N(x) =ϕ−1(NB/ϕ(B)(x)). This is an multiplicative analogue of ψ .Lemma 9.3.1. If x∈E∗ and Ux denote the set y∈A whose reduction modulo p is x, then N is a contractiblemap of Ux for the p-adic topology.Proof. Note that N induces the identity on E and thus fixes Ux. On the other hand, if y ≡ 1 mod pk, wehaveN(y)≡ 1+ϕ−1TrB/ϕ(B)(y−1) = 1+ pψ(y−1) mod p2k,which implies in particular that N(y)−1 ∈ pk+1A. We deduce that if y1,y2 two elements of Ux verifyingy1− y2 ∈ pkA, then N(y1)−N(y2) = N(y2)(N(y−12 y1)−1) ∈ pk+1A, which proves the lemma.Corollary 9.3.2.i) If x ∈ E, there exists an unique element xˆ ∈ A whose image modulo p is x and N(xˆ) = xˆ.ii) If x and y are two elements in E, the x̂y = xˆyˆ.Proof. i) follows from the above lemma if x 6= 0 and completeness of Ux for the p-adic topology. On theother hand, N(pkA) ⊂ ppkA, this proves that 0 is the only element of y in pA satisfying N(y) = y. Thuswe prove the uniqueness. ii) follows from the uniqueness in i).Remark 9.3.3. There are two multiplicative maps from E to A˜, namely the map x→ xˆ and the Techmullermap [x]. We have xˆ 6= [x] unless x ∈ Fp.Lemma 9.3.4. Let K be a finite extension of Qp and d = [BK : BQp ] = [K∞ : Qp(µp∞))]. If n(K) is thesmallest integer n ≥ 2 such that there exist e1, ...,ed ∈ A˜†,rnK such that ϕ(e1), ...,ϕ(ed) form a basis ofA˜†,rn+1K over A˜†,rn+1Qpand if n≥ n(K), then N(A˜†,rn+1K )⊂ A˜†,rnK .Proof. By definition of n(K), if n ≥ n(K) and x ∈ A†,rn+1K , we can write x in the form x = ∑di=1 xiϕ(ei)where xi ∈ A†,rn+1Qp. On the other hand, we can write xi in the form xi = ∑p−1j=0 xi, j[ε] j where xi, j =ϕ(ψ([ε]− jxi)) and corollary 5.3.3 and proposition 7.2.1 show that we have xi, j ∈ ϕ(A†,rnQp ). We hencededuce that the coordinate y j = ∑di=1 xi, jϕ(ei) of x in basis 1, [ε], ..., [ε]p−1 of B over ϕ(B) belongs to to48A†,rn+1K ∩ϕ(B) = ϕ(A†,rnK ). On the other hand, NB/ϕ(B) is the determinant of the multiplication by x in Bconsidered as a vector space of dimension p over ϕ(B), therefore the determinant of the matrixy0 [ε]pyp−1 · · · [ε]py1y1 y0. . ........... . . [ε]pyp−1yp−1 yp−2 · · · y0.We deduce that NB/ϕ(B) belongs to ϕ(A†,rnK ) Together with the relation N = ϕ−1 ◦NB/ϕ(B), we completethe proof.Corollary 9.3.5. If x∈E+K , then xˆ∈A†,rn(K)K . Moreover, if K is a unramified extension over Qp and x∈E+K ,then xˆ ∈ A+K = AK ∩A+Qp=OK [[pi]].Proof. Let v ∈A†K whose image in EK is x and let n≥ n(K) such that v ∈A†,rnK . Let (vk)k∈N the sequenceof elements in AK defined by v0 = v and vk = N(vk−1) if k ≥ 1. By lemma 9.3.1, the sequence tends to xˆin AK as k tends to +∞. On the other hand, lemma 9.3.4, implies that vk ∈ A†,rnK for k ∈ N and since A†,rnKis relatively compact in A†,rn+1K , which implies that xˆ ∈A†,rk+1 and the result follows by using lemma 9.3.4by descending A†,rn+1K to A†,rn(K)K .In the case where K is unramified over Qp, the reduction modulo p induces a surjection from A+K toE+K and since A+K is a closed subring of AK fixed by N, a similar argument shows that v ∈ A+K implies thatxˆ ∈ A+K .9.4 Generalized Coleman’s power seriesLet us recall the construction of the classical Coleman’s power series.Proposition 9.4.1. Let F be a finite unramified extension of Qp. If u = (u(n))n∈N is an element of theprojective limit lim←−O∗Fn of O∗Fn with respect to the norm map, there exists a unique power series Colu(T )in OF [[T ]]∗ such that we have Colϕ−nu (ε(n)−1) = u(n) for all n ∈ N.Proof. See [8].Lemma 9.4.2. If K is a finite extension of Qp and n≥ n(K), then the diagramA†,rn+1KN //ϕ−(n+1)A†,rnKϕ−nKn+1[[t]]NKn+1/Kn//// Kn[[t]]is commutative.49Proof. By definition, NB/ϕ(B) (resp. NKn+1/Kn(ϕ−(n+1)(x))) is the determinant of the multiplication by x(resp. ϕ−(n+1)(x)) over B (resp. Kn+1[[t]]) considered as a ϕ(B)-vector space (resp. Kn[[t]]-module) andthe commutativity of the diagram follows from the fact ϕ(n+1) is a ring homomorphism and ϕ−n ◦N =ϕ−(n+1) ◦NB/ϕ(B).Denote the map θn the homorphism θ ◦ϕ−n where θ : BdR→ Cp and ϕ−n : B†,rn → BdR.Lemma 9.4.3. If u = (u(n)) ∈ lim←−OKn and n≥ n(K), then θn(ι̂K(u)) = u(n).Proof. By the preceeding lemma, (θn(ι̂K(u)))n≥n(K) belongs to lim←−OKn . On the other hand, since [ιK(u)]−ι̂K(u) ∈ A˜†,rn(K)K ∩ pA˜, which implies that if n ≥ n(K), then νp(θn([ιK(u)]− θn(ι̂K(u))) ≥ 1− 1pn−n(K) andsince νp(θn([ιK(u)]−u(n))≥ 1p if n large enough, we show that (θn(ι̂K(u))n≥n(K) has same image as u inE+K (c.f. proposition 4.1.1), so it is equal.Proposition 9.4.4. Let K be a finite extension of Qp, F = K∞∩Qurp and eK = [K∞ : F∞].i) If eK = 1 and u ∈ lim←−OKn , then ι̂K(u) = Colu(pi).ii) When eK ≥ 2, there exist Laurent series f0, · · · , feK−1 ∈OF((T )) converges in the annulus 0 < νp(x)<1(p−1)pn(K)−1such that, if n≥ n(K), then (un)eK + f ϕ−neK−1(ε(n)−1)(u(n))eK−1 + · · ·+ f ϕ−n0 (ε(n)−1) = 0.Proof. By corollary 9.3.5, ι̂K(u) ∈ A+F if u ∈ lim←−OKn . In particular, there exists f ∈ OF [[T ]] such thatι̂K(u) = f (pi). On the other hand, by applying lemma 9.4.3 to the map θn, we obtain u(n) = f ϕ−n(ε(n)−1),which shows f = Colu by the characterization of Colu.ii) By corollary 9.3.5, ι̂K(u) ∈ A†,rn(K)K . On the other hand, A†,rn(K)K is of dimension eK over A†,rn(K)F(by the definition of n(K)); so we can find elements f˜0, · · · , f˜e−1 ∈ A†,rn(K)F such that we have ι̂K(u)eK+f˜e−1ι̂K(u)eK−1+ · · ·+ f˜0 = 0, by lemma 9.4.3, we obtain the result.9.5 The map LogQp(1) and Exp∗QpLemma 9.5.1. If u ∈ EK , the sequence (ϕ−n(ι̂K(u)))pnconverges to [ιK(u)] in A˜ and B+dR.Proof. Since ι̂K(u) ∈ A†,rn(K) with image ιK(u) in E, it can be written in the form [ιK(u)]+∑+∞k=1 pk[xk],where xk are elements of E˜ satisfying νE(xk) ≥ −kpn(K). We have the formula vn = ϕ−n(ι̂K(u)) =[ιK(u)p−n]+∑+∞k=1 pk[xp−nk ] and the congruence vpnn ≡ [ιK(u)] mod pn+1A˜, thus it converges in A˜.Let α an element in E˜+ verifying νE(α) = p−1p , thus (p[α])i tends to 0 in B+dR as i tends to +∞. Ifn ≥ n(K) + 1, the above formula shows that vn belongs to the subring A (c.f. section 5.4) of B+dR ofelements of the form y =∑+∞i=0 yi(p[α])i, where yi are elements in A and we have vn− [ιK(u)p−n] ∈ p[α]A. Wededuce that vpnn tends to [ιK(u)] in A and a fortiori in B+dR.Proposition 9.5.2. Let K be a finite extension of Qp and u ∈ lim←−O∗Kn .50i) LogQp(1)(κ(u)) = t−1 log[ιK(u)]ii) If n≥ n(K), then Tn(LogQp(1)(κ(u))) = t−1 logϕ−n(ι̂K(u)).iii) Exp∗Qp(κ(u)) = ι̂K(u)−1∂ ι̂K(u), where ∂ is the derivation (1+pi) ddpi (see 7.2).Proof. By construction of the Kummer map, if un is any element in E˜+ satisfying u(0)n = u(n), then τ 7→(1− τ)( log[un]t (1)) is a 1-cocycle on GKn with values in Qp(1) whose image in H1(Kn,Qp(1)) is equal toκn(u(n)). Since we suppose that u(n) ∈O∗Kn , we have log[un]∈Bcris andlog[un]t (1)∈Bϕ=1cris ⊗Qp(1), provingthat κn(u(n)) ∈ H1e (Kn,Qp(1)). Hence we deduce the formulaLogQp(1)(κ(u)) = limn→+∞ pn log[un]t(1) = t−1 limn→+∞log([un]pn).Finally, we have νp(θ([ιK(u)p−n])−θ([un])) ≥ 1p if n large enough, therefore [un]pn tends to [ιK(u)] as ntends to +∞. We complete i).By i) and lemma 9.5.1, we haveTn(LogQp(1)(κ(u))) = t−1 limm→+∞Tn(pm log(ϕ−m(ι̂K(u)))).On the other hand, if m ≥ n, we have Tn = TrKm[[t]]/Kn[[t]] ◦ Tm and sicne ϕ−m(ι̂K(u))) ∈ Km[[t]] and therestriction of Tm on Km[[t]] is multiplication by p−m, we obtain the formulaTn(LogQp(1)(κ(u))) =t−1 limm→+∞TrKm[[t]]/Kn[[t]](log(ϕ−m(ι̂K(u))))=t−1 limm→+∞log(NKm[[t]]/Kn[[t]](ϕ−m(ι̂K(u))))and this completes ii) by using lemma 9.4.2.Note that t−1 is a generator of DdR(Qp(1)). ii) and theorem 8.3.3 implies that if n is large enough, wehaveϕ−n(Exp∗Qp(κ(u))) =t−1(tddt(pn logϕ−n((ι̂K(u))))=pnddt (ϕ−n(ι̂K(u))ϕ−n(ι̂K(u))=ϕ−n(ι̂K(u)−1∂ ι̂K(u))by lemma 7.2.3 ,which completes iii).519.6 Cyclotomic units and Coates-Wiles homomorphismsExample 9.6.1. Let K = Qp, V = Qp(1) and u = (ζpn−1ζpn)n≥1 ∈ lim←−O∗Fn . Then its Coleman’s power seriesis Colu(T ) = 1+TT . By iii) of proposition 9.5.2, we have Exp∗Qp(κ(u)) = (∂Colu(T )Colu(T ))(pi) = 1pi . On the otherhand, sinceϕ−1(pi)−1 = ([ε1p ]−1)−1 =1ε(1) exp(t/p)−1,by iii) of theorem 8.2.1, we haveExp∗Qp,F1(κ(u)) =1pTrF1/Qpϕ−1(1pi )=1p ∑ζ p=1,ζ 6=11ζ exp(t/p)=−1t( t1− exp(t)−t/p1− exp(t/p))=+∞∑k=1(1− p−k)ζ (1− k)(−t)k−1(k−1)!.Thus by ii) of theorem 8.2.1,exp∗Qp(1+k)∗(∫ΓF1χ−kκ(µ)) =0 if k ≤ 0(1− p−k)ζ (1− k) (−t)k−1(k−1)! if k ≥ 1.Example 9.6.2. Let K = Qp, V = Qp(1) and u = (ζ apn−1ζpn−1)n≥1 ∈ lim←−O∗Fn , where a ∈ Z. Then its Coleman’spower series is Colu(T ) =(1+T )a−1T . By iii) of proposition 9.5.2, we have Exp∗Qp(κ(u)) = (∂Colu(T )Colu(T ))(pi) =a(1+pi)a(1+T )a−1 −1+TT . On the other hand, sinceϕ−1(pi)−1 = ([ε1p ]−1)−1 =1ε(1) exp(t/p)−1,by iii) of theorem 8.2.1, we haveExp∗Qp,F1(κ(u)) =1pTrF1/Qpϕ−1(Exp∗Qp(κ(u)))=a−1+1p ∑ζ p=1,ζ 6=1aζ expat/p−1 −1ζ exp t/p=a−1+−1t( at1− exp(at)−at/p1− exp(at/p)−t1− exp(t)+t/p1− exp(t/p))=+∞∑k=1(1− p−k)(ak−1)ζ (1− k)(−t)k−1(k−1)!.52Thus by ii) of theorem 8.2.1,exp∗Qp(1+k)∗(∫ΓF1χ−kκ(µ)) =0 if k ≤ 0(ak−1)(1− p−k)ζ (1− k) (−t)k−1(k−1)! if k ≥ 1.Example 9.6.3. Let K = Qp(ζd), V = Qp(1) and ε be a Dirichlet character of conductor d ≥ 1 prime top. Set u =(−1G(ε−1) ∑bmoddε−1(b)ζ bd ζpn−1)n≥1 ∈ lim←−O∗Kn , then we haveColu(T ) =−1G(ε−1) ∑bmoddε−1(b)ζ bd (1+T )−1and thusExp∗Qp(κ(u)) =−1G(ε−1) ∑bmoddε−1(b)ζ bd (1+pi)−1.Hence we have,Exp∗Qp,K1(κ(u)) =p−1TrK1/Kϕ−1(Exp∗Qp(κ(u)))=p−1−1G(ε−1) ∑zp=1,z6=1∑bmoddε−1(b) 1ζ b/pd zexp(t/p)−1=−1G(ε−1) ∑bmoddε−1(b)(11−ζ bd exp(t)− p−111−ζ b/pd exp(t/p))= ∑bmoddε(b)exp(bt)1− exp(dt)− p−1ε(p)ε(b)exp(bt/p)1− exp(dt/p)=+∞∑k=1(1− ε(p)p−k)L(1− k,ε) tk−1(k−1)!and thusexp∗Q∗p(1+k)(∫ΓK1χ−kκ(µ)) =0 if k ≤ 0(1− ε(p)p−k)L(1− k,ε) tk−1(k−1)! if k ≥ 1.This allow us to define a homomorphism CWk,n from H1Iw(K,V ) to Kn⊗DdR(V ) by puttingCWk,n(µ) = ∂k(TKn(LogV (µ))).for each n ∈N and k ∈ Z, the homomorphism is a generalization of Coates-Wiles homomorphism and wehave the following theorem by proposition 8.3.1.Theorem 9.6.4. If µ ∈ H1Iw(K,V ), if n ∈ N and k ∈ Z, thenCWk,n(µ) =−exp∗(∫ΓKnχ(x)−kµ).53Remark 9.6.5. The maplim←−O∗Qp(µpn )→ H1Iw(Qp,Qp(1))→Qp, u 7→ exp∗Q∗p(1+k)(∫ΓF1χ−kκ(µ))is just the classical Coates-Wiles homomorphism ([7] section 2.6).54Chapter 10(ϕ ,Γ)-modules and differential equations10.1 The rings Bmax and B˜+rigThe ring B+max is defined byB+max = {∑n≥0anωnpn| an ∈ B˜+ is a sequence converges to 0},and Bmax = B+max[1t ]. It is closely related to Bcris but tends to be more amenable. One could replace ω byany generator of ker(θ) in A˜+. The ring Bmax injects canonically into BdR and, in particular, it is endowedwith the induced Galois action and filtration, as well as with a continuous Frobenius ϕ , extending the mapϕ : B˜+→ B˜+. Note that ϕ deos not extend continuously to BdR. We set B˜+rig = ∩+∞n=0ϕn(B+max).We call a representation V of GK is crystalline if it is Bcris-admissible, which is equivalent to Bmax-admissible or B˜+rig[1t ]-admissible (because ∩+∞n=0ϕn(B+max) = ∩+∞n=0ϕn(B+cris) and the periods of crystallinerepresentations live in finite dimensional F-vector subspaces of Bmax, stable by ϕ and so in fact in∩∞n=0ϕn(B+max[1t ]); that is, the F-vector spaceDcris(V ) = (Bcris⊗Qp V )GK = (Bmax⊗Qp V )GK = (B˜+rig[1t]⊗Qp V )GKis of dimension d = dimQp(V ). Then Dcris(V ) is endowed with a Frobenius ϕ induced by that of Bmaxand (BdR⊗Qp V )GK = DdR(V ) = K⊗F Dcris(V ) so that a crystalline representation is also de Rham andK⊗F Dcris(V ) is a filtered K-vector space.If V is a p-adic representation, we say V is Hodge-Tate, with Hodge Tate weights h1, ...,hd , if we havea decomposition Cp⊗Qp V ∼= ⊕dj=1Cp(h j). We say that V is positive if its Hodge-Tate weights are allnegative. By using the map θ : B+dR→ Cp, it is easy to see that a de Rham representation is Hodge-Tateand that the Hodge-Tate weights of V are those integers h such that Fil−hDdR(V ) 6= Fil−h+1DdR(V ).5510.2 The structure of D(T )ψ=1Recall in section 4, we introduced (ϕ,Γ)-modules and their relation with Galois representations. Let usnow set K = F (i.e. we are working in an unramified extension of Qp). We say that a p-adic representationV of GF is of finite height if D(V ) has a basis over BF made up of elements of D+(V ) = (B+⊗Qp V )HF .A result of [10, proposition III.2] shows that V is of finite height if and only if D(V ) has a sub-B+F -modulewhich is free of rank d, and stable by ϕ . Let us recall the main result of [10, theorem 1] regardingcrystalline representation of GF :Theorem 10.2.1. If V is a crystalline representation of GF , then V is of finite height.Let V be a crystalline representation of GF and let T denote a GF stable lattice of V . The followingproposition is proved in [2, proposition II.1.1]Proposition 10.2.2. If T is a lattice in a positive crystalline representation V , then there exists a uniquesub-A+F -module N(T ) of D+(T ), which satisfies the following conditions:1. N(T ) is an free A+F -module of rank d = dimQpV ;2. the action of ΓF preserves N(T ) and is trivial on N(T )/piN(T );3. there exists an integer r ≥ 0 such that pirD+(T )⊂ N(T ).Moreover, N(T ) is stable by ϕ , and the B+F -module N(V ) = B+F ⊗A+F N(T ) is the unique sub-B+F -moduleof D+(V ) satisfying the corresponding conditions.The A+F -module N(T ) is called the Wach module associated to T .Notice that N(T (−1)) = piN(T )⊗ e−1. When V is no longer positive, we can therefore define N(T )as pi−hN(T (−h))⊗eh for h large enough so that V (−h) is positive. Using the results of [2, III.4], one canshow that:Proposition 10.2.3. If T is a lattice in a crystalline representation V of GF , whose Hodge-Tate weightsare in [a;b], then N(T ) is the unique sub-A+F -module of D+(T )[1/pi] which is free of rank d, stable by ΓFwith the action of ΓF being trivial on N(T )/piN(T ) and such that N(T )[1/pi] = D+(T )[1/pi].Finally, we have ϕ(pibN(R))⊂ pibN(T ) and pibN(T )/ϕ∗(pib) is killed by qb−a, where q = ϕ(ω). Theconstruction T 7→ N(T ) gives a bijection between Wach modules over A+F which are lattices in N(V ) andGalois lattices T in V .Indeed D(V )ψ=1 is not very far from being included in N(V ):Theorem 10.2.4. If V is a crystalline representation of GF , whose Hodge-Tate weights are in [a;b], thenD(V )ψ=1 ⊂ pia−1N(V ). In addition, if V has no quotient isomorphic to Qp(a), then actually D(V )ψ=1 ⊂piaN(V ).Proof. See [1, Theorem A.3].5610.3 p-adic representations and differential equationsIn this paragraph, we recall some of the results of [3], which allow us to recover Dcris(V ) from the (ϕ,Γ)-module associated to V . Let H αF ′ be the set of power series f (T ) = ∑k∈Z akTk such that ak is a sequence(not necessarily bounded) of elements of F ′, and such that f (T ) is holomorphic on the p-adic annulus{p−1/α ≤ |T |< 1}.For r≥ r(K) (c.f. proposition 7.2.1), define B˜†,rrig,K as the set of f (piK) where f (T ) ∈HekrF ′ . Obviously,B†,rK ⊂ B˜†,rrig,K and the second ring is the completion of the first one for the natural Fre´chet topology. If V isa p-adic representation, letD†,rrig(V ) = B˜†,rrig,K⊗B†,rKD†,r(V ) and D†rig(V ) = (B˜†rig)HK ⊗B†KD†(V ).One of the main technical tools of [3] is the construction of a large ring B˜†rig, which contains B˜+rig andB˜†. This ring is a bridge between p-adic Hodge theory and the thoery of (ϕ,Γ)-modules.As a consequence of the above two inclusions, we have:Dcris(V )⊂ (B˜†rig[1t]⊗Qp V )GK and D†rig(V )[1t]⊂ (B˜†rig[1t]⊗Qp V )HK .One of the main result of [3] is:Theorem 10.3.1. If V is a p-adic representation of GK then Dcris(V ) = (D†rig(V )[1t ])ΓK . If V is positive,then Dcris(V ) = D†rig(V )ΓK .Proof. See [3, theorem 3.6].Note that B˜†,rrig,K is the completion of B†,rK for the ring’s natural Fre´chet topology and that B†rig,K is theunion of the B˜†,rrig,K . Similarly, there is a natural Fre´chet topology on B˜†,r, B˜†,rrig is the completion of B˜†,r forthat topology and B˜†rig = ∪r≥0B˜†,rrig . Actually, one can show that B˜+rig ⊂ B˜†,rrig for any r and there is an exactsequence0 // B˜+ // B˜+rig⊕ B˜†,r // B˜†,rrig // 0which one can take as providing a definition of B˜†,rrig .Recall that if n≥ 0 and rn = pn−1(p−1), then there is a well-defined injective map ϕ−n : B˜†,rn → B+dR(c.f. section 7.2), and the map extends to an injective map ϕ−n : B˜†,rnrig → B+dR (see [3, corollary 2.13]).Let B+rig,F be the set of f (pi) where f (T ) =∑k≥0 akT k with ak ∈ F , and such that f (T ) is holomorphicon the p-adic open unit disk. Set D+rig(V ) = B+rig,F ⊗B+F D+(V ). One can show the following refinement oftheorem 10.3.1:Proposition 10.3.2. We have Dcris(V ) = (D+rig(V )[1/t])ΓF and if V is positive then Dcris(V ) = D+rig(V )ΓF .Indeed if N(V ) is the Wach module associated to V , then N(V )⊂ D+(V ) when V is positive and it isshown in [1, II.2] that under that hypothesis, Dcris(V ) = (B+rig,F ⊗B+F N(V ))ΓF .5710.4 The Fontaine isomorphism revisitedThe purpose of this section is to recall the constructions in section 5.2 and extend them a little bit. Let Vbe a p-adic representation of GK . Recall in section 5.2, we constructed a map h1K,V : D(V )ψ=1→H1(K,V )such that when ΓK is torsion free, it gives rise to an exact sequence:0 // D(V )ψ=1ΓKh1K,V // H1(K,V ) // (D(V )ψ−1 )ΓK // 0We shall extend h1K,V to a map h1K,V : D†rig(V )ψ=1→ H1(K,V ).Lemma 10.4.1. If r is large enough and γ ∈ ΓK then1− γ : D†,rrig(V )ψ=0→ D†,rrig(V )ψ=0is an isomorphismProof. We first show that 1− γ is injective. By theorem 10.3.1, an element in the kernel of 1− γ wouldbe in Dcris(V ) and therefore in Dcris(V )ψ=0, which is obviously 0.To prove surjectivity. Recall that by iii) of proposition 7.3.2, if r is large enough and γ ∈ ΓK thern1−γ : D†,r(V )ψ=0→D†,r(V )ψ=0 is an isomorphism whose inverse is uniformly continuous for the Fre´chettopology of D†,r(V ).In order to show the surjectivity of 1− γ it is therefore enough to show that D†,r(V )ψ=0 is dense inD†,rrig(V )ψ=0 for the Fre´chet topology. For r large enough, D†,r(V ) has a basis in ϕ(D†,r/p(V )) so thatD†,r(V )ψ=0 =(B†,rK )ψ=0 ·ϕ(D†,r/p(V ))D†,rrig(V )ψ=0 =(B˜†,rrig,K)ψ=0 ·ϕ(D†,r/prig (V )).The fact that D†,r(V )ψ=0 is dense in D†,rrig(V )ψ=0 for the Fre´chet topology will therefore follow from thedensity of (B†,rK )ψ=0 in (B˜†,rrig,K)ψ=0. The last statement follows from the facts that by definition B†,r/pK isdense in B˜†,r/prig,K and that(B†,rK )ψ=0 =⊕p−1i=1 [ε]iϕ(B†,r/pK ) and (B˜†,rrig,K)ψ=0 =⊕p−1i=1 [ε]iϕ(B˜†,r/prig,K ).Lemma 10.4.2. The following maps are all surjective and the kernel is Qp1−ϕ : B˜†→ B˜†, 1−ϕ : B˜+rig→ B˜+rig and 1−ϕ : B˜†rig→ B˜†rig58Proof. Since B˜+rig ⊂ B†rig and B˜† ⊂ B˜†rig it is enough to show that (B˜†rig)ϕ=1 = Qp. If x ∈ (B˜†rig)ϕ=1, then[3, proposition 3.2] shows that actually x ∈ (B˜+rig)ϕ=1, and therefore x ∈ (B˜+rig)ϕ=1 = (B+max)ϕ=1 = Qp by[9, proposition III 3.5].The surjectivity of 1−ϕ : B˜+rig → B˜+rig results from the surjectivity of 1−ϕ on the first two spacessince by [3, lemma 2.18], one can write α ∈ B˜+rig as α = α++α− with α+ ∈ B˜+rig and α− ∈ B˜†.The surjectivity of 1−ϕ : B˜+rig → B˜+rig follows from the facts that 1−ϕ : B+max → B+max is surjective([9, proposition III 3.1]) and that B˜+rig = ∩∞n=0ϕn(B+max).The surjectivity of 1− ϕ : B˜† → B˜† follows from the facts that 1− ϕ : B˜→ B˜ is surjective (it issurjective on A˜ as can be seen by reducing modulo p and using the fact that E˜ is algebraically closed) andthat if β ∈ B˜ is such that (1−ϕ)β ∈ B˜†, then β ∈ B˜†.If x = ∑+∞i=0 pi[xi] ∈ A˜, let us set wk(x) = infi≤k νE(xi) ∈ R∪{+∞}. The definition of B˜†,r shows thatx ∈ B˜†,r if and only if limk→+∞ wk(x)+prp−1 k =+∞. A short computation shows that wk(ϕ(x)) = pwk(x)and that wk(x+ y)≥ inf(wk(x),wk(y)) with equality if wk(x) 6= wk(y).It is then clear thatlimk→+∞wk((1−ϕ)x)+prp−1k =+∞=⇒ limk→+∞wk(x)+p(r/p)p−1k =+∞and so if x ∈ A˜ is such that (1−ϕ)x ∈ B˜†,r then x ∈ B˜†,r/p and likewise for x ∈ B˜ by multiplication by asuitable power of p. This shows the second fact.Proposition 10.4.3. If y ∈ D†rig(V )ψ=1 and ΓK is torsion free, there exists b ∈ B˜†rig⊗Qp V such that (γ −1)(ϕ−1)b = (ϕ−1)y and the formulah1K,V (y) = log0p(γ)[σ 7→σ −1γ−1 y− (σ −1)b]then defines a map h1K,V : D†rig(V )ψ=1ΓK7→H1(K,V ) which does not depend either on the choice of generatorγ of ΓK or on the particular solution b, and if y ∈D(V )ψ=1 ⊂D†rig(V )ψ=1, then h1K,V (y) coincides with thecocycle constructed in section 5.2.Proof. Our construction closely follows section 5.2; to simplify the notations, we may assume thatlog0p(γ) = 1. The fact that h1K,V is independent of the choice of γ is same as lemma 5.1.2.Let us start by showing the existence of b∈ B˜†rig⊗Qp V . If y∈D†rig(V )ψ=1, then (ϕ−1)y∈D†rig(V )ψ=0.By lemme 10.4.1, there exists x∈D†rig(V )ψ=0 such that (γ−1)x = (ϕ−1)y. By lemma 10.4.2, there existsb ∈ B˜†rig⊗Qp V such that (ϕ−1)b = x.Recall that we define h1K,V (y) ∈ H1(K,V ) by the formula:h1K,V (y)(σ) =σ −1γ−1 y− (σ −1)b.59Notice that, a priori, h1K,V (y) ∈ H1(K, B˜†rig⊗Qp V ), but(ϕ−1)h1K,V (y)(σ) =σ −1γ−1 (ϕ−1)y− (σ −1)(ϕ−1)b=σ −1γ−1 (γ−1)x− (σ −1)x=0,so that h1K,V (y)(σ) ∈ (B†rig)ϕ=1⊗Qp V =V . In addition, two different choices of b differ by an element of(B˜†rig)ϕ=1⊗Qp V =V , and therefore give rise to two cohomologous cocycles.It is clear that if y ∈ D(V )ψ=1 ⊂ D†rig(V )ψ=1, then h1K,V coincide with the cocycle constructed insection 5.2, as can be seen by their identical construction, and it is immediate that if y ∈ (γ − 1)D†rig(V ),then h1K,V (y) = 0.Lemma 10.4.4. We have corKn+1/Kn ◦h1Kn+1,V = h1Kn,V .Proof. Same as lemma Iwasawa algebra and power seriesGiven a finite unramified extension F of Qp, denote by Λ(ΓF) (resp. Λ(Γ1F) where Γ1F = Gal(F∞/F1)) theIwasawa algebra Zp[[ΓF ]] (resp. Zp[[Γ1F ]]).LetH = { f ∈Qp[∆][[X ]] | f convergs on the open unit disk},and define H (ΓF) to be the set of f (γ−1) with f (X) ∈H and γ a topological generator of Γ. We mayidentify Λ(ΓF)⊗Qp with the subring of H (ΓF) consisting of power series with bounded coefficients.Note that H (Γ) may be identified with the continuous dual of the space of locally analytic functions onΓF , with multiplication corresponding to convolution, implying that its definition is independent of thechoice of generator γF (c.f. section 1.2).The action of ΓF on B+rig,Qp gives an isomorphism of H (ΓF) with (B+rig,Qp)ψ=0 via the Mellin trans-form [20, corollary B.2.8]M :H (ΓF)→(B+rig,Qp)ψ=0f (γ−1) 7→ f (γ−1)(pi+1).In particular, Λ(ΓF) corresponds to (A+Qp)ψ=0 under M. Similarly, we define H (Γ1F) as the subring ofH (ΓF) defined by power series over Qp, rather than Qp[∆]. Then, H (Γ1F) (resp. Λ(Γ1F)) corresponds to(1+pi)ϕ(B+rig,Qp) (resp. (1+pi)ϕ(A+Qp)) under M.6010.6 Iwasawa algebras and differential equationsBy [3, proposition 2.24], we have maps ϕ−n : B˜†,rnrig →B+dR whose restriction to B+rig,F satisfies ϕ−n(B+rig,F)⊂Fn[[t]] and which can be characterized by the fact that pi maps to ε(n) exp(t/pn)−1.Recall if z ∈ Fn((t))⊗F Dcris(V ), we denote the constant coefficient of z by ∂V (z) ∈ Fn⊗F Dcris(V ).Lemma 10.6.1. If y ∈ (B+rig,F [1/t]⊗F Dcris(V ))ψ=1, then for any m≥ n≥ 0, the elementp−mTrFm/Fn∂V (ϕ−m(y)) ∈ Fn⊗Dcris(V )does not depend on m and we havep−mTrFm/Fn∂V (ϕ−m(y)) =p−n∂V (ϕ−n(y)) if n≥ 1(1− p−1ϕ−1)∂V (y) if n = 0Proof. Recall that if y = t−l ∑+∞k=0 akpik ∈ B+rig,F [1/t]⊗F Dcris(V ), thenϕ−m(y) = pmlt−l+∞∑k=0ϕ−m(ak)(ε(m) exp(t/pm)−1)k,and that by the definition of ψ , ψ(y) = y means that:ϕ(y) = 1p ∑ζ p=1y(ζ (1+T )−1).The lemma then follows from the fact that if m≥ 2, then the conjugates of ε(m) under Gal(Fm/ Fm−1) arethe ζε(m), where ζ p = 1, while if m = 1, then the conjugates of ε(1) under Gal(F1/F) are the ζ , whereζ p = 1 but ζ 6= 1.Recall that since F is an unramified extension of Qp, ΓF ' Z∗p and that ΓFn = Gal(F∞/Fn) is the set ofelements γ ∈ ΓF such that χ(γ) ∈ 1+ pnZp.The Iwasawa algebra of ΓF isΛOF =Zp[[ΓF ]]∼=Zp[∆F ]⊗Zp Zp[[ΓF1 ]], and we setH (ΓF)=Qp[∆F ]⊗QpH (Γ1F) where H (Γ1F) is the set of f (γ−1) with γ ∈ Γ1F and where f (X) ∈Qp[[X ]] is convergent on thep-adic open unit disk. We define ∇i ∈H (ΓF) by∇i =log(γ)logp(χ(γ))− i.We will also use the operator ∇0/(γn−1), where γn is a topological generator of ΓnF . It is defined by theformula∇0γn−1=log(γn)logp(χ(γn))(γn−1)=1logp(χ(γn))∑i≥1(1− γn)i−1i,61or equivalently by∇0γn−1= limη∈ΓnF ,η→1η−1γn−11logp(χ(η)).It is easy to see that ∇0/(γn−1) acts on Fn by 1/ logp(χ(γn)).The algebra H (ΓF) acts on B+rig,F and one can easily check that∇i = tddt− i = log(1+pi)∂ − i, where ∂ = (1+pi) ddpi .In particular, ∇0B+rig,F ⊂ tB+rig,F and if i≥ 1, then∇i−1 ◦ · · · ◦∇0 ⊂ tiB+rig,F .Lemma 10.6.2. If n≥ 1, then ∇0/(γn−1)(B+rig,F)ψ=0 ⊂ (t/ϕn(pi))(B+rig,F)ψ=0 so that if i≥ 1, then∇i−1 ◦ · · · ◦∇1 ◦∇0γn−1(B+rig,F)ψ=0 ⊂ (tϕn(pi))i(B+rig,F)ψ=0.Proof. Since ∇i = t ·d/dt− i, the second claim follows easily from the first one. By the standard proper-ties of p-adic holomorphic functions, what we need to do is to show that if x ∈ (B+rig,F)ψ=0, then∇0γn−1x(ε(m)−1) = 0for all m≥ n+1.On the other hand, up to a scalar factor, one has for m≥ n+1:∇0γn−1x(ε(m)−1) = TrFm/Fnx(ε(m)−1),which can be seen from the fact that∇0γn−1= limη∈ΓnF ,η→1η−1γn−1·1logp(χ(η)).On the other hand, the fact ψ(x) = 0 implies that for every m≥ 2, TrFm/Fm−1x(ε(m)−1) = 0. This completesthe proof.Finally, let us point out that the actions of any element of H (ΓF) and ϕ commute. Since ϕ(t) = pt,we also see that ∂ ◦ϕ = pϕ ◦∂ .We will henceforth assume that logp(χ(γn)) = pn, and in addition ∇0/(γn−1) acts on Fn by p−n.6210.7 Bloch-Kato’s exponential maps: Three explicit reciprocity formulasIn this section, we explain the results of Berger in [1] on explicit reciprocity formulas when V is a crys-talline representation of an unramified field.Recall HK = Gal(Qp/K∞), let ∆K be the torsion subgroup of ΓK = GK/HK = Gal(K∞/K) and letΓ1K = Gal(K∞/K(µp)), so that ΓK ' ∆K ×Γ1K . Let ΓK = Zp[[ΓK ]] and H (ΓK) = Qp[∆K ]⊗Qp H (Γ1K)where H (Γ1K) is the set of f (γ1− 1) with γ1 ∈ ΓK1 and where f (T ) ∈ Qp[[T ]] is a power series whichconverges on the p-adic unit disk.When F is an unramified extension of and V is a crystalline representation of GF , Perrin-Riou hasconstructed in [18] a period map ΩV,h which interpolates the expF,V (k) as k runs over the positive integers.It is crucial ingredient in the construction of p-adic L-funtions, and is a vast generalization of Coleman’sisomorphism.The main result of [18] is the construction, for a crystalline representation of V of GF of a family ofmaps (parameterized by h ∈ Z):ΩV,h : (H (ΓF)⊗Qp Dcris(V ))∆=0→H (ΓF)⊗ΛF H1Iw(F,V )/VHF ,whose main property is that they interpolate Bloch-Kato’s exponential map. More precisely, if h, j 0,then the diagram:(H (ΓF)⊗Qp Dcris(V ( j)))∆=0ΩV ( j),h //Ξn,V ( j)H (ΓF)⊗ΓF H1Iw(F,V ( j))/V ( j)HFprFn ,V ( j)Fn⊗F Dcris(V )(h+ j−1)!×exp∗Fn ,V ( j) // H1(Fn,V ( j)).is commutative where ∆ and Ξ are two maps whose definition is rather technical (see section 10.9 for aprecise definition).Using the inverse of Perrin-Riou’s map, one can then associate to an Euler system a p-adic L-function.For example, if one starts with V = Qp(1), then Perrin-Riou’s map is the inverse of the Coleman isomor-phism and one recovers Kubota-Leopoldt p-adic L-functions (See section 11.2).The goal of this section is to give formulas for expK,V , exp∗K,V ∗(1) and ΩV,h in terms of the (ϕ,Γ)-module associated to V .10.8 The Bloch-Kato’s exponential map and its dual revisitedRecall in section 8.1, we defined the Bloch-Kato’s exponential map and its dual. The goal of this paragraphis to compute Bloch-Kato’s exponential map and its dual in terms of the (ϕ,Γ)-module of V. Let h≥ 1 bean integer such that Fil−hDcris(V ) = Dcris(V ).63Recall that we have seen that Dcris(V ) = (D+rig[1/t])ΓF and by [2, II.3], there is an isomorphismB+rig,F [1/t]⊗F Dcris(V ) = B+rig,F [1/t]⊗F D+rig(V ).If y ∈ B+rig,F ⊗F Dcris(V ), then the fact that Fil−hDcris(V ) = Dcris(V ) implies by result of [2, II.3] thatthy ∈ D+rig(V ), so that ify =d∑i=0yi⊗di ∈ (B+rig,F ⊗F Dcris(V ))ψ=1,then∇h−1 ◦ · · ·∇0(y) =d∑i=0th∂ hyi⊗di ∈ D+rig(V )ψ=1.One can apply the operator h1Fn,V to ∇h−1 ◦ · · ·∇0(y), then we have:Theorem 10.8.1. If y ∈ (B+rig,F ⊗F Dcris(V ))ψ=1, thenh1Fn,V (∇h−1 ◦ · · ·∇0(y)) = (−1)h−1(h−1)!expFn,V (p−n∂V (ϕ−n(y))) if n≥ 1expF,V ((1− p−1ϕ−1)∂V (y)) if n = 0Proof. Because the diagramFn+1⊗F Dcris(V )expFn+1 ,V //TrFn+1/Fn⊗idH1(Fn+1,V )corFn+1/FnFn⊗Dcris(V )expFn ,V // H1(Fn,V )is commutative, it is enough to prove the theorem under the assumption that ΓnF is torsion free. Let usset yh = ∇h−1 ◦ · · · ◦∇0(y). Since we are assuming for simplicity that χ(γn) = pn, the cocycle h1Fn,V (yh) isdefined by:h1Fn,V (yh)(σ) =σ −1γn−1yh− (σ −1)bn,hwhere bn,h is a solution of the equation (γn−1)(ϕ−1)bn,h = (ϕ−1)yh. In lemma 10.6.2 above, we provethat∇i−1 ◦ · · · ◦∇1 ◦∇0γn−1(B+rig,F)ψ=0 ⊂ (tϕn(pi))i(B+rig,F)ψ=0.It is then clear that if one setszn,h = ∇h−1 ◦ · · · ◦∇0γn−1(ϕ−1)y,thenzn,h ∈ (tϕn(pi))h(B+rig,F)ψ=0⊗F Dcris(V )⊂ ϕn(pi−h)D+rig(V )ψ=0 ⊂ D†rig(V )ψ=0.64Let q = ϕ(pi)/pi . By lemma 10.8.2 below, there exists an element bn,h ∈ ϕn−1(pi−h)B˜+rig ⊗QpV suchthat(ϕ−ϕn−1(qh))(ϕn−1(pih)bn,h) = ϕn(pih)zn,h,so that (1−ϕ)bn,h = zn,h with bn,h ∈ ϕn−1(pi−h)B˜+rig⊗Qp.If we set wn,h = ∇h−1 ◦ · · · ◦∇0γn−1 y, then wn,h and bn,h ∈ Bmax⊗Qp V and the cocycle h1Fn,V (yh) is thengiven by the formula h1Fn,V (yh)(σ) = (σ −1)(wn,h−bn,h). Now (ϕ−1)bn,h = zn,h and (ϕ−1)wn,h = zn,has well, so that wn,h−bn,h ∈ Bϕ=1max ⊗Qp V .We can also writeh1Fn,V (yh)(σ) = (σ −1)(ϕ−n(wn,h)−ϕ−n(bn,h)).Since we know that bn,h ∈ ϕn−1(pi−h)B+max⊗Qp V, we have ϕ−n(bn,h) ∈ B+dR⊗Qp V .The definition of Bloch-Kato exponential gives rise to the following construction: if x ∈ DdR(V ) andx˜ ∈ Bϕ=1max ⊗Qp V is such that x− x˜ ∈ B+dR⊗Qp V then expK,V (x) is the class of the coclycle g 7→ g(x˜)− x˜.The theorem therefore follows from the fact that:ϕ−n(wn,h)− (−1)h−1(h−1)!p−n∂V (ϕ−n(y) ∈ B+dR⊗Qp V,since we already know that ϕ−n(bn,h) ∈ B+dR⊗Qp V.In order to show this, first notice thatϕ−n(y)−∂V (ϕ−n(y)) ∈ tFn[[t]]⊗F Dcris(V ).We can therefore write∇0γn−1ϕ−n(y) = p−n∂V (ϕ−n(y))+ tz1and a simple recurrence shows that∇i−1 ◦ · · · ◦∇01− γnϕ−n(y) = (−1)i−1(i−1)!p−n∂V (ϕ−n(y))+ t izi,with zi ∈ Fn[[t]]⊗F Dcris(V ). By taking i = h, we see thatϕ−n(wn,h)− (−1)h−1(h−1)!p−n∂V (ϕ−n(y)) ∈ B+dR⊗Qp V,Since we choose h such that thDcris(V )⊂ B+dR⊗Qp V .Lemma 10.8.2. If α ∈ B˜+rig, the there exists β ∈ B˜+rig such that(ϕ−ϕn−1(qh))β = α.65Proof. By [3, proposition 2.19], the ring B˜+ is dense in B˜+rig for the Fre´chet topology. Hence, if α ∈ B˜+rig,then there exists α0 ∈ B˜+ such that α−α0 = ϕn(pih)α1 with α1 ∈ B˜+rig.The map ϕ−ϕn−1(qh) : B˜+→ B˜+ is surjective because ϕ−ϕn−1(qh) : A˜+→ A˜+ is surjective, as canbe seen by reducing modulo p and using the fact that E˜ is algebraically closed and that E˜+ is its ring ofintegers.One can therefore write α0 = (ϕ−ϕn−1(qh))β0. Finally by lemma 10.4.2, there exists β ∈ B˜+rig suchthat α1 = (ϕ−1)β1, so that ϕn(pih)α1 = (ϕ−ϕn−1(qh))(ϕn−1(pih)β1).Theorem 10.8.3. If y∈ (D†rig(V ))ψ=1 and y∈D+rig(V )[1/t] (so that in particular y∈ (B+rig,F [1/t]⊗FDcris(V ))ψ=1),thenexp∗Fn,V ∗(1)(h1Fn,V (y)) =p−n∂V (ϕ−n(y)) if n≥ 1(1− p−1ϕ−1)∂V (y) if n = 0Proof. Since the following diagramH1(Fn+1,V )exp∗Fn+1 ,V∗(1) //corFn+1/FnFn+1⊗Dcris(V )TrFn+1/Fn⊗idH1(Fn,V )exp∗Fn ,V∗(1) // Fn⊗Dcris(V )is commutative, we only need to prove the theorem when ΓnF is torsion free by lemma 10.8.1. We thenhave (assuming that χ(γn) = pn for simplicity) :h1Fn,V (y)(σ) =σ −1γn−1y− (σ −1)b,where (γn−1)(ϕ−1)b = (ϕ−1)y. Recall that B˜†rig = ∪r≥0B˜†,rrig . Since b ∈ B˜†rig⊗Qp V , there exists m 0such that b ∈ B˜†,rmrig ⊗Qp V and that the map ϕ−m embeds B˜†,rmrig into B+dR. we can then writeh1(y)(σ) = σ −1γn−1ϕ−m(y)− (σ −1)ϕ−m(b),and ϕ−m(b) ∈ B+dR⊗Qp V . In addition, ϕ−m(y) ∈ Fm((t))⊗F Dcris(V ) and γn−1 is invertible on tkFm⊗FDcris(V ) for every k 6= 0 This shows that the cocycle h1Fn,V is cohomologous in H1(Fn,BdR⊗Qp V ) toσ 7→ σ −1γn−1(∂V (ϕ−m(y)))which is itself cohomologous (since γn−1 is invertible on FTrFm/Fn=0m ) toσ 7→ σ −1γn−1(pn−mTrFm/Fn∂V (ϕ−m(y)))= σ 7→ p−n logp(χ(σ))pn−mTrFm/Fn∂V (ϕ−m(y)).66It follows from this and proposition 8.1.2 and lemma 10.6.1 thatexp∗Fn,V ∗(1)(h1Fn,V (y)) = p−mTrFm/Fn∂V (ϕ−m(y)) =p−n∂V (ϕ−n(y)) if n≥ 1(1− p−1ϕ−1)∂V (y) if n = 0.10.9 Perrin-Riou’s big exponential mapBy using the results of the previous paragraphs, we can give a uniform formula for the image of an elementy ∈ (B+rig,F ⊗F Dcris(V ))ψ=1 in H1(Fn,V ( j)) under the composition of the following maps:(B+rig,F ⊗F Dcris(V ))ψ=1 ∇h−1◦···◦∇0 // D†rig(V )ψ=1 ⊗e j // D†rig(V ( j))ψ=1h1Fn ,V ( j) // H1(Fn,V ( j))Here e j is a basis of Qp( j) such that e j+k = e j⊗ ek so that if V is a p-adic representation, then we havecompatible isomorphisms of Qp-vector spaces V →V ( j) given by v 7→ v⊗ e j.Theorem 10.9.1. If y∈ (B+rig,F⊗F Dcris(V ))ψ=1, and h≥ 1 is an integer such that Fil−hDcris(V )=Dcris(V ),then for all j with h+ j ≥ 1, we have :h1Fn,V ( j)(∇h−1 ◦ · · ·∇0(y)⊗ e j) =(−1)h+ j−1(h+ j−1)!×expFn,V ( j)(p−n∂V ( j)(ϕ−n(∂− jy⊗ t− je j)) if n≥ 1expF,V ( j)((1− p−1ϕ−1)∂V ( j)(∂− jy⊗ t− je j)) if n = 0while if h+ j ≤ 0, then we have:exp∗Fn,V ∗(1− j)(h1Fn,V (∇h−1 ◦ · · ·∇0(y)⊗ e j)) =1(−h− j)!p−n∂V ( j)(ϕ−n(∂ jy⊗ t− je j)) if n≥ 1(1− p−1ϕ−1)∂V ( j)(∂− jy⊗ t− je j) if n = 0Proof. If h+ j ≥ 1, then we have the following commutative diagram:D+rig(V )ψ=1 ⊗e j // D+rig(V ( j))ψ=1(B+rig,F ⊗F Dcris(V ))ψ=1 ∂− j⊗t− je j//∇h−1◦···◦∇0OO(B+rig,F ⊗F Dcris(V ( j)))ψ=1.∇h+ j−1◦···◦∇0OOand the theorem is then a straightforward consequence of theorem 10.8.1 applied to ∂− jy⊗ t− je j, h+ j67and V ( j).On the other hand, if h+ j ≤ 0, and ΓnF is torsion free, then theorem 10.8.3 shows thatexp∗Fn,V∗(1− j)(h1Fn,V ( j)(∇h−1 ◦ · · · ◦∇0(y)⊗ e j)) = p−n∂V ( j)(ϕ−n(∇h−1 ◦ · · · ◦∇0(y)⊗ e j))in Dcris(V ( j)), and a short computation involving Taylor series shows thatp−n∂V ( j)(ϕ−n(∇h−1 ◦ · · · ◦∇0(y)⊗ e j)) = (−h− j)!−1 p−n∂V ( j)(ϕ−n(∂− jy⊗ t− je j)).Finally, to get the case n = 0, one just needs to use the corresponding statement of theorem 10.8.3 orequivalently corestrict.Remark 10.9.2. The notation ∂− j is not injective on B+rig,F (it is surjective by integration) but it can bechecked that it leads to no ambiguity in the formulas above.We will now use the above result to give a construction of Perrin-Riou’s exponential map. If f ∈B+rig,F⊗Dcris(V ), we define ∆( f ) to be the image of⊕hk=0∂ k( f )(0) in⊕hk=0(Dcris(V ))/(1− pkϕ)(k). Thereis then an exact sequence of Qp⊗Zp ΛF -modules (cf [18, section 2.2]):0−→⊕hk=0tkDcris(V )ϕ=p−k−→ (B+rig,F ⊗Dcris(V ))ψ=1 1−ϕ−−→(B+rig,F)ψ=0⊗F Dcris(V )∆−→⊕hk=0Dcris(V )1− pkϕ (k)−→ 0.If f ∈ ((B+rig,F)ψ=0⊗F Dcris(V ))∆=0, then by the above exact sequence there existsy ∈ (B+rig,F ⊗Dcris(V ))ψ=1such that f = (1−ϕ)y, and since ∇h−1 ◦· · ·∇0 kills⊕h−1k=0tkDcris(V )ϕ=p−kwe see that ∇h−1 ◦· · ·∇0(y) doesnot depend upon the choice of such y unless Dcris(V )ϕ=p−h6= 0.Definition 10.9.3. Let h≥ 1 be an integer such that Fil−hDcris(V )=Dcris(V ) and such that Dcris(V )ϕ=p−h=0. One deduces from the above construction a well-defined mapΩV,h :((B+rig,F)ψ=0⊗F Dcris(V ))∆=0→ D+rig(V )ψ=1given by ΩV,h( f ) = ∇h−1 ◦ · · ·∇0(y), where y ∈ (B+rig,F ⊗Dcris(V ))ψ=1 is such that f = (1−ϕ)y.If Dcris(V )ϕ=p−h6= 0 then we get a mapΩV,h :((B+rig,F)ψ=0⊗F Dcris(V ))∆=0→ D+rig(V )ψ=1/V GF=χh.68Theorem 10.9.4. If V is a crystalline representation and h ≥ 1 is such that we have Fil−hDcris(V ) =Dcris(V ), then the mapΩV,h :((B+rig,F)ψ=0⊗F Dcris(V ))∆=0→ D+rig(V )ψ=1/V HFwhich takes f ∈ ((B+rig,F)ψ=0⊗F Dcris(V ))∆=0 to ∇h−1 ◦ · · ·∇0((1−ϕ)−1 f ) is well defined and coincideswith Perrin-Riou’s exponential map.Proof. The map ΩV,h is well defined because as we seen above the kernel of 1−ϕ is killed by ∇h−1 ◦· · · ◦∇0, except for thDcris(V )ϕ=p−h, which is mapped to copies of Qp(h) ∈V HF .The fact that ΩV,h coincides with Perrin-Riou’s exponential map follows directly from theorem 10.9.1above applied to those j’s for which h + j ≥ 1, and the fact that by [18, theorem 3.2.3], the ΩV,h areuniquely determined by the requirement that they satisfy the following diagram for h, j 0:(H (ΓF)⊗Qp Dcris(V ( j))∆=0ΩV ( j),h //Ξn,V ( j)H (ΓF)⊗ΓF (H1Iw(F,V ( j)/V ( j)HF )prFn ,V ( j)Fn⊗F Dcris(V )(h+ j−1)!expFn ,V ( j) // H1(Fn,V ( j)).Here Ξn,V ( j)(g) = p−n(ϕ⊗ϕ)−n( f )(ε(n)−1) where f is such that(1−ϕ) f = g(γ−1)(1+pi) ∈ (B+rig,F ⊗F Dcris(V ))ψ=0and the ϕ on the left of ϕ ⊗ϕ is the Frobenius on B+rig,F while the ϕ on the right is the Frobenius onDcris(V ).Note that by theorem 6.1.2, we have an isomorphism D(V )ψ=1 ' H1Iw(F,V ) and therefore we get amap H (ΓF)⊗ΛF H1Iw(F,V )→ D†rig(V )ψ=1. On the other hand, there is a mapH (ΓF)⊗Qp Dcris(V ( j))→ (B+rig,F ⊗F Dcris(V ))ψ=0which sends ∑ fi(γ−1)⊗di to ∑ fi(γ−1)(1+pi)⊗di. These two maps allow us to compare the diagramabove with the formulas given by theorem 10.9.1.Remark 10.9.5. By the above remarks, if V is a crystalline representation and h≥ 1 is such that Fil−hDcris(V )=Dcris(V ) and Qp(h) 6⊂V , then the mapΩV,h :((B+rig,F)ψ=0⊗F Dcris(V ))∆=0→ D+rig(V )ψ=1which takes f ∈ ((B+rig,F)ψ=0⊗F Dcris(V ))∆=0 to ∇h−1 ◦· · ·∇0((1−ϕ)−1 f ) is well defined, without havingto kill the ΛF -torsion of H1Iw(F,V ).69Remark 10.9.6. It is clear from theorem 10.9.1 that we have:ΩV,h(x)⊗ e j =ΩV ( j),h+ j(∂ jx⊗ t− je j) and ∇h ◦ΩV,h(x) =ΩV,h+1(x)and following Perrin-Riou, one can use these formulas to extend the definition of ΩV,h to all h ∈ Z bytensoring all H (ΓF)-modules with the field of fractions of H (ΓF)10.10 The explicit reciprocity formulaRecall we have a map H (ΓF)→ (B+rig,Qp)ψ=0 which sends f (γ − 1) to f (γ − 1)(1+ pi), and that thismap is a bijection and whose inverse is the Mellin transform, and that if g(pi) ∈ (B+rig,Qp)ψ=0, then g(pi) =M(g)(1+ pi). If f ,g ∈ (B†rig,Qp)ψ=0 then we define f ∗ g by the formula M( f ∗ g) = M( f )M(g). Let[−1] ∈ ΓF be the element such that χ([−1]) =−1, and let ι be the involution of ΓF which sends γ to γ−1.The operator ∂ j on (B+rig,Qp)ψ=0 corresponds to Tw j on ΓF (Tw j is defined by Tw j(γ) = χ(γ j)γ). We willmake use of the facts that ι ◦∂ j = ∂− j ◦ ι and [−1]◦∂ j = (−1) j∂ j ◦ [−1].If V is a crystalline representation, then the natural mapsDcris(V )⊗F Dcris(V ∗(1))−→ Dcris(Qp(1))TrF/Qp−−−−→Qpallow us to define a perfect pairing [·, ·]V : Dcris(V )×Dcris(V ∗(1)) which we extend by linearity to[·, ·]V : (B+rig,F ⊗Dcris(V ))ψ=0× (B+rig,F ⊗Dcris(V∗(1)))ψ=0→ (B+rig,Qp)ψ=0by the formula [ f (pi)⊗d1,g(pi)⊗d2]V = ( f ∗g)(pi)[d1,d2]V .We can also define a semi-linear pairing (with respect to ι)〈·, ·〉V : D+rig(V )ψ=1×D+rig(V )ψ=1→ (B+rig,Qp)ψ=0by the formula〈·, ·〉V = lim←− ∑τ∈ΓF/ΓnF〈τ−1(h1Fn,V (y1)),h1Fn,V ∗(1)(y2)〉Fn,V· τ(1+pi)where the pairing 〈·, ·〉Fn,V is given by the cup product:〈·, ·〉Fn,V : H1(Fn,V )×H1(Fn,V∗(1))→ H2(Fn,Qp(1))∼= Qp.The pairing 〈·, ·〉V satisfies the relation 〈γ1x1,γ2x2〉V = γ1ι(γ2)〈x1,x2〉V , where γ1,γ2 ∈ ΓF . Perrin-Riou’sexplicit reciprocity formula is then:Theorem 10.10.1. If x1 ∈ (B+rig,F ⊗F Dcris(V ))ψ=0 and x2 ∈ (B+rig,F ⊗F Dcris(V∗(1)))ψ=0, then for every h,70we have(−1)h〈ΩV,h(x1), [−1] ·ΩV ∗(1),1−h(x2)〉V =−[x1, ι(x2)]V .Proof. By the theory of p-adic interpolation, it is enough to prove that if xi = (1− ϕ)yi with y1 ∈(B+rig,F ⊗F Dcris(V ))ψ=1 and y2 ∈ (B+rig,F ⊗F Dcris(V∗(1)))ψ=1, then for all j 0;(∂− j(−1)h〈ΩV,h(x1), [−1] ·ΩV ∗(1),1−h(x2)〉V)(0) =−(∂− j[x1, ι(x2)]V )(0).The above formula is equivalent to:(1) (−1)h+ j〈h1F,V ( j)ΩV ( j),h+ j(∂− jx1⊗t− je− j),h1F,V ∗(1− j)ΩV ∗(1− j),1−h− j(∂jx2⊗ tje− j)〉F,V ( j)=[∂V ( j)(∂− jx1⊗ t− je j),∂V ∗(1− j)(∂ jx2⊗ t je− j))]V ( j).By combining theorems 10.9.1 and 10.9.4 with remark 10.9.6, we see that for j 0:h1F,V ( j)ΩV ( j),h+ j(∂− jx1⊗ t− je j) = (−1)h+ j−1expF,V ( j)((h+ j−1)!(1− p−1ϕ−1)∂V ( j)(∂− jy1⊗ t− je j)),and thath1F,V ∗(1− j)ΩV ∗(1− j),1−h− j(∂jx2⊗ tje− j)=(exp∗F,V ∗(1− j))−1(h+ j−1)!−1((1− p−1ϕ−1)∂V ∗(1− j)(∂ jy2⊗ t je− j)).Using the fact that by definition, if x ∈ Dcris(V ( j)) and y ∈ H1(F,V ( j)) then[x,exp∗F,V ∗(1− j)y]V ( j) = 〈expF,V ( j)x,y〉F,V ( j),we see that〈h1F,V ( j)ΩV ( j),h+ j(∂− jx1⊗ t− je j),h1F,V ∗(1− j)ΩV ∗(1− j),1−h− j(∂jx2⊗ tje− j)〉F,V ( j) (10.1)= (−1)h+ j−1[(1− p−1ϕ−1)∂V ( j)(ϕ− jy1⊗ t− je j),(1− p−1ϕ−1)∂V ∗(1− j)(∂ jy2⊗ t je− j)]V ( j).It is easy to see that under [ , ], the adjoint of (1− p−1ϕ−1) is 1−ϕ and that if xi = (1−ϕ)yi, then∂V ( j)(∂− jx1⊗ t− je j) =(1−ϕ)∂V ( j)(∂− jy1⊗ t− je j),∂V ∗(1− j)(∂ jx2⊗ t je− j) =(1−ϕ)∂V ∗(1− j)(∂ jy2⊗ t je− j),So that (10.1) implies (1), and this proves the theorem.71Chapter 11Perrin-Riou’s big regulator mapLet F be a finite unramified extension over Qp and V a continuous p-adic representation of GF , which iscrystalline with Hodge-Tate weights ≥ 0 and with no quotient isomorphic to the trivial representation. In[19], Perrin-Riou construct a big logarithm mapL ΓFF,V : H1Iw(F,V )−→H (ΓF)⊗Qp Dcris(V )which interpolates the values of Bloch-Kato’s dual exponential and logarithm maps for V ( j), j ∈ Z, overeach Fn.In this section, we follow [17, Appendix B] to adapt Berger’s explicit formulas to construct Perrin-Riou’s big logarithm and use it to calculate Kubota-Leopoldt p-adic L-function.11.1 Perrin-Riou’s big logarithm mapLet V be a positive crystalline representation of Gal(F∞/F) and x ∈H (ΓF)⊗ΛF H1Iw(F,V ). We write x jfor the image of x in H1Iw(F,V (− j)), and x j,n for the image of x j in H1(Fn,V (− j)). If we identify x withits image in D(V )ψ=1, then x j corresponds to the element x⊗ e− j ∈ D(V )ψ=1⊗ e− j = D(V (− j))ψ=1.Since V is positive, we may interpret x as an element of the module (B+rig,F [1/t]⊗Dcris(V ))ψ=1.We shall assume:x ∈ (B+rig,F ⊗B+F N(V ))ψ=1 ⊂ (B+rig,F [1/t]⊗F Dcris(V ))ψ=1. (11.1)The condition is satisfied if V has no quotient isomorphoic to Qp (c.f. theorem 10.2.4).Recall in section 7.2, we define ∂ to be the differential operator (1+pi) ddpi (orddt ) on B+rig,F and wehave a map∂V ◦ϕ−n : B+rig,F [1/t]⊗F Dcris(V )→ Fn⊗F Dcris(V )which sends pik⊗d to the constant coefficient of (ζn exp(t/pn)−1)k⊗ϕ−n(d) ∈ Fn((t))⊗F Dcris(V ).72For m ∈ Z, define Γ∗(m) to be the leading term of Taylor series expansion of Γ(x) at x = m; thusΓ∗(m) =j! if n≥ 0(−1)− j−1(− j−1)! if n≤−1Proposition 11.1.1. DefineR j,n(x) =1j!×p−n∂V (− j)(ϕ−n(∂ jx⊗ t je− j)) if n≥ 1(1− p−1ϕ−1)∂V (− j)(∂ jx⊗ t je− j) if n = 0Then we haveR j,n(x) =exp∗Fn,V ∗(1− j)(x j,n) if j ≥ 0logFn,V (− j)(x j,n) if j ≤−1Proof. This result is essentially a minor variation on thoerem 10.9.1. The case j ≥ 0 is immediate fromtheorem 10.8.1 applied with V replaced by V (− j) and x by x⊗ e− j, using the formula∂V (− j)(ϕ−n(x⊗ e− j)) =1j!∂V (− j)(ϕ−n(∂ jx⊗ t je− j)).For the formula for j ≤−1, we choose h such that Fil−hDcris(V ) = Dcris(V ). The element ∂ jx⊗ t je− jlies in (B+rig,F ⊗F Dcris(V (− j)))ψ=1. Applying theorem 10.8.1 with V ,h and x replaced by V (− j), h− j,and ∂ jx⊗ t− je j, we see thatΓ∗( j+1)R j,n(x) = Γ∗( j−h+1) logFn,V (− j)[(∇0 ◦ · · · ◦∇h−1x) j,n].For x ∈H (ΓF)⊗ΛF H1Iw(F,V ), we have(∇rx) j,n = ( j− r)x j,n,so se have(∇0 ◦ · · · ◦∇h−1x) j,n = ( j)( j−1) · · ·( j−h+1)x j,nas require.For ω a finite order character on ΓF of conductor n, we denoteG(ω) = ∑σ∈ΓF/ΓFnω(σ)ζσpn .the Gauss sum of ω .73Proposition 11.1.2. If x is as above, and L ΓFV (x) is the unique element of H (ΓF)⊗F Dcris(V ) such thatL ΓFV (x) · (1+pi) = (1−ϕ)x, then for any j ∈ Z we have(1−ϕ)∂V (− j)(ϕ−n(∂ jx⊗ t je− j)) =L ΓFV (x)(χj)⊗ t je− j,while for any finite order character ω of ΓF of conductor n≥ 1, we have(∑σ∈ΓF/ΓnFω(σ)−1σ)·∂V (− j)(ϕ−n(∂ jx⊗ t je− j) = G(ω)ϕ−n(L ΓFV (x)(χjω)⊗ t je− j).Proof. We note thatL ΓFV (− j)(∂jx⊗ t je− j) = Tw j(LΓFV (x))⊗ tje− j,so it suffices to prove the result for j = 0. Suppose we have x = ∑k≥0vkpik where vk ∈ Dcris(V ). Then∂V (ϕ−n(x)) = ∑k≥0ϕ−n(vk)(ζpn−1)k.On the other hand,∂V (ϕ−n((1−ϕ)x)) = ∑k≥0ϕ−n(vk)(ζpn−1)k−∑k≥0ϕ1−n(vk)(ζpn−1−1)k.Applying the operator eω = ∑σ∈ΓF/ΓnFω(σ)σ , we have for n≥ 1eω ·∂V (ϕ−n(x)) = eω ·∂V (ϕ−n((1−ϕ)x)),since eω is zero on Fn−1((t)).However, since the map ∂V ◦ϕ−n is a homomorphism of ΓF -modules, we haveeω ·∂V (ϕ−n((1−ϕ)x)) =eω ·∂V (ϕ−n(L ΓFV (x) · (1+pi)))=ϕ−n(L ΓFV (x)) · eω∂F(ϕ−n(1+pi))=G(ω)ϕ−n(L ΓFV (x)(ω)).This completes the proof of the proposition for j = 0.Definition 11.1.3. Let x ∈ H1Iw(F,V ) If η is any continuous character of ΓF , denote by xη the image of xin H1Iw(F,V (η−1)). If n≥ 0, denote by xη ,n the image of xη in H1(Fn,V (η−1)).Thus xχ j,n = x j,n in the previous notation. The next lemma is valid for arbitrary de Rham representa-tions of GF (with no restriction on Hodge-Tate weights):74Lemma 11.1.4. For any finite-order character ω factoring through ΓF/ΓnF , with values in a finite exten-sion E/F, we have∑σ∈ΓF/ΓnFω(σ)−1exp∗Fn,V ∗(1)(x0,n)σ = exp∗Fn,V (ω−1)∗(1)(xω,0)and∑σ∈Γ/Γnω(σ)−1 logFn,V (x0,n)σ = logFn,V (ω−1)(xω,0)where we identify DdR(V (ω−1))∼= (E⊗F Fn⊗F Dcris(V ))Γ=ω .Proof. This follows from the compatibility of the maps exp∗ and log with the corestriction maps (c.f.Theorem 10.8.1 and 10.8.3).Combining the three results above, we obtain:Theorem 11.1.5. Let j ∈ Z and let x satisfy (11.1). Let η be a continuous character of ΓF of the formχ jω , where ω is a finite-order character of conductor n.i) If j ≥ 0, we haveL ΓFV (x)(η) = j!×(1− p jϕ)(1− p−1− jϕ−1)−1(exp∗F,V (η−1)∗(1)(xη ,0)⊗ t− je j)if n = 0G(ω)−1 pn(1+ j)ϕn(exp∗F,V (η−1)∗(1)(xη ,0)⊗ t− je j)if n≥ 1.ii) If j ≤−1, we haveL ΓV (x)(η) =(−1)− j−1(− j−1)!×(1− p jϕ)(1− p−1− jϕ−1)−1(logF,V (η−1)(xη ,0)⊗ t− je j)if n = 0G(ω)−1 pn(1+ j)ϕn(logF,V (η−1)(xη ,0)⊗ t− je j)if n≥ 1.In both cases, we assume that (1− p−1− jϕ−1) is invertible on Dcris(V ) when η = χ j.11.2 Cyclotomic units and Kubota-Leopoldt p-adic L-functionsThe relation between Coleman’s power series and the Perrin-Riou’s big logarithm map is given by thefollowing diagram:lim←−O∗Fnκ //ColH1Iw(F,Zp(1))L ΓF,Qp(1)OF [[pi]]∗(1− ϕp ) logOF [[pi]]ψ=0 // H (Γ)⊗Qp Dcris(F,Qp(1))75If we identify Dcris(F,Qp(1)) with F via the basis vector t−1⊗ e1, then the bottom map sends f ∈OF [[pi]]ψ=0 to ∇0 ·M−1( f ), where ∇0 = logγlogχ(γ) for any non-identity element γ ∈ Γ1 and M is the Mellintransform defined in section 10.5. Thus the image of the bottom map is precisely ∇0 ·ΛOF (Γ)⊂HF(Γ);and if we definehF(u) = ∇−10 ·LΓF,Qp(1)(κ(u)) ∈ ΛOF (Γ),then we haveM(hF(u)) = (1−ϕp) logColu(u).By the calculation in section 9.6, we can use theorem 11.1.5 to calculate the Kubota-Leopoldt p-adicL-functions.Example 11.2.1. (Kubota-Leopoldt p-adic zeta-function) Let K = Qp, V = Qp(1) andu = (ζpn−1ζpn)n≥1 ∈ lim←−O∗Qp(µpn ).Then by the calculation in section 9.6, we havehF(u)(χk) =χk(∇−10 ) · k! ·(1− pkϕ)(1− p−1−kϕ−1 )(exp∗Qp,V ∗(1− j)(uk,0)⊗ t−kek)=1kk! ·(1− pkϕ)(1− p−1−kϕ−1)((1− p−k)ζ (1− k) tk−1(k−1)!⊗ t−kek)=(1− pk−1)ζ (1− k)t−1and for ω a finite order character of Γ of conductor n, we havehF(u)(χkω) =χk(∇−10 ) · k! ·G(ω)−1 pn(1+k)ϕn(exp∗Qp,V (η−1)∗(1)(uη ,0)⊗ t−kek)=1kk! ·G(ω)−1 pn(1+k)ϕn(p−(n+1)kG(ω)L(1− k,ω) tk−1(k−1)!⊗ t−kek)=L(1− k,ω)t−1Example 11.2.2. (Kubota-Leopoldt p-adic L-function) Let K = Qp(ζd), V = Qp(1) and ε is a Dirichletcharacter of conductor d ≥ 1 prime to p. Set u =(−1G(ε−1) ∑0≤a≤d−1 ε(a)−1 ζ ad ζpnζ ad ζpn−1)n≥1. Then by calcula-76tion in section 9.6, we havehF(u)(χk) =χk(∇−10 ) · k! ·(1− pkϕ)(1− p−1−kϕ−1)−1(exp∗K,V ∗(1− j)(uk,0)⊗ t−kek)=1kk! ·(1− pkϕ)(1− p−1−kϕ−1)−1((1− ε(p)p−k)L(1− k,ε) tk−1(k−1)!⊗ t−kek)=(1− ε(p)pk−1)L(1− k,ε)t−1and for ω a finite order character of Γ of conductor n, we havehF(u)(χkω) =χk(∇−10 ) · k! ·G(ω)−1 pn(1+k)ϕn(exp∗Qp,V (η−1)∗(1)(uη ,0)⊗ t−kek)=1kk! ·G(ω)−1 pn(1+k)ϕn(p−(n+1)kG(ω)L(1− k,ωε) tk−1(k−1)!⊗ t−kek)=L(1− k,ωε)t−177Bibliography[1] L. Berger. Bloch and kato’s exponential map: three explicit formulas. Doc. Math. Extra, Extra:99–129, 2003. → pages 56, 57, 63[2] L. Berger. Limites de repre´sentations cristallines. Compositio Math., 140(6):1473–1498, 2004. →pages 2, 56, 64[3] L. B. Berger. Repre´sentations p-adiques et e´quations diffe´rentielles. Invent. Math., 148(2):219–284,2002. → pages 57, 59, 61, 66[4] S. Bloch and K. Kato. L-Functions and Tamagawa Numbers of Motives, pages 333–400. Birkha¨userBoston, 2007. → pages 2[5] F. Cherbonnier and P. Colmez. Repre´sentations p-adiques surconvergentes. Invent. Math., 133:581–611, 1998. → pages 2, 39, 40[6] F. Cherbonnier and P. Colmez. The´orie d’iwasawa des repre´sentations p-adiques d’un corps local. J.A.M.S, 12:241–268, 1999. → pages 2[7] J. Coates and R. Sujatha. Cyclotomic Fields and Zeta values. Springer monographs in mathematics.Springer-Verlag, Berlin, 2006. → pages 54[8] R. Coleman. Division values in local fields. Invent. Math., 53:91–116, 1979. → pages 1, 2, 49[9] P. Colmez. The´orie d’iwasawa des repre´sentations de de rham d’un corps local. Annals of Math.,148:485–571, 1998. → pages 41, 42, 43, 45, 46, 59[10] P. Colmez. Repre´sentations cristallines et repre´sentations de hauteur finie. J. Reine Angew. Math,514:119–143, 1999. → pages 56[11] P. Colmez. Arithmetique de la fonction zeta. Journees mathematiques, X-UPS:37–164, 2002. →pages 2[12] P. Colmez. Fonctions d’une variable p-adique. Aste´risque, pages 61–153, 2010. → pages 2, 3, 4, 5[13] J. Fontaine and Y. Ouyang. Theory of p-adic galois represntations. http://math.stanford.edu/ con-rad/252Page/handouts/galoisrep.pdf. → pages 20[14] L. Herr. Sur la cohomologie galoisienne des corps p-adiques. Bulletin de la Socie´te´ Mathe´matiquede France, 126:563–600, 1998. → pages 2, 24, 26, 2778[15] S. J. Neukirch, A and W. K. Cohomology of number fields, volume 126 of Grundleh- ren derMathematischen Wissenschaften. Springer-Verlag, Berlin, 2000. → pages 33[16] K. Kato. Lectures on the approach to Iwasawa theory for Hasse-Weil L-functions via BdR. Part I,volume 1553, pages 50–163. Springer Berlin Heidelberg, 1993. → pages 42[17] D. Loeffler and S. L. Zerbes. Iwasawa theory and p-adic l-functions over ${\mathbb z} {p}ˆ{2}$-extensions. International Journal of Number Theory, 10(08):2045–2095, 2015/02/19 2014.→ pages2, 72[18] B. Perrin-Riou. The´orie d’iwasawa des re´presentations p-adiques sur un corps local. Invent. Math.,115:81–149, 1994. → pages 2, 36, 63, 68, 69[19] B. Perrin-Riou. Fonctions l p-adiques des representations p-adiques. Aste´risque, pages 81–149,1995. → pages 72[20] B. Perrin-Riou. The´orie d’Iwasawa des repre´sentations p-adiques semi-stables, volume 84.Me´moires de la SMF, 2001. → pages 6079


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