中国科学技术大学 欧阳毅--中文主页--科研.pdf

Introduction to Iwasawa Theory Yi Ouyang Department of Mathematical Sciences Tsinghua University Beijing, China 100084 Email: youyang@math.tsinghua.edu.cn Contents 1 Modules up to pseudo-isomorphism 1 2 Iwasawa modules 7 3 Zp -extensions 14 4 Iwasawa theory of elliptic curves 21 0 Chapter 1 Modules up to pseudo-isomorphism Let A be a commutative noetherian integrally closed domain. Let K be the quotient field of A. Let P (A) = {℘ ∈ Spec(A) | ht ℘ = 1} be the set of prime ideals of A of height 1. Then for every ℘ ∈ P (A), A℘ is a discrete valuation ring. Theorem 1.1. \ A= A℘ . ℘∈P (A) Proof. See [7, Theorem 11.5, page 81]. This is a well known theorem about normal noetherian domains. For an A-module M , we let M + := HomA (M, A). Thus there is a pairing M + × M → A, (α, m) 7→ α(m) which induces a homomorphism of A-modules ϕM : M → M ++ . Definition 1.2. An A-module M is called reflexive if the canonical map ϕM : M −→ M ++ = HomA (HomA (M, A), A) m 7−→ (ϕM (m) : α 7→ α(m)) is an isomorphism. Remark. M + is always torsion free, thus M is reflexive implies that M is torsion free. 1 Assume M is a finitely generated torsion free A-module, then M ,→ M℘ ,→ M℘ ⊗A℘ K = M ⊗A K := V and M + ,→ (M + )℘ ,→ (M + )℘ ⊗A℘ K = M + ⊗A K = V ∧ where V ∧ = HomK (V, K) is the dual of V . One has M + = {λ ∈ V ∧ | λ(m) ∈ A for all m ∈ M }, (M + )℘ = {λ ∈ V ∧ | λ(m) ∈ A℘ for all m ∈ M℘ } = (M℘ )+ , where M℘ is regarded as an A℘ -module. Lemma 1.3. LetTM be a finitely generated torsion free A-module, then (1) M + = M℘+ . ℘∈P (A) T (2) M ++ = M℘ . ℘∈P (A) T (3) M is reflexive if and only if M = M℘ . ℘∈P (A) Proof. (1) ⊆ is trivial. If λ ∈ T M℘+ , then for all m ∈ M , λ(m) ∈ A℘ for ℘∈P (A) ℘ ∈ P (A), hence λ(m) ∈ A and λ ∈ M + . (2) since M℘ = M℘++ for ht ℘ = 1 (A℘ is a discrete valuation ring). (3) follows from (2). Corollary 1.4. If M is a finitely generated A-module, then M + is reflexive. Definition 1.5. A finitely generated A-module M is called pseudo-null if the following two equivalent conditions are fulfilled: (1) M℘ = 0 for all prime ideals ℘ in A of height ht(℘) ≤ 1, i.e., Supp(M ) = {℘ ∈ Spec(A) | M℘ 6= 0} ⊆ {℘ ∈ Spec(A) | ht(℘) ≥ 2}. (2) If ℘ is a prime ideal with annA (M ) ⊆ ℘, then ht(℘) ≥ 2. Recall that annA (M ) = {a ∈ A | aM = 0}. Remark. (i) For the equivalence of the two conditions: M℘ = 0 if and only if there exists s ∈ A\℘, such that sM = 0, which is equivalent to annA (M ) * ℘. (ii) A pseudo-null module is torsion since M(0) = M ⊗A K = 0. (iii) If A is a Dedekind domain, then M is pseudo-null if and only if M = 0. (iv) If A is a 2-dimensional noetherian integrally closed local ring with finite residue field, then M is pseudo-null if and only if M is finite. Indeed, let m be the maximal ideal of A, if M is finite, there exists r ∈ N such that mr M = 0, thus Supp(M ) ⊆ {m}. On the other hand, if Supp(M ) ⊆ {m},then there exists r ∈ N such that mr M = 0, thus mr ⊆ annA (M ), therefore M is a finitely generated A/mr -module, hence finite. 2 Definition 1.6. Let f : M → N be a homomorphism of finitely generated Amodules M and N . f is called a pseudo-isomorphism if both ker f and coker f are pseudo-null, equivalently, if the induced homomorphisms f℘ : M℘ −→ N℘ S are isomorphisms for all ℘ ∈ P (A) {0}. We write a pseudo-isomorphism f as ≈ f : M −→ N or f : M ∼ N . Lemma 1.7. Let M be a finitely generated A-torsion module, if 0 6= α ∈ A such that Supp(A/αA) is disjoint to Supp(M ) ∩ P (A), then α : M −→ M, m 7−→ αm is a pseudo-isomorphism. Proof. This is clear since α is a unit of A℘ for every ℘ ∈ Supp(M ) ∩ P (A). From now on, we set TA (M ) : the torsion submodule of M ; FA (M ) = M/TA (M ) : the maximal torsion free quotient of M. Proposition 1.8. Let M be a finitely generated A-module. Then (1) There exists a pseudo-isomorphism M ≈ f : M −→ TA (M ) FA (M ). (2) There exists {℘i }i∈I ⊆ P (A), ni ∈ N, and a pseudo-isomorphism M ≈ g : TA (M ) −→ A/℘ni i i∈I where {℘i , ni } are uniquely determined by TA (M ) up to re-numbering. Proof. (1) Let {℘1 , · · · , ℘h } = Supp(M ) ∩ P (A). If h = 0, then TA (M ) is pseudo-null, the homomorphism (0,can) f : M −−−−→ TA (M ) M FA (M ) is a pseudo-isomorphism. If h > 0, let S= h \ h /[ A\℘i = A ℘i . i=1 i=1 −1 then S A is a semi-local Dedekind domain with maximal ideals S −1 ℘i , i = 1, · · · , h. Thus S −1 A is a principal ideal domain by the approximation theorem, 3 and S −1 TA (M ) is a direct summand of S −1 M and is the torsion module of S −1 M . Since M is finitely generated, then HomS −1 A (S −1 M, S −1 TA (M )) = S −1 HomA (M, TA (M )). Thus there exists f0 : M → TA (M ) and s0 ∈ S such that f0 : S −1 M −→ S −1 TA (M ) s0 is the projection of S −1 M onto S −1 TA (M ), hence f0 | −1 = IdS −1 TA (M ) . s0 S TA (M ) Thus there exists s1 ∈ S, f1 = s1 f0 such that f1 |TA (M ) = s1 s0 IdTA (M ) . Let f = (f1 , can) : M −→ TA (M ) ⊕ FA (M ) by the commutative diagram 0 −−−−→ TA (M ) −−−−→  f | y 1 TA (M ) −−−−→ FA (M ) −−−−→ 0 M  f y 0 −−−−→ TA (M ) −−−−→ TA (M ) ⊕ FA (M ) −−−−→ FA (M ) −−−−→ 0 By Lemma 1.7, f1 |TA (M ) is a pseudo-isomorphism, the snake lemma implies that f is also a pseudo-isomorphism. (2) By the structure theorem of finitely generated modules over a principal ideal domain, there exists an isomorphism   ri h M M ∼ n = g0 : S −1 TA (M ) −→ S −1 E = S −1  A/℘i ij  i=1 j=1 for some uniquely determined E= ri h M M n A/℘i ij . i=1 j=1 Using HomS −1 A (S −1 TA (M ), S −1 E) = S −1 HomA (TA (M ), E), again we obtain g : TA (M ) → E and s ∈ S, such that g = sg0 . Again using the previous lemma, g is a pseudo-isomorphism. ≈ Remark. (i) If M , N are torsion modules, then f : M → N implies that there ≈ exists g : N → M . 4 In general, this is not true. For example, let A = Zp [[T ]] = Λ, N = Λ and 6≈ ≈ M = ker (N → Z/pZ). Then M → N but N → M . (ii) If the exact sequence of finitely generated A-torsion modules 0 −→ M 0 −→ M −→ M 00 −→ 0 satisfies that the associated sets of prime ideals of height 1 of M 0 and M 00 are ≈ disjoint. Then M → M 0 ⊕ M 00 . Proposition 1.9. Let M be a finitely generated torsion free A-module. Then there exists an injective pseudo-isomorphism of M onto a reflexive A-module M 0. Proof. Consider the homomorphism ϕM : M → M ++ . One notes that: (1) M℘ ∼ = M℘++ for ht ℘ ≤ 1. In particular, ker ϕM ⊗A K = 0, hence ker ϕM is torsion. As M is torsion free, ker ϕM = 0; (2) M ++ is reflexive. Proposition 1.10. Let A be an n-dimensional regular local ring, 2 ≤ n < ∞. Let {p1 , · · · , pn } be a regular system of parameters generating the maximal ideal of A. Let p0 := 0. Then for a finitely generated A-module M , the following two assertions are equivalent: (1) For every i = 0, · · · , n − 2, the A/(p0 , · · · , pi )-module M/(p0 , · · · , pi )M is reflexive. (2) M is a free A-module. In particular, a reflexive A-module M over a 2-dimensional regular local ring A is free. Proof. We only need to show (1)⇒ (2). From (1), M is reflexive, hence torsion free. Let ϕ : Ar  M be a minimal free presentation of M . Consider the diagram p1 0 −−−−→ Ar −−−−→ Ar −−−−→ (A/p1 )r −−−−→ 0    ϕ̃ ϕ ϕ y y y p1 0 −−−−→ M −−−−→ M −−−−→ M/p1 M −−−−→ 0 Assume M/p1 is a free A/p1 -module, then by Nakayama Lemma and the minimality of r, ϕ̃ is an isomorphism. Hence p1 : ker ϕ → ker ϕ is an isomorphism. By Nakayama again, ker ϕ = 0. We get M is a free A-module. Thus we only need to show (*) M/p1 is a free A/p1 -module Note that (i) A/p1 is a regular local ring of dimension n − 1; 5 (ii) Let p̃i = pi + p1 A, then {p̃2 , · · · , p̃n } is a regular system of parameter of A/p1 . Thus (1) holds for (A/p1 , M/p1 ). By induction, we only need to check (*) for n = 2. In this case, A/p1 is regular of dimension 1, hence a discrete valuation ring and an integral domain. Thus HomA (M + , A/p1 ) is torsion free, therefore M/p1 = M ++ /p1 = HomA (M + , A) ⊗ A/p1 ,→ HomA (M + , A/p1 ) is also torsion free over the discrete valuation ring A/p1 , which must be free. Theorem 1.11 (Structure Theorem). Let A be a 2-dimensional regular local ring and M be a finitely generated A-module. Then there exists finitely many primes ℘i of height 1, natural numbers ni for each i, nonnegative integer r and a pseudo-isomorphism M ≈ f : M −→ Ar ⊕ (A/℘ni i ) , i∈I ℘i , ni and r are uniquely determined by r = dimK M ⊗A K, {℘i |i ∈ I} = Supp M ∩ P (A). 6 Chapter 2 Iwasawa modules In this chapter, we let K be a finite extension of Qp and let O be the ring of integers of K, let π be a uniformizing parameter of O. Let k = O/(π) be the residue field of O. Then k is a finite extension of Fp . As a convention, we write Λ = Zp [[T ]]. For f (T ) = a0 + a1 T + · · · + ai T i + · · · ∈ O[[T ]], f 6= 0, set µ(f ) = min{ordπ (ai )}, λ(f ) = min{i : ordπ (ai ) = µ(f )}. Lemma 2.1 (Division Lemma). Suppose f = a0 + a1 T + · · · ∈ O[[T ]] but π - f , i.e. µ(f ) ∈= / 0. Let n = λ(f ). Then any g ∈ O[[T ]] can be uniquely written as g = qf + r where q ∈ O[[T ]], and r ∈ O[T ] is a polynomial of degree at most n − 1. Proof. First we show the uniqueness. If qf + r = 0, we need to show that q = r = 0. If not, we may assume that π - q or π - r. But 0 = qf + r mod π implies that π | r and therefore π | qf . Since π - f , we have π | q, contradiction! For the existence, we have two proofs. First proof: We let τn = τ be the O-linear map ∞ X bi T i 7−→ i=0 ∞ X bi T i−n . i=n Note that (i) τ (T n h) = h for h ∈ O[[T ]]. (ii) τ (h) = 0 if and only if h is a polynomial of degree ≤ n − 1. Write f = πP (T ) + T n U (T ), where P (T ) is a polynomial of degree at most n − 1 and U (T ) is a unit in O[[T ]]. For any g ∈ O[[T ]], let  j ∞ P 1 X j j (−1) π τ ◦ ◦ τ (g) ∈ O[[T ]]. q(T ) = U j=0 U 7 Then τ (qf ) = τ (πqP ) + τ (T n qU ) = πτ (qP ) + qU and  πτ (qP ) =τ  = ∞ X ∞ πP X U  P U (−1)j π j τ ◦ j=0 j−1 j (−1)  π j=1 P τ◦ U  j ◦ τ (g) j ◦ τ (g) =τ (g) − qU. Thus τ (qf ) = τ (g). Second proof: Note that k[[T ]] is a discrete valuation ring, it has a simple division algorithm. We let ḡ(T ) be the reduction of g(T ) modulo π. Since f¯(T ) = T n · (unit) in k[[T ]], we have ḡ(T ) = q̄(T )f¯(T ) + r̄(T ) for suitable q̄(T ) ∈ k[[T ]] and r̄(T ) ∈ k[T ] of degree ≤ n − 1. Let q1 (T ) ∈ O[[T ]], r1 (T ) ∈ O[T ] (of the same degree of r̄(T )) be liftings of q̄(T ) and r̄(T ) respectively. Then g(T ) = f (T )q1 (T ) + r1 (T ) + πg1 (T ) for some g1 (T ) ∈ O[[T ]]. Apply the same procedure for g1 , we get g(T ) = f (T )q1 (T ) + r1 (T ) + π(f (T )q20 (T ) + r20 (T ) + πg2 (T )) = f (T )q2 (T ) + r2 (T ) + π 2 g2 (T ) where q2 = q1 mod π, r2 = r1 mod π. Repeat the process, we get g(T ) = f (T )qn (T )+rn (T )+π n gn (T ), qn+1 = qn mod π n , rn+1 = rn mod π n . By taking the limits, the desired result is obtained. Corollary 2.2. If π - f ∈ O[[T ]] (i.e., µ(f ) = 0), then O[[T ]]/(f ) is a free O-module of rank n = λ(f ) with basis {T i : i < n}. Definition 2.3. A distinguished polynomial (or Weierstrass polynomial ) F (T ) ∈ O[T ] is a polynomial of the form F (T ) = T n + an−1 T n−1 + · · · + a0 , ai ∈ (π). We note that an Eisenstein polynomial is an irreducible distinguished polynomial. 8 Corollary 2.4. Let F be a distinguished polynomial, then ∼ = O[T ]/F O[T ] −→ O[[T ]]/F O[[T ]]. Theorem 2.5 (Weierstrass Preparation Theorem). Let f ∈ O[[T ]], f 6= 0. Then f can be uniquely written as f = π µ P (T )U (T ) where µ = µ(f ), P (T ) is a distinguished polynomial of degree n = λ(f ), U (T ) is a unit in O[[T ]]. As a consequence, O[[T ]] is a factorial domain. Proof. One may assume π - f . Write f = a0 + a1 T + · · · + an T n + · · · with π - an and π | ai for i < n. By the division lemma, T n = q(T )f (T ) + r(T ) with deg r < n and q(T ) ∈ O[[T ]]. One has r(T ) = 0 mod π. Therefore f (T )q(T ) = T n − r(T ) := P (T ) = T n mod π, we have q(T )an = 1 mod π and 1 × q(T ) := U (T ) ∈ (O[[T ]]) . Thus in this case f (T ) = U (T )P (T ). The uniqueness follows from the division lemma, since T n = U (T )−1 f (T ) + (T n − P (T )). Remark. For π - f , then O[[T ]]/(f (T )) ∼ = O[T ]/(P (T )). Thus P (T ) is the characteristic polynomial of the linear transformation T : O[[T ]]/(f ) → O[[T ]]/(f ). Corollary 2.6. There are only finitely many x ∈ Cp , |x| < 1 such that f (x) = 0. Proof. This is an easy exercise. ) Lemma 2.7. Let P be a distinguished polynomial. If Pg(T (T ) ∈ O[[T ]], g(T ) ∈ ) O[T ], then Pg(T (T ) ∈ O[T ]. Proof. Let g(T ) = P (T )f (T ), f ∈ O[[T ]]. For any root x ∈ Cp of P (T ), 0 = P (x) = xn + multiple of π, one has |x| < 1, hence f (x) converges and g(x) = 0. Continue this process, we get P (T ) | g(T ) as polynomials, hence f (T ) ∈ O[T ]. Let Γ = Zp = limn Z/pn Z. As a profinite group, Γ is compact and pro←− cyclic. Let γ be a topological generator of Γ, i.e., Γ = hγi. Let Γn = hγ pn i be the unique closed subgroup of index pn of Γ, then Γ/Γn is cyclic of order pn generated by γ + Γn . One has an isomorphism   n ∼ O[Γ/Γn ] −→ O[T ]/ (1 + T )p − 1 γ mod Γn 7−→ (1 + T ) n mod (1 + T )p − 1 Moreover, if m ≥ n ≥ 0, the natural map Γ/Γm → Γ/Γn induces a natural map φm,n : O[Γ/Γm ] → O[Γ/Γn ], which is compatible with the isomorphism. We let   pn O[[Γ]] = lim O[Γ/Γn ] = lim O[T ]/ (1 + T ) − 1 . ← − ← − n n 9 Note that O is a topological ring, compact and complete with π-adic topology, so are the rings O[Γ/Γn ], thus O[[Γ]] is endowed with the product topology of π-adic topology, it is also compact and π-complete. The ring O[[Γ]] is called the Iwasawa algebra and its modules are called Iwasawa modules. Theorem 2.8. One has a topological isomorphism O[[T ]] −→ O[[Γ]], T 7−→ γ − 1 where O[[T ]] is a compact topological ring complete with (π, T )-topology. n Proof. Write ωn (T ) = (1+T )p −1. ωn is a distinguished polynomial. Moreover, n n ωn+1 (T ) = (1 + T )p (p−1) + · · · + (1 + T )p + 1 ∈ (p, T ) ⊆ (π, T ), ωn (T ) thus ωn (T ) ∈ (p, T )n+1 for n ≥ 0. By Corollary 2.4, for every n ∈ N, we have a projection ∼ ∼ O[[T ]]  O[[T ]]/(ωn ) → O[T ]/(ωn ) → O[Γ/Γn ] which is compatible with the transition map. By the universal property of projective limits, then we have a continuous homomorphism O[[T ]] → O[[Γ]], T 7→ γ − 1. T On one hand ker  ⊆ n (ωn ) ⊆ n (p, T )n+1 = 0, thus  is injective. On the other hand, O[[T ]] is compact, hence the image is closed, it is also dense since at every level the map is surjective, hence  is also surjective. : T From now on let O = Zp and Λ = Zp [[T ]]. Let m = (p, T ) be the maximal ideal of Λ. We identify Zp [[Γ]] and Λ by the above Theorem, though we should keep in mind that this isomorphism depends on the choice of the topological n generator γ of Γ. Write ωn (T ) = (1 + T )p − 1 and νn,e (T ) = ωn (T )/ωe (T ). Lemma 2.9. If f and g are relatively prime to each other, then |Λ/(f, g)| < ∞. Proof. Let h ∈ (f, g) be of minimal degree. we show that h = ps (up to Z∗p ). If not, h = ps H for deg H ≥ 1. By the division algorithm, f = Hq + r, thus ps r ∈ (f, g), contradiction! Proposition 2.10. The prime ideals of Λ are (0), m = (p, T ), (p), (P ) where P are irreducible distinguished polynomials in Λ. Proof. First all in the list are prime ideals. Let ℘ be a prime ideal of Λ and h ∈ ℘ be of minimal degree. Then h = ps H with H = 1 or distinguished (up to Z∗p ). If H 6= 1, then it must be irreducible by minimality. Then (f ) ⊆ ℘ where f = p or an irreducible distinguished polynomial. If (f ) = ℘, we are done. If 10 not, there exists g ∈ ℘ such that f, g are relatively prime. By the above lemma, |Λ/℘| ≤ |Λ/(f, g)| < ∞. Therefore pN ∈ ℘ for N  0, which implies p ∈ ℘ since ℘ is prime; also there exists a pair i < j, such that T i − T j ∈ ℘, as 1 − T j−i is a unit, T i ∈ ℘, hence T ∈ ℘. Thus (p, T ) ⊆ ℘. Theorem 2.11 (Structure Theorem for Iwasawa modules). For any finitely generated Λ-module M , ≈ M → Λr ⊕ s M Λ/pmi ⊕ i=1 t M n Λ/Fj j j=1 where r = rank M , mi (i = 1, · · · s), Fj and nj (j = 1, · · · , t) are uniquely determined by M . t Q Definition 2.12. FM = j=1 n Fj j is called the characteristic polynomial of M . If M is a torsion module, we define the Iwasawa invariants of M by λ(M ) = s X mi , µ(M ) = i=1 X nj deg Fj = deg FM . j Remark. The isomorphism of Zp [[Γ]] and Zp [[T ]] depends on the choice of γ. Therefore if a finitely generated Iwasawa module M is considered as a Λ-module, the corresponding Fj and FM depend on the choice of γ, but λ(M ) and µ(M ) are independent invariants. Lemma 2.13 (Topological Nakayama’s Lemma). Let M be a compact Λ-module. Then the following are equivalent: (1) M is finitely generated over Λ; (2) M/T M is a finitely generated Zp -module; (3) M/(p, T )M is a finitely dimensional Fp -vector space. Proof. (1) ⇒ (2) ⇒ (3) are easy. Assuming (3), let x1 , · · · , xn generate M/(p, T )M as Fp -vector space. Let N = Λx0 + · · · + Λxn ⊆ M , then N + (p, T )M M M = = (p, T ) . N N N Thus M/N = (p, T )n M/N for all n > 0. Consider a small neighborhood U of 0 in M/N . Since (p, T )n → 0 in Λ, for any z ∈ M/N , there exists a neighborhood Uz of z and some nz such that (p, T )nz Uz ⊆ U . But M/N is compact, then (p, T )n M/N ⊆ U for n  0, hence M/N = ∩(p, T )n M/N = 0 and M = N is finitely generated over Λ. Theorem 2.14. Let X be a compact Λ-module. Then (1) X = 0 ⇔ X/T X = 0 ⇔ X/mX = 0. (2) X is a finitely generated Λ-module ⇔ X/T X is a finitely generated Zp -module ⇔ X/mX is a finite dimensional Fp -vector space. Moreover, for 11 a finitely generated Λ-module X, the minimal number of generators of X is dimFp (X/mX). (3) If X/T X is finite, then X is a torsion Λ-module. (4) If we replace T by any distinguished polynomial in (1), (2) and (3), the corresponding assertions still hold. Proof. (1) and (2) are Nakayama’s Lemma. For (3), by (2), X is a finitely generated Λ-module. Let x1 , · · · , xd be a set of generators. Suppose X/T X has exponent pk , then pk xi ∈ T X for 1 ≤ i ≤ d. Write d X pk xi = T aij (T )xj , j=1 and let A = (pk δij − T aij (T ))i, j and g(T ) = det A. Then g(T )xi = 0 for all i = 1, · · · , d but g(0) = pdk 6= 0, hence X is torsion. (4) follows similarly. Lemma 2.15. Let g be a distinguished polynomial of degree d prime to ωn /ωe for every n > e. Then for n  0, |Λ/(g, ωn )| = pdn+O(1) . Proof. We know Λ/(g, ωn ) is finite for n  0 by Lemma 2.9. Write V = Λ/(g(T )). Since T d = pQ(T ) mod g, by induction, then for k ≥ d, T k = p · poly. mod g. Therefore for pn ≥ d, n (1 + T )p = 1 + p · poly. mod g and n+1 (1 + T )p = (1 + p · poly.)p = 1 + p2 · poly. mod g, n+2 −1 ωn+2 (T ) (1 + T )p = n+1 ωn+1 (T ) (1 + T )p −1 n+1 =(1 + T )(p−1)p =p(1 + p · poly.) n+1 + · · · + (1 + T )p mod g. +1 n+2 (T ) n Thus ω ωn+1 (T ) acts as p · unit on V for p ≥ d. n+2 (T ) For n0 > e, pn ≥ d and n ≥ n0 , then ωn+2 V = ω ωn+1 (T ) (ωn+1 V ) = pωn+1 V , and |V /ωn+2 V | =|V /pV | · |pV /pωn+1 V | = |V /pV | · |V /ωn+1 V | =pd(n−n0 +1) |V /ωn0 +1 V | = pnd+c . This finishes the proof. 12 Lemma 2.16. For a Λ-module M , let MΓ = M/T M and M Γ = M γ=1 . If there is exact sequence 0 −→ M 0 −→ M −→ M 00 −→ 0, then there is a long exact sequence 0 → M 0Γ → M Γ → M 00Γ → MΓ0 → MΓ → MΓ00 → 0. Proof. Apply the snake lemma to the commutative diagram 0 −−−−→ M 0 −−−−→ M −−−−→ M 00 −−−−→ 0       γ−1y γ−1y γ−1y 0 −−−−→ M 0 −−−−→ M −−−−→ M 00 −−−−→ 0 with exact rows. n Remark. If replacing γ by γ p , we shall have corresponding results. Proposition 2.17. Let M be a finitely generated torsion Λ-module such that n M/ωn M is finite for all n ≥ 0. Then for n  0, |M/ωn M | = pµ(M )p +λ(M )+O(1) where λ(M ) and µ(M ) are Iwasawa invariants. Proof.LBy the above we can replace M by a torsion Λ-module of the
lemma, Lt s form i=1 Λ/pki ⊕ j=1 (Λ/fj (T )mj ). Now just apply Lemma 2.15. 13 Chapter 3 Zp-extensions Definition 3.1. A Zp -extension is a Galois extension K∞ /K whose Galois group is isomorphic to the ring of p-adic integers Zp . Proposition 3.2. There are exactly one sub-extension Kn of K inside K∞ with Galois group Gal(Kn /K) ∼ = Z/pn Z cyclic of order pn . Proof. This follows easily from the fundamental theorem of Galois Theory, as the only closed subgroups of Zp are 0 and pn Zp for n ∈ N. Proposition 3.3. Let K be a number field, then Zp -extensions over K are unramified outside p. Proof. Let v be a prime of K not lying above p. We need to show the inertia subgroup I of v is 0. if not, I = pn Zp for some n ∈ N. By local class field theory, UKv  I = pn Zp is surjective, but UKv = finite groups × Za` for a ∈ N and ` 6= p, this is impossible. Lemma 3.4. Let K be a number field. Then there exists at least one prime ramified in K∞ /K, and there exists n ≥ 0 such that every prime which is ramified in K∞ /K is totally ramified in K∞ /Kn . Proof. This is an easy exercise. × Suppose K is a number field. Let E = OK be the group of global units. Let E1 = {x ∈ E | x ≡ 1 mod ℘ for all ℘ | p}. Let U1,℘ be the group of local units congruent to 1 mod ℘. Then we have an injective diagonal map Y ψ:E→U = U℘ ,  7→ (, · · · , ) ℘|p such that ψ(E1 ) ⊆ U1 = Q ℘|p U1,℘ . 14 Lemma 3.5. (1) ψ(E1 ) = U1 ∩ K × Q Uv . v-p (2) ψ(E) = U ∩ K × Q Uv . v-p Proof. (1). ⊆ is easy. For ⊇, we write Un = Q Un,v , where Un,v is the group of v|p local units congruent to 1 mod v n , then K× Y v-p Uv = \ (K × n Y Uv Un ), ψ(E1 ) = \ ψ(E1 )Un . n v-p Q It suffices to show that U1 ∩K × v-p Uv Un ⊆ ψ(E1 )Un . For any element xu0 un ∈ Q Q U1 ∩ K × v-p Uv Un where x ∈ K × , u0 ∈ v-p Uv and un ∈ Un , we have x ∈ E1 and for v - p, (xu0 )v = 1. Then xu0 un = ψ(x)un ∈ ψ(E1 )Un . The proof of (2) is similar to (1). Conjecture 3.6 (Leopoldt Conjecture). rankZ E1 = rankZp E1 ⊗Z Zp = rankZp ψ(E). Leopoldt Conjecture is true for abelian number fields. Let δ = rankZ E1 − rankZp E1 ⊗Z Zp . Then δ ≥ 0 and δ = 0 if Leopoldt Conjecture holds. Example 3.7. Note that 7, 13 are independent over Z, but log3 13/ log3 7 ∈ Z3 , thus h7, 13iZ3 = h7iZ3 . Theorem 3.8. Let K̃ be the composite of all Zp -extensions of K inside K ab . Then Gal(K̃/K) ∼ = Zpr2 +1+δ where r2 is the number of complex embeddings of K and δ is the Leopoldt defection. Proof. Since K̃/K is unramified outside p, we first consider the maximal abelian extension F of K unramified outside p. Let H be the maximal unramified abelian extension of K inside F , i.e. the Hilbert class field of K. Write Jk the group of ideles of K and IK the ideal class group of K. By Class field theory, then Y Uv , Gal(F/K) = JK /K × v-p Gal(H/K) ∼ = IK = JK /K × Y Uv . v Write V = K × Q v-p Uv . We have Gal(F/H) = K × Y Uv /V = U V /V ∼ = U/(U ∩ V ). v 15 Note that U = U1 × (finite group), then U/U ∩ V and U1 /(U1 ∩ V ) differ by a [K:Q] finite group. Note that U1 ∼ , then by Lemma 3.5 = (finite group) × Zp U1 /U1 ∩ V = U1 /ψ(E1 ) ∼ = finite × Zrp2 +1+δ . Thus K× Y v and hence Uv /K × Y Uv ∼ = finite × Zrp2 +1+δ v-p Gal(F/K) = finite. Zpr2 +1+δ Suppose the quotient is of order N . Write J 0 = Gal(F/K) = Jk /V . Then N Zrp2 +1+δ ⊆ N J 0 ⊆ Zrp2 +1+δ , 0 thus N J 0 ∼ = {x ∈ J 0 | N x = 0}, then = Zrp2 +1+δ as Zp -modules. Let JN 0 0 ∼ 0 0 J /JN = N J . JN is a finite group with order ≤ N : otherwise, there exist 0 with the same image at J 0 /Zpr2 +1+δ , then x − x0 ∈ distinct elements x, x0 ∈ JN Zrp2 +1+δ and N (x − x0 ) = 0, contradiction! 0 By definition, the fixed field of JN must be K̃ and we get the Theorem. Theorem 3.9 (Iwasawa). Let K = K0 ⊆ · · · ⊆ Kn ⊆ K∞ be a tower of Zp extensions. Let pen be the exact p-power dividing h(Kn ), the order of ideal class group of Kn . Then there exist integers λ ≥ 0, µ ≥ 0 and ν such that en = λn + µpn + ν for n sufficiently large. Let Gal(K∞ /K) = Γ. We fix a topological generator γ0 of Γ. For every n ∈ N, let Ln be the maximal unramified S abelian p-extension of Kn . By the maximality, Ln /K is Galois. Let L = Ln . Then L/K is also n≥0 Galois. Write Xn = Gal(Ln /Kn ), X = Gal(L/K∞ ) and G = Gal(L/K). We 16 have the following diagram: Γ fff L ffffff  f f f f f Xffff  ffffff  f f f  f f  fffff  K∞ f       Γn L jjjj
n j j j
Xn jjj
jjjj

j j
j
jjjj

Kn

G






Γ/Γn L0 z zz z zz zz K For n  0, then all primes which are ramified in K∞ /K are totally ramified in K∞ /Kn . Then for n  0, Kn+1 ∩ Ln = Kn and Xn = Gal(Ln /Kn ) ∼ = Gal(Ln Kn+1 /Kn+1 ), thus a quotient of Xn+1 . Moreover Xn ∼ Gal(L K /K = n ∞ ∞) and [ lim Xn = Gal( Ln K∞ /K∞ ) = Gal(L/K∞ ) = X. ←− Since Xn is an abelian p-group, there is an Zp -action on Xn , since Gal(Ln /K) is Galois, Xn is also equipped with an Γ/Γn -action: let γ ∈ Γ/Γn , let γ̃ be any lifting of γ in Gal(Ln /K), then for x ∈ Xn , xγ = γ̃xγ̃ −1 is independent of the choices of the lifting. Then Xn is a Zp [Γn ]-module. Passing to the limit, we see X = lim Xn is a compact lim Zp [Γn ] = Zp [[Γ]] = Λ-module. ←− ←− We make the following assumption at first: (*) All primes ramified in K∞ /K are totally ramified. Let ℘1 , · · · , ℘s be primes of K which ramify in K∞ /K. Fix ℘˜i of L lying above ℘i , let Ii ⊆ G be the inertia group. Since L/K∞ is unramified, Ii ∩ X = 1. Since K∞ /K is totally ramified at ℘i , Ii ∼ = G/X = Γ, thus G = Ii X = XIi , i = 1, · · · , s. We identify I1 with Γ. Let σi be a topological generator of Ii , then σi = ai σ1 for some ai ∈ X. Lemma 3.10. With the assumption (*). Then G0 = [G, G] = X γ0 −1 = T X. 17 Proof. Let a = αx, b = βy for α, β ∈ Γ and x, y ∈ X. Then aba−1 b−1 =αxβyx−1 α−1 y −1 β −1 = xα αβyx−1 α−1 y −1 β −1 =xα αβyx−1 β −1 α−1 βy −1 β −1 = xα (yx−1 )αβ (y −1 )β =xα(1−β) y β(α−1) . Let β = 1 and α = γ0 , then y γ0 −1 ∈ G0 , hence X γ0 −1 ⊆ G0 . On the other hand, write β = γ0c for c ∈ Zp , then c xα(1−β) = xα(1−γ0 ) ∈ X γ0 −1 P c
n β(α−1) since 1 − γ0c = 1 − (1 + T )c = 1 − ∈ X γ0 −1 , n T ∈ T Λ. Similarly y 0 γ0 −1 hence G ⊆ X . Lemma 3.11. With the assumption. Let Y0 = hT X, a2 , · · · , as i. Let νn = pn ωn /ω0 = (1+TT) −1 and let Yn = νn Y0 . Then Xn ∼ = X/Yn for n ≥ 0. Proof. For n = 0, L0 /K is the maximal unramified abelian p-extension of K, thus the maximal abelian unramified extension inside the Galois extension L/K, by Galois theory, Gal(L/L0 ) is the closed subgroup generated by Ii for 1 ≤ i ≤ s and G0 , i.e., Gal(L/L0 ) = I1 Y0 and X0 = G/I1 Y0 = I1 X/I1 Y0 = X/Y0 . n For general n, just replace K by Kn , γ0 by γ0p and Y0 by Yn . Theorem 3.12. X is a finitely generated torsion Λ-module. Proof. First with the assumption. To show that X is finitely generated is equivalent to showing that Y0 is finitely generated. But Y /ν1 Y is finite and ν1 ∈ (p, T ), by Nakayama’s Lemma, Y is a finitely generated Λ-module. Moreover Y and X are torsion by Theorem 2.14. In general, suppose all primes ramified in K∞ /K are totally ramified in K∞ /Ke . Replace K by Ke , then for n ≥ e, Xn = X/νn,e Ye where νn,e = ωn /ωe and Ye is the corresponding Y0 for the extension K∞ /Ke . Similarly we can show that Ye is finitely generated and hence X is finitely generated. Lemma 3.13. Let M1 ∼ M2 be two finitely generated Λ-modules with a given pseudo-isomorphism. If |M1 /νn,e M1 | < ∞ for all n ≥ e. Then there exist some constant c and some n0 ≥ e, such that |M1 /νn,e M1 | = pc |M2 /νn,e M2 | for n ≥ n0 . 18 Proof. Consider the diagram 0 −−−−→ νn,e M1 −−−−→ M1 −−−−→ M1 /νn,e M1 −−−−→ 0       φy φ00 φ0n y ny 0 −−−−→ νn,e M2 −−−−→ M2 −−−−→ M2 /νn,e M2 −−−−→ 0 by the snake lemma, we have an exact sequence 0 → ker φ0n → ker φ → ker φ00n → coker φ0n → coker φ → coker φ00n → 0. We have (1) |ker φ0n | ≤ |ker φ|; (2) | coker φ00n | ≤ | coker φ|; (3) | coker φ0n | ≤ | coker φ|; (4) |ker φ00n | ≤ |ker φ|| coker φ|. Now for m ≥ n, we have (a) |ker φ0n | ≥ |ker φ0m |; (b) | coker φ00n | ≤ | coker φ00m |; (c) | coker φ0n | ≥ | coker φ0m |. (3) and (c) needs a little more explanation, others are easy. For (c), let νm,e y ∈ νm,e M2 , let z ∈ νn,e M2 be a representative of νn,e y in coker φ0n . Then νn,e y−z = φ(νn,e x) for νn,e x ∈ νn,e M1 and νm,e y is represented by νm,n z in coker φ0m . The proof of (3) is similar. By (2) and (b), the sizes of coker φ00n ’s are non-decreasing with an upper bound | coker φ|, when n  0, | coker φ00n | will be stable. Similarly the sizes of ker φ0n and coker φ0n will be stable when n  0, hence also the size of ker φ00n by the long exact sequence. Proof of Theorem 3.9. By Theorem 3.12, X∼E= s M Λ/p ki
⊕ i=1 t M (Λ/fj (T )mj ) j=1 where fj (T )’s are irreducible distinguished polynomials. By the above lemma 3.13, |Xn | = |X/νn,e Y0 | is equal to |E/νn,e E| up to a bounded factor. Note that n e n |Λ/(pki , νn,e | = pki (p −p ) = pki p +c . We have to compute |Λ/(νn,e , fj (T )mj )|. Let g be a distinguished polynomial of degree d. Write V = Λ/(g(T )). As in the proof of Lemma 2.15, for n0 > e, pn ≥ d and n ≥ n0 , then νn+2,e V = ωn+2 (T ) ωn+1 (T ) (νn+1,e V ) = pνn+1,e V , and |V /νn+2,e V | =|V /pV | · |pV /pνn+1,e V | = |V /pV | · |V /νn+1,e V | =pd(n−n0 +1) |V /νn0 +1,e V | = pnd+c . 19 m Plug the above result in the case g = Fj j , we have when n  0, n |E/νn,e E| = pµp +λn+c P P with µ = µ(E) = ki and λ = λ(E) = mj deg Fj . We just showed that the module X is a finitely generated torsion Λ-module. Here we give more examples of Iwasawa modules. Hereafter we consider the following special case: K = Q(ζp ), Kn = Q(ζpn+1 ) and K∞ = Q(ζp∞ ). We let ∆ = Gal(Q(ζp )/Q) ∼ = (Z/pZ)× . Then Gal(K∞ /Q) = ∆ × Γ. Let En be the group of global units of Kn and Cn be the subgroup generated by ζpn+1 and ζpn+1 − 1 as Gal(Kn /Q)-module, which is called the group of cyclotomic units. We Q recall the map ψ maps En into the finitely generated Zp [Kn /Q]-module UKn ℘ . Let En = ψ(En ) and Cn = ψ(Cn ). Let ℘|p E∞ = lim En , ←− C∞ = lim Cn ←− n∈N n∈N with the transition maps given by the norm map. Then E∞ and C∞ are finitely generated Zp [[Gal(K∞ /Q)]] = Λ[∆]-modules. For any character χ : ∆ → Z× p χ χ and a Λ[∆]-module M , let M χ = eχ M be the χ-part of M . Then E∞ , C∞ and (E∞ /C∞ )χ are finitely generated Λ-modules. Recall En /Cn are finite for all /C∞ n ∈ N, and En /Cn = ωEn∞(E/C) , then E∞ /C∞ is Λ-torsion and so is (E∞ /C∞ )χ . Similarly X is a Λ[∆]-module and X χ is Λ-torsion. Then the Iwasawa Main Conjecture is the following theorem of Mazur-Wiles: Theorem 3.14 (Main Conjecture). If χ is even (i.e., χ(−1) = 1), χ 6= 1, then (Char X χ ) = (Char(E∞ /C∞ )χ ). The main conjecture has another equivalent form. By the proof of Theorem 3.8, we know for any number field K, the maximal abelian pro-p extension of K unramified outside p has Zp -rank r2 (K) + 1 + δ(K). In a Zp -extension K∞ /K, let Mn (resp. M∞ ) be the maximal abelian pro-p extension of Kn (resp. K∞ ) unramified outside p. Then K∞ ⊂ Mn ⊂ M∞ . Let Xn = Gal(Mn /K∞ ), X∞ = Gal(M∞ /K∞ ). Then X∞ is a finitely generated Λ-module since Xn = X∞ /ωn X∞ is finitely generated as Zp -module. Back to the special case. Then δ(K) = 0 and X∞ is of Λ-rank r2 (K) + 1, and there is an action of ∆ on X∞ . One can show that if χ is even, Xχ∞ is a torsion Λ-module. On the other hand, the p-adic L-function Lp (s, χ) is given by Lp (1 − s, χ) = g((1 + T )s − 1) for some g(T ) ∈ Λ. Then Theorem 3.15 (Equivalent form of Main Conjecture). For χ even, χ 6= 1, (Char(Xχ∞ )) = (g(T )). 20 Chapter 4 Iwasawa theory of elliptic curves Let K be any number field. For an elliptic curve E defined over K, the theorem of Mordell-Weil claims that the set of K-rational points E(K) of E is a finitely generated abelian group, that is E(K) = Zr ⊕ T for T the torsion group of E(K) and r the rank of E(K). The study of r(E(K)) is a major problem in the arithmetic of elliptic curve. For example, the famous Birch-Swinnerton-Dyer Conjecture claims that this rank equals the order of zeroes of L(E, s), the L-function of E, at s = 0, and gives a conjectural relation about the leading terms of L(E, s). Let F∞ /F be a Zp -extension and Fn be the n-th layer. Let E be an elliptic curve defined over F . One can ask how rank E(Fn ) varies as n varies. We shall study this question in this chapter. First let us introduce the definitions of Selmer groups and Shafarevich groups. Let L be a field of characteristic 0 and E be an elliptic curve defined over L. Let L be an algebraic closure of L. Let GL = Gal(L/L). We write H i (L, −) for the cohomology group H i (GL , −). For the exact sequence [n] 0 −→ E[n] −→ E −→ E −→ 0, taking the Galois cohomology, one has (4.1) 0 −→ E(L) κ −→ H 1 (L, E[n]) −→ H 1 (L, E)[n] −→ 0, nE(L) where the Kummer map κ is defined as follows: For b ∈ E(L), choose a ∈ E(L) such that na = b, then κ(b) is the cohomological class associated to the cocycle κ(b)(σ) = aσ − a, ∀σ ∈ GL . 21 Let v be a place of L, then we get a local exact sequence analogue to (4.1). If we regard GLv as a subgroup of GL , then the restriction maps from H 1 (L, −) to H 1 (Lv , −) yield the following commutative diagram: 0 −−−−→ E(L) nE(L)   y κ −−−−→ H 1 (L, E[n]) −−−−→ H 1 (L, E)[n] −−−−→ 0     y y κη E(Lv ) 0 −−−−→ nE(L −−−−→ H 1 (Lv , E[n]) −−−−→ H 1 (Lv , E)[n] −−−−→ 0 v) The n-th Selmer group of E over L is the group \ SelE (L)[n] = ker (H 1 (L, E[n]) → H 1 (Lv , E(Lv ))[n]). v The Shafarevich-Tate group of E over L is the group \ XE (L) = ker (H 1 (L, E(L)) → H 1 (Lη , E(Lv )). v Easily by diagram chasing, these two groups and the Mordell-Weil group are related by the following important fundamental exact sequence (4.2) 0 → E(L)/nE(L) → SelE (L)[n] → XE (L)[n] → 0. For every pair (n, m) such that n ≤ m, we have the following commutative diagram 0 −−−−→ E(L) nE(L)   y −−−−→ H 1 (L, E[n]) −−−−→ H 1 (L, E)[n] −−−−→ 0     y y E(L) 0 −−−−→ mE(L) −−−−→ H 1 (L, E[m]) −−−−→ H 1 (L, E)[m] −−−−→ 0 where the vertical maps are natural injections. Furthermore, the local analogue of the above diagram also holds and the restriction maps are compatible with the diagrams. Passing to the limit, we have κ 0 −−−−→ E(L) ⊗ Q/Z −−−−→ H 1 (L, E(L)tors ) −−−−→ H 1 (L, E) −−−−→ 0       y y y κ 0 −−−−→ E(Lv ) ⊗ Q/Z −−−v−→ H 1 (Lv , E(Lv )tors ) −−−−→ H 1 (Lv , E) −−−−→ 0 The Selmer group of E over L is the group \ SelE (L) = ker (H 1 (L, E(L)tors ) → H 1 (Lv , E(Lv ))). v One has the exact sequence (4.3) 0 → E(L) ⊗ Q/Z → SelE (L) → XE (L) → 0. 22 Let p be a prime number, then the p-primary Selmer group is given by \ SelE (L)p = ker (H 1 (L, E[p∞ ]) → H 1 (Lv , E(Lv ))[p∞ ]) v =ker H 1 (L, E[p∞ ] → Y H 1 (Lv , E[p∞ ]) v ! Im κv and one has an exact sequence 0 → E(L) ⊗ Qp /Zp → SelE (L)p → XE (L)p → 0. Put HE (Lv ) = H 1 (Lv , E[p∞ ]) , Im κv Denote by PE (L) the product of HE (Lv ) for all primes v of L. Then SelE (L)p = ker (H 1 (L, E[p∞ ]) → PE (L)). Put GE (L) = Im (H 1 (L, E[p∞ ]) → PE (L)), then one has an exact sequence (4.4) 0 → SelE (L)p → H 1 (L, E[p∞ ]) → GE (L) → 0. Suppose furthermore that the extension L/F is a Galois extension. Write G = Gal(L/F ). For every intermediate field F 0 of L/F , write G(L/F 0 ) = Gal(L/F 0 ). One has the following commutative diagram with exact rows 0 / SelE (F 0 )p 0  / Sel (L)G(L/F 0 ) p E / H 1 (F 0 , E[p∞ ]) sL/F 0  / GE (F 0 ) hL/F 0 / H 1 (L, E[φ∞ ])G(L/F 0 )  /0 gL/F 0 / GE (L)G(L/F 0 ) where the vertical maps sL/F 0 , hL/F 0 and gL/F 0 are natural restrictions. The snake lemma then gives the exact sequence: (4.5) 0 → ker sL/F 0 → ker hL/F 0 → ker gL/F 0 → coker sL/F 0 → coker hL/F 0 . Theorem 4.1 (Mazur’s Control Theorem). If F∞ /F is a Zp -extension, assuming that E has good ordinary reduction at all primes of F lying over p. Let Fn be the n-th layer of the Zp extension. Then the natural maps sn = sF∞ /Fn : SelE (Fn )p −→ SelE (F∞ )Γp n have finite kernels and cokernels, whose orders are bounded as n → ∞. We first give some consequences of Mazur’s Control Theorem: 23 Corollary 4.2. Suppose E is an elliptic curve defined over F such that E has good, ordinary reduction at all primes lying above p. If E(F ) and XE (F ) are both finite, then SelE (F∞ )p is Λ-cotorison. Consequently, rankZ E(Fn ) is bounded as n varies. Proof. Let X = Hom(SelE (F∞ )p , Qp /Zp ). Then X is an Λ-module. Moreover, X/T X = Hom(SelE (F∞ )Γp , Qp /Zp ) is finite since SelE (F )p is finite, thus X is a finitely generated Λ-torsion module, hence SelE (F∞ )p is Λ-cotorison. Now X/XZp -tors ∼ = Zλp , thus (SelE (F∞ )p )div ∼ = (Qp /Zp )λ and (SelE (Fn )p )div ∼ = tn (Qp /Zp ) for some tn ≤ λ. Since E(Fn ) ⊗ Qp /Zp ,→ (SelE (Fn )p )div through the Kummer map, we have rank E(Fn ) ≤ λ. Corollary 4.3. Suppose E is an elliptic curve defined over F such that E has good, ordinary reduction at all primes lying above p. If E(Fn ) and XE (Fn ) are finite for all n, then there exist λ, µ ≥ 0, depending only on E and F∞ /F , such that n |XE (Fn )p | = pλn+µp +O(1) . Proof. From the assumption, SelE (Fn )p are finite. Let X = Hom(SelE (F∞ )p , Qp /Zp ). Then |X/ωn X| = | SelE (F∞ )Γp n | < ∞ for all n, thus X is a finitely generated torn sion Λ-module. Apply Proposition 2.17, we get |X/ωn X| = pλ(X)n+µ(X)p +O(1) . The result then follows. Corollary 4.4. Suppose E is an elliptic curve defined over F such that E has good, ordinary reduction at all primes lying above p. Let r = corankΛ (SelE (F∞ )p ) = rankΛ X, then corankZp SelE (Fn )p = rpn + O(1). Proof. Let X = Hom(SelE (F∞ )p , Qp /Zp ). Then X is a finitely generated Λmodule, say pseudo-isomorphic to Λr × Y × Z for Y a free Zp -module of finite rank and Z a torsion group of bounded components. Since X/ωn X is the Pontragin dual of SelE (F∞ )Γp n , and the size of latter one differs from | SelE (Fn )p | by a finite bounded value, then corankZp SelE (Fn )p = rankZp X/ωn X = rpn + O(1). We shall not give a complete proof of the control theorem here (cf. Greenberg [3]). One has to use the exact sequence 0 → ker sn → ker hn → ker gn → coker sn → coker hn , then to study ker sn and coker sn , it suffices to study ker hn , coker hn and ker gn . The first two are easy by the inflation-restrction exact sequence, but the third one needs more analysis. One needs to study the local restriction rv : H 1 (Fn,v , E[p∞ ]) H 1 (Lη , E[p∞ ]) −→ , Im κv Im κη 24 for every place v. For v - p, it is easy. For v | p, it is more difficult. Here we only prove Theorem 4.6, which will be key to the study of the local maps. One can use Tate’s duality theorem for local fields to prove Theorem 4.6, but we give a proof using methods of Iwasawa theory. We first have: Lemma 4.5. Let K be a finite extension over Qp and let F/K be a finite abelian extension with Galois group ∆. Let χ : ∆ → Z∗p be a character of ∆. Let MF be a maximal abelian p-extension over F . Then MF /K is Galois and ( [K : Qp ] + 1, if χ = 1, rankZp Gal(MF /F )χ = [K : Qp ], otherwise. Proof. MF /K is Galois since MF is maximal. By class field theory, the isomorphism n lim F × /F ×p −→ Gal(MF /F ) ← − n is ∆-equivariant. Recall that F × = hπF i × UF , the p-completion of hπF i is a copy of Zp , with a trivial action of ∆, the pcompletion of UF is isomorphic to OF = OK [∆] × µp∞ (F ). Thus ( [K : Qp ] + 1, if χ = 1, χ rankZp Gal(MF /F ) = [K : Qp ], otherwise. Theorem 4.6. Let Kv be a finite extension of Qp . Suppose that A is a Gkv module and that A ∼ = Qp /Zp as a group. Then H 1 (Kv , A) is a cofinitely generated Zp -module of Zp -corank ( 1, if A = µp∞ or A = Qp /Zp ; = [Kv : Qp ] + 0, otherwise. Proof. GKv acts on A ∼ = Qp /Zp through a character ψ : GKv → Aut(Qp /Zp ) ∼ = × ZP : for any g ∈ GKv and a ∈ A, ga = ψ(g)a. If A = Qp /Zp (i.e. ψ = 1) or A = µp∞ (i.e. ψ is the cyclotomic character), the theorem is easy to check following from Lemma 4.5 and Kummer theory. We suppose now A is not Qp /Zp or µp∞ , there are two cases: H (1) Im ψ is finite. Let H = ker ψ, then G = GKv /H is finite. Let F = Kv be the field fixed by H, then Gal(F/Kv ) = G is a finite abelian group. We consider the inflation-restriction sequence 0 → H 1 (G, A) → H 1 (Kv , A) → H 1 (H, A)G → H 2 (G, A). 25 1 If p 6= 2, then Z× p is pro-cyclic, G is cyclic in this case and |H (G, A)| = 2 |H (G, A)|. Suppose G is generated by σ and ψ(σ) = a ∈ A, then H 1 (G, A) = 1 N A/(a − 1)A. Note that a 6= 1 and A/(a − 1)A is finite, so H (G, A) and 2 n H (G, A) are both finite. If p = 2, then G = Im ψ = Z/2 Z × Z/2Z or Z/2n Z. In each case one can verify that H 1 (G, A) and H 2 (G, A) are finite. Now H acts trivially on A, then H 1 (H, A) = Hom(H, A) = Hom(Gal(Kv /F )ab , A) = Hom(Gal(MF /F ), A). Thus H 1 (H, A)G = HomG (Gal(MF /F ), A) = Hom(Gal(MF /F )χ , Qp /Zp ) where χ is the restriction of ψ at G. The theorem follows from Lemma 4.5. ker ψ and G = Gal(F∞ /Kv ), Note that (2) Im ψ is infinite. Let F∞ = Kv G∼ = Im ψ ,→ Z∗p , one can write G ∼ = ∆ × Γ, where ∆ is a subgroup of Z/(p − 1)Z or Z/2Z if p = 2. Let F = FΓ∞ . Again we need to consider the inflationrestriction sequence 0 → H 1 (G, A) → H 1 (Kv , A) → H 1 (F∞ , A)G → H 2 (G, A). First consider the spectral sequence H p (∆, H q (Γ, A)) ⇒ H p+q (G, A). For n = p + q = 2, as Zp has cohomological dimension 1, H 2 (Γ, A) = 0. If prime p 6= 2, the order of ∆ is prime to p, H 1 (Γ, A) and Aγ are p-groups, hence H 1 (∆, H 1 (Γ, A)) = 0 and H 2 (∆, AΓ ) = 0, thus H 2 (G, A) = 0. If p = 2 and ∆ trivial, again H 2 (G, A) = 0; if ∆ = Z/2Z, one can get H 2 (G, A) ∼ = Z/2Z, but easy to see it is finite. For n = p + q = 1, for ∆ = 1, easily to see H 1 (G, A) = H 1 (Γ, A) is finite; for prime p 6= 2 or ∆ = 1, we have H 1 (∆, H 0 (Γ, A)) = 0 and H 0 (∆, H 1 (Γ, A)) = 0; for p = 2 and ∆ ∼ = Z/2Z 6= 1, both are again finite. Thus H 1 (G, A) is finite. So we have corankZp H 1 (Kv , A) = corankZp H 1 (F∞ , A)G . Γn . Fix an algebraic closure Qp of Qp . Let Mn be the maximal Let Fn = F∞ abelian pro-p extension of Fn and M∞ be the maximal abelian pro-p extension of F∞ . Let X = Gal(M∞ /F∞ ) and Xn = Gal(Mn /Fn ). By Lemma 4.5, ( 1, χ = 1; Gal(Mn /Fn )χ = [Kv : Qp ]pn + 0, χ 6= 1. Hence Gal(Mn /F∞ )χ = [Kv : Qp ]pn . Write ψ∆ and ψΓ the restrictions of ψ on ∆ and Γ. Then corankZp H 1 (Kv , A) = corankZp H 1 (F∞ , A)G = corankZp Hom(Gal(Qp /F∞ ), A)G = corankZp HomG (X, A) = rankZp X ψ = rankZp (X ψ∆ )ψΓ = rankZp X ψ∆ X ψ∆ = rankZp (γ0 − ψ(γ0 )) T −b 26 where b = ψ(γ0 ) − 1 ∈ pZp . We need to study X, X ψ∆ . Note for p 6= 2, X ψ∆ = eψ∆ X for eχ the idempotent element of χ. Note that Mn is the maximal abelian sub-extension inside M∞ /Fn , thus Gal(M∞ /Mn ) = Gal(M∞ /Fn )0 . By the exact sequence 1 −→ X −→ Gal(M∞ /Fn ) −→ Γn → 1 n then any element in Gal(M∞ /Fn ) is of the form αx for α = γ̃0p m and x ∈ X. Let αx, βy ∈ Gal(M∞ /Fn ), then αxβyx−1 α−1 y −1 β −1 = xα(1−β) y (α−1)β , we have Gal(M∞ /Fn )0 = ωn X. Since X is compact, ωn X is closed and Gal(M∞ /Mn ) = ωn X and Gal(Mn /F∞ ) = X/ωn X. By Nakayama Lemma, X is a finitely generated Λ-module of rank [Kv : Qp ]|∆|. Moreover, X/ωn X is ∆-equivariant, Gal(Mn /F∞ )χ = (X/ωn X)χ = X χ /ωn X χ , then X χ is a finitely generated Λ-module of rank [Kv : Qp ]. By Class field theoy, since p∞ | [F∞ : Qp ], GF∞ has p-adic cohomological dimension 1, hence H 1 (F∞ , Qp /Zp ) is a divisible group. Thus X = H 1 (F∞ , Qp /Zp )∧ is torsion free as Zp -module. Thus X has no nonzero finite Λ-submodules. Let Y = XΛ−tors and W = X/Y . Then W is torsion free and 0 −→ Y X W −→ −→ −→ 0 ωn Y ωn X ωn W is exact by snake lemma. W has Λ-rank [Kv : Qp ]|∆| and hence W/ωn W has Zp -rank Kv : Qp ]|∆|pn , the same as the Zp -rank of X/ωn X. Therefore Y /ωn Y is finite and must be isomorphic to a subgroup of (X/ωn X)Zp −tors = µp∞ (Fn ). On one hand, if µp∞ (F∞ ) is finite, then Y = limn Y /ωn Y is finite and hence ←− Y = 0. On the other hand, if Y is infinite, then Y = limn Y /ωn Y is pro-cyclic ←− and therefore ∼ = Zp as a Zp -module. Suppose W → Λr is a quasi-isomorphism, then 0 → W → Λr → B → 0 is exact and B is a finite Λ-module, by snake lemma again, (W/ωn W )Zp −tors is bounded by ker (ωn : B → B), which equals B when n  0. Therefore if µp∞ (Fn ) is unbounded, then Y /ωn Y is also unbounded and Y is infinite. Hence if µp∞ ⊂ F∞ , then Y ∼ = Tp (µp∞ ). Now we can finish the proof of the Theorem. We have corankZp H 1 (Kv , A) = rankZp X ψ∆ /(T − b) for T − b a distinguished polynomial of degree 1. As X is quasi-isomorphic to Λ[Kv :Qp ]|∆| if µp∞ ( F∞ , or Tp (µp∞ ) ⊕ Λ[Kv :Qp ]|∆| . 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