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Iwasawa theoretic studies on K-groups and Selmer groups

March 2017

Takahiro Kitajima

AThesisfortheDegreeofPh.D.inScience

Iwasawa theoretic studies on K-groups and Selmer groups

March 2017

Graduate School of Science and Technology Keio University

Takahiro Kita jima Contents

1 Introduction 2 1.1 Background ...... 2 1.2 Iwasawa theory ...... 3 1.2.1 Iwasawa theory for ideal class groups ...... 3 1.2.2 Iwasawa theory for elliptic curves ...... 4 1.3 Main results ...... 6 1.3.1 The plus and the minus Selmer groups for elliptic curves at supersingular primes in the cyclotomic Zp-extension 6 1.3.2 Orders of even K-groups of rings of integers in the cyclotomic Zp-extension of Q ...... 7 1.4 Outline ...... 8

2 Iwasawa modules 9

3 K-groups of rings of integers 12 3.1 K-groups of rings ...... 12 3.2 Lemmas ...... 15 3.3 The orders of even K-groups ...... 22

4 The plus and the minus Selmer groups 26 4.1 Definitions ...... 26 4.2 The formal groups and the norm subgroups ...... 28 4.2.1 Honda’stheory ...... 29 4.2.2 The formal groups associated to E ...... 30 4.2.3 The norm subgroups ...... 36 4.2.4 The plus and the minus local conditions ...... 44 4.2.5 More on the plus and the minus local conditions . . . . 50 4.3 Finite Λ-submodules ...... 52 4.3.1 Finite Λ-submodules of Sel(F ,E[p∞])∨ ...... 52 ∞ 4.3.2 Finite Λ-submodules of Sel±(F ,E[p∞])∨ ...... 54 ∞

1 Chapter 1

Introduction

In number theory the K-groups of the ring of integers of a number field and the Selmer groups of an elliptic curve are important objects, and have been studied intensively. In Iwasawa theory, we study the behavior of these objects in Zp-extensions. The aim of this thesis is such Iwasawa theoretic study of these objects.

1.1 Background

We begin with some historical background of our research. The K-groups of the ring of integers are generalizations of the ideal class groups. We briefly explain ideal class groups. The ideal class group Cl(L) of a number field L is the quotient group Cl(L)=I(L)/P (L), where I(L) is the group of fractional ideals of L and P (L)isitssubgroupofprincipal ideals of L.ItisknownthatCl(L)isafiniteabeliangroup,however,itis quite hard in general to compute the exact order of Cl(L)foreachL.Ideal class groups are widely investigated even today. There exists a classical and remarkable formula, the analytic class number formula, which relates the ideal class groups to the Dedekind zeta function. The explicit statement is as follows. Let L be a number field, ζL(s)the Dedekind zeta function of L, h(L)theorderofCl(L), Reg(L)theregulator of L,andw(L)thenumberoftherootsofunityinL.Thenwehave

(r1+r2 1) h(L)Reg(L) lim s− − ζL(s)= , s 0 → − w(L) where r1 and 2r2 are the numbers of real embeddings L" R and complex → embeddings L" C,respectively.Thisformulameansthatthespecialvalues of the Dedekind→ zeta functions know the orders of the ideal class groups. As

2 arefinementoftheanalyticclassnumberformula,thereisaconjecturein Iwasawa theory, which is called the Iwasawa Main Conjecture.

1.2 Iwasawa theory

We introduce Iwasawa theory in this section. We review some basic known properties on ideal class groups in the first subsection, and those on Selmer groups of an elliptic curve in the second subsection. We also mention some problems on Selmer groups of an elliptic curve in Iwasawa theory.

1.2.1 Iwasawa theory for ideal class groups The Iwasawa Main Conjecture for ideal class groups describes a more pre- cise relation between the “p-part” of ideal class groups and the “p-adic” L-functions for each prime number p,thantheanalyticclassnumberfor- mula. Let p be a prime number, L atotallyrealnumberfield,F aCM extension of L such that F/L is abelian, χ an odd character of Gal(F/L), F the cyclotomic Zp-extension of F ,and F the Galois group of the ∞ ∞ maximal abelian pro-p unramified extension ofX F .WedefineaGal(F/L)- ∞ module χ by χ = Zp[Imageχ]onwhichGal(F/L)actsviaχ; σx = χ(σ)x O O for σ Gal(F/L)andx χ. Note that χ is a discrete valuation ring and∈ we denote by π aprimeelementof∈O O . Assume for simplicity Oχ F L = L.Thentheχ-part Mχ of a Zp[Gal(F/L)]-module M is defined ∩ ∞ by Mχ = M Zp[Gal(F/L)] χ.PutΛχ = χ[[Gal(F /F )]]. Then the χ-part ⊗ O O ∞ F ,χ of F is known to be finitely generated Λχ-torsion. By the structure X ∞ X ∞ theorem for Λχ-modules, there exist irreducible distinguished polynomials fj, nonnegative integers s, t, mi, nj and a homomorphism

s t mi nj F ,χ Λχ/(π ) Λχ/(fj ) (1.2.1) X ∞ → ⊕ i=1 j=1 ! ! with finite kernel and finite cokernel. We note that the target of the homo- morphism (1.2.1) is uniquely determined by F ,χ.Wedefinethecharacter- X ∞ istic ideal of F ,χ by X ∞ s t χ mi nj CharΛ( F )= π fj . X ∞ "i=1 j=1 $ # #

Let p,χ Λχ be the p-adic L-function of Deligne and Ribet, constructed fromL the∈ Stickelberger elements. Then the Iwasawa Main Conjecture claims

3 that

CharΛχ ( F ,χ)=( p,χ) Λχ X ∞ L ⊂ for any odd character χ such that χ = ω where ω is the Teichm¨uller character. ̸ This conjecture was proved when L = Q by Mazur and Wiles (cf. [22]), and for general totally real field L and for p>2 by Wiles (cf. [41]). As an application of the Iwasawa Main Conjecture, we have the following formula which describes the relation between the Dedekind zeta function and the K-groups. Let L be a totally real number field, L the ring of integers in L,andK ( )them-th K-group of definedO by Quillen. Then for m OL OL each even number m Z>0,wehave ∈

am(L) #K2m 2( L) ζL(1 m)= 2 − O − ± wm(L)

m for some am(L) Z,wherewm(L)=max t Z>0 Gal(L(µt)/L) =1 . ∈ { ∈ | } Next we refer a classical result on ideal class groups concerning non- existence of nontrivial finite submodules. We denote the completed group ring Zp[[Gal(F /F )]] by Λ. Then F ∞ ∞ is known to be a finitely generated torsion module over Λ. X If F is a CM-field, the complex conjugation J acts naturally on F . J=1 JX= ∞1 Further if p is odd, then F is decomposed into F = F F − , J=1 X ∞ J= 1 X ∞ X ∞ ⊕X ∞ where F = x F Jx = x and F − = x F Jx = x . X ∞ { ∈X ∞ | } X ∞ { ∈X ∞ | − } J= 1 Theorem 1.2.1 (Iwasawa). If F is a CM-field, then F − has no nontrivial finite Λ-submodules. X ∞

If a given Λ-module M has no nontrivial finite Λ-submodules as in The- orem 1.2.1, then we can understand more precise structure of M than the J= 1 general structure theorem. Actually Iwasawa studied F − throughly using this non-existence of nontrivial finite Λ-submodules.X The∞ non-existence of nontrivial finite Λ-submodules is a basic and important property in Iwasawa theory.

1.2.2 Iwasawa theory for elliptic curves Let E be an elliptic curve defined over Q such that E has good reduction at aprimenumberp,i.e.thereducedcurveE mod p is nonsingular. Denote ap =1+p #E(Fp). We say that E has good ordinary reduction at p if − a / 0(modp), and E has good supersingular% reduction at p if a 0(mod p ≡ p ≡ % 4 p). Note that if E has good supersingular reduction at p and p 5, then ≥ ap = 0 by the Hasse bound ap 2√p. | |≤ We put F0 = Q(µm)forsimplicity.LetF /F0 be the cyclotomic Zp- ∞ extension and Fn the n-th layer. Denote Λ= Zp[[Gal(F /F0)]]. The ∞ Selmer groups for E over F are subgroups of the Galois cohomology group 1 ∞ H (F ,E[p∞]) defined by certain “local conditions”. The Pontryagin dual ∞ of the Selmer group is a corresponding object to the ideal class group in Iwasawa theory for elliptic curves. When E has good ordinary reduction at p,thePontryagindualofthe p-primary Selmer group Sel(F ,E[p∞]) of E over F (see Section 4.1 for ∞ ∞ the definition) is conjectured to be Λ-torsion. This conjecture was proved in our setting by Kato. Greenberg and Hachimori-Matsuno proved that if E has ordinary reduction at p and E(F0)[p]=0thenthePontryagindualof Sel(F ,E[p∞]) has no nontrivial finite Λ-submodules (cf. [7]). ∞ On the contrary, when E has good supersingular reduction at p,the Pontryagin dual of the p-primary Selmer group of E over F is no longer ∞ Λ-torsion. In fact, it is known that the Pontryagin dual Sel(F ,E[p∞])∨ of ∞ the p-primary Selmer group has positive Λ-rank; more precisely, we have

rankΛ Sel(F ,E[p∞])∨ [F0 : Q] ∞ ≥ (cf. [6, Theorem 1.7]). So we cannot formulate the Iwasawa Main Conjecture in the classical style in terms of the p-primary Selmer group. To avoid this dif- + ficulty, Shin-ichi Kobayashi [17] defined new Selmer groups Sel (F ,E[p∞]) ∞ and Sel−(F ,E[p∞]), which we call the plus and the minus Selmer groups, ∞ when ap =0,andF0 = Q(µp), where µp denotes the group of p-th roots of unity. He actually proved that the Pontryagin duals Sel±(F ,E[p∞])∨ are ∞ Λ-torsion and formulated the Iwasawa Main Conjecture in the classical style. Adrian Iovita and Robert Pollack [10] generalized definitions of the plus and the minus Selmer groups to the case when p 1(modm)andap =0.Fur- ther Byoung Du Kim [11] generalized them to≡ the case when GCD(p, m)=1 and ap =0. Byoung Du Kim studied in [12] the problem on the existence of nontrivial finite Λ-submodules of Sel±(F ,E[p∞])∨,however,hisresultwasnotcom- ∞ pletely general. In particular, he assumed further the congruence condition p 1(modm)whenhestudiedtheplusSelmergroup.Wealsonotethat ≡ he did not consider the case when µp F0. The aim of this thesis is to remove⊂ those conditions. In this thesis, we consider more general F0.Inparticular,weremovetheassumptiononthe congruence condition p 1(modm). We also consider the case when µ ≡ p ⊂ F0.

5 1.3 Main results

We study two different objects, K-groups and Selmer groups, and get two different results. The main result of this thesis is the result on Selmer groups. So we explain our result on Selmer groups at first in 1.3.1, and that on K- groups in 1.3.2. § § 1.3.1 The plus and the minus Selmer groups for elliptic curves at supersingular primes in the cyclotomic Zp-extension In this section, we state our main theorem on the non-existence of nontrivial finite Λ-submodules of the Pontryagin duals of the plus and the minus Selmer groups. As in Section 1.2.2, we explain our results in the following setting. Let E be an elliptic curve defined over Q such that E has good supersingular reduction at a prime number p with ap =0.LetF = Q(µm)suchthat GCD(p, m)=1,F0 = F (µp)andF the cyclotomic Zp-extension of F0. ∞ In this setting, our main result (Theorem 4.3.8) is the following general- ization of a result of Byoung Du Kim.

Theorem 1.3.1 (Corollary 4.3.9). Both Sel±(F ,E[p∞])∨ have no nontrivial ∞ finite Λ-submodule.

Remark 1.3.2. (1) Our proof works as well for more general elliptic curves and base fields than the above. The precise setting will be explained in Section 4.1, and we will get the same conclusion, i.e. non-existence of nontrivial finite Λ-submodule.

(2) We can also remove the condition GCD(p, m)=1,sincetheSelmer groups Sel±(F ,E[p∞]) are objects over F and the non-existence of ∞ ∞ nontrivial finite Λ-submodules is independent of the choice of a base field F0. (3) By this theorem (and the above remark), we find that the plus and the minus Selmer groups are unconditionally in “good” situation as the ideal class groups (see Theorem 1.2.1).

To prove the above theorem, we study the Λ-module structures of the plus and the minus local conditions E±(k ) Qp/Zp.Thekeyoftheproof ∞ of Theorem 1.3.1 is to determine the explicit Λ⊗-module structures of them as the following theorem.

6 Theorem 1.3.3 (Theorem 4.2.37). Let E be an elliptic curve defined over Qp with ap =0, k = Qp(µm) such that GCD(p, m)=1, k0 = k(µp) and k the cyclotomic Zp-extension of k0. We denote the completed group ring ∞ Zp[[Gal(k /k)]] by Λk. Then we have isomorphisms ∞ [k:Qp] ∨ ⊕ E−(k ) Qp/Zp = Λk , ∞ ⊗ ∼ + [k:Qp] δ ∨ ⊕ &E (k ) Qp/Zp' = Λk (Λk/I)⊕ , ∞ ⊗ ∼ ⊕ of Λk-modules, where& I =ker(Λk ' Zp) is the augmentation ideal, and →

0 if [k : Qp] / 0(mod 4), δ = ≡ (2 otherwise. Remark 1.3.4. In the setting of [12], the Pontryagin duals of the plus and the minus local conditions E±(k ) Qp/Zp are Λ-free. By our theorem, we ∞ find that this Λ-freeness breaks down⊗ in general.

1.3.2 Orders of even K-groups of rings of integers in the cyclotomic Zp-extension of Q We formulate a problem on the orders of even K-groups and state our theo- rem on this problem. For that purpose, we briefly review some known properties on K-groups. Let A be a Dedekind domain and Km(A)them-th Quillen K-group for any integer m 0. The precise definition of the Quillen K-group of a ring will be ≥ introduced in Section 3.1. We know that K0(A)isisomorphictoZ Cl(A)(cf. [23]), Thus the ideal class group of A is isomorphic to the torsion⊕ subgroup of K0(A);

Cl(A) ∼= K0(A)tors. When A is a ring of integers, For any m 0, K ( )isafinitely ≥ m OL generated abelian group (cf. Quillen [27]). Thus the torsion part Km( L)tors is finite for any m 0. We will study the torsion parts of the K-groupsO as ≥ an analogue of the ideal class groups. In this thesis, we study Km(A)tors in the case when A varies over the class of the rings of integers. Let F be a number field and F the composite field of all the cyclotomic Zp-extensions of F .Weconsiderthefollowingproblem. ) Problem 1.3.5. Assume that F is totally real and m 0 is an integer. Is ≥ #K ( ) L/F : finite extension,L F { m OL tors | ⊂ } bounded? ) 7 Remark 1.3.6. When m is even, Km( L)isfinite(cf.Borel[2]),andthus we have K ( ) = K ( ). O m OL tors m OL Remark 1.3.7. (1) The case when m =0istheproblemonidealclass groups, which is posed by John Coates. This case is quite hard.

(2) The case when m =1istrue.InfactK ( )isknowntobeisomorphic 1 OL to × (cf. [1]). Hence K ( ) = µ for all intermediate fields L of OL 1 OL tors 2 F/F.

In) this thesis, we study the case when m 2(mod4)andF = Q.We prove that the problem is false in this case. We≡ now state our result more precisely. Let p be a prime number. Let B /Q be the cyclotomic Zp-extension and ∞ Bn the n-th layer. Theorem 1.3.8 (Theorem 3.3.3). Let m be an even number 2 and h = ≥ n #K2m 2( Bn ). Then hn n 0 is unbounded. − O { | ≥ } Furthermore, there are infinitely many prime numbers which divide hn for some n Z 0, i.e. ∈ ≥

# ℓ : prime number there exists n Z>0 such that ℓ hn = . { | ∈ | } ∞ Remark 1.3.9. This theorem shows that the behavior of the K-groups

K2m 2( Bn )iscompletelydifferentfromtheconjecturalbehavioroftheideal − O class groups Cl(Bn).

1.4 Outline

The outline of this thesis is as follows. In Chapter 2, we review some properties of Iwasawa algebras and Iwasawa modules. In Chapter 3, we prove our main theorem on the orders of even K-groups K2m 2( Bn ). In − Chapter 4, we study the Λ-module structures of the plus and theO minus Selmer groups Sel±(F ,E[p∞])∨ and the plus and the minus local conditions ∞ (E±(k ) Qp/Zp)∨.Weprovethattheclassicalp-primary Selmer group ∞ ⊗ Sel(F ,E[p∞])∨ has no nontrivial finite Λ-submodules under our setting. ∞ Finally we prove our main theorem on the non-existence of nontrivial finite Λ-submodules of the plus and the minus Selmer groups Sel±(F ,E[p∞])∨. ∞

8 Chapter 2

Iwasawa modules

In this chapter, we review some properties of Iwasawa algebras and Iwasawa modules. Let p be a prime number, Γa topological group which is isomorphic to the additive group Zp. Definition 2.0.1 (Completed group ring). The completed group ring of Γ over Zp is the topological inverse limit

Zp[[Γ]] = lim Zp[Γ/Γn], ←−n where n runs through positive integers, and Γn the unique subgroup of Γof index pn.

Proposition 2.0.2 (Serre [29]). Let γ be a topological generator of Γ( Zp). There is a non-canonical isomorphism ≃

Zp[[X]] ∼ Zp[[Γ]], −→ X γ 1, 1−→ − of topological rings.

By this proposition, we identify the completed group ring Zp[[Γ]] with the formal power series ring Zp[[X]] using any fixed generator γ,anddenotethe ring by Λ.

Definition 2.0.3 (Distinguished polynomial). Apolynomialf(X) Zp[X] is said to be a distinguished polynomial if it is of the form ∈

s s 1 f(X)=X + as 1X − + + a1X + a0 − ··· with coefficients a0,...,as 1 contained in the maximal ideal (p)ofZp. −

9 Theorem 2.0.4 (Structure theorem for Λ-modules). Let M be a finitely generated Λ-module. Then there exist irreducible distinguished polynomials fj Zp[X], nonnegative integers r, s, t, positive integers mi,nj, and a homo- morphism∈

s t r m nj M Λ⊕ Λ/p i Λ/f −→ ⊕ ⊕ j i=1 j=1 ! ! with finite kernel and cokernel. The integers r, mi,nj and the prime ideals (fj) of Λ are uniquely determined by M. This theorem guarantees the following quantities are well-defined.

Definition 2.0.5 (Λ-rank, Iwasawa invariants, characteristic ideal). With the notation of Theorem 2.0.4, we call

rankΛ(M)=r the Λ-rank of M, s

µ(M)= mi the Iwasawa µ-invariant of M, i=1 *t λ(M)= nj deg(fj) the Iwasawa λ-invariant of M, j=1 * t Char (M)= f nj ( Λ) the characteristic ideal of M. Λ j ⊂ "j=1 $ # Proposition 2.0.6. Let

0 M M M 0 −→ 1 −→ 2 −→ 3 −→ be an exact sequence of Λ-modules. Then we have

rankΛ(M2)=rankΛ(M1)+rankΛ(M3).

Proof. Let Q(Λ) be the quotient field of Λ. We have

rank (M )=dim (M Q(Λ)) Λ i Q(Λ) i ⊗Λ for i =1, 2, 3, and the sequence

0 M Q(Λ) M Q(Λ) M Q(Λ) 0 −→ 1 ⊗Λ −→ 2 ⊗Λ −→ 3 ⊗Λ −→ is exact. Thus we get the conclusion.

10 We denote MΓn the module of Γn-coinvariants of M:

M = M/I M M/ω (X)M, Γn Γn ≃ n where IΓn is the augmentation ideal

I = σ 1 σ Γ , Γn ⟨ − | ∈ n⟩Λ and ω (X)=(1+X)pn 1. n − Proposition 2.0.7 (Invariant-coinvariant exact sequences). Let

0 M M M 0 −→ 1 −→ 2 −→ 3 −→ be an exact sequence of Λ-modules. Then there are exact sequences of Zp- modules

0 M Γn M Γn M Γn −→ 1 −→ 2 −→ 3 (M ) (M ) (M ) 0 −→ 1 Γn −→ 2 Γn −→ 3 Γn −→ for all n 0. ≥

Proof. Since all Γn are pro-cyclic, we can take topological generators γn of Γn for all n,andwehaveexactsequences

γn 1 0 M Γn M − M (M ) 0 −→ i −→ i −→ i −→ i Γn −→ for i =1, 2, 3. The result follows from the snake lemma.

Proposition 2.0.8 (Topological Nakayama’s lemma). Let M be a Λ-module.

(1) The following three conditions are equivalent.

(i) M is a finitely generated Λ-module.

(ii) MΓ is a finitely generated Zp-module.

(iii) M/(p, X)M is a finite-dimensional Fp-vector space. (2) If M is a finitely generated Λ-module, then

(p, X)M = M M =0. ⇐⇒

11 Chapter 3

K-groups of rings of integers

In this chapter we study the orders hn of even K-groups K2m 2( Bn ). In − 3.1, we review the definition of higher K-groups after Daniel Quillen.O 3.2 is§ a preparation for 3.3. In particular, we estimate the absolute values§ of the generalized Bernoulli§ numbers for each valuation. In 3.3, we give an § upper bound on the p-exponent of the quotient hn/hn 1. Finally we prove − the main theorem (Theorem 3.3.3).

3.1 K-groups of rings

In this section, we review Quillen’s Q-construction for exact categories, and define higher K-groups for rings as a special case. Main references of this section are [25, 26, 33, 40].

Definition 3.1.1 (Small, essentially small categories). Acategory is said to be small if the class of objects of forms a set. A Acategory is said to be essentiallyA small if it is equivalent to a small category. A

Example 3.1.2. Let R be a ring with unit. The category (R)offinitely generated projective R-modules is essentially small. IndeedP (R)hasaset of isomorphism classes of objects. P

For a nonnegative integer n,denoten the category with objects 0, 1,...,n , with exactly one morphism i j for each i j.Denote∆ the following{ cat- } → ≤ egory; the objects are n for all n Z 0,andthemorphismsareallfunctors ≥ m n. ∈ →Asimplicialsetisacontravariantfunctor∆ Sets. →

12 Definition 3.1.3 (Geometric realization of a simplicial set). Let : ∆ Sets be a simplicial set. The geometric realization of is definedF to be→ the quotient space |F| F

= ( (n) ∆ ) / , |F| F × n ∼ "n 0 $ +≥ n+1 where ∆n = (t0,...,tn) R ti 0, ti =1 ,and (n)isregarded as a discrete{ space for each∈ n 0.| The≥ equivalence} relationF is defined as ≥ , ∼ follows: for each f Hom∆(m, n), let f Cont(∆m, ∆n)betheinduced continuous map. Then∈ for any x ∆ and∈y (n), we set ∈ m ∈F % ( (f)(y),x) (y, f(x)), F ∼ where ( (f)y, x) (m) ∆m,and(y, f%(x)) (n) ∆n.Let be the equivalenceF relation∈F so generated,× and the∈ quotientF × space, with∼ the |F| quotient topology. %

Definition 3.1.4 (The nerve of a small category). The nerve N of a small category is the simplicial set ∆ Sets defined by the followingA data. Its n-simplicesA are diagrams in of the→ form A A A A . 0 −→ 1 −→ ···−→ n Definition 3.1.5 (Classifying space of a small category). The classifying space of a small category is defined to be the geometric realization of N and is denoted by B ; A A A B = N . A | A| Definition 3.1.6 (Exact categories). An exact category is an additive category embedded as a full subcategory of an abelian categoryA ,such A B that is closed under extension in i.e. if 0 A′ A A′′ 0isan A B → → → → exact sequence in with A′, A′′ ,thenA is isomorphic to an object of . A ∈B B An exact sequence in an exact category is defined to be an exact sequence in whose term lies in . A B A An epimorphism q : A A′′ in is said to be admissible if there → q A is an exact sequence 0 A′ A A′′ 0in .Amonomorphism → → → → A i : A′ A in is said to be admissible if there is an exact sequence →i A 0 A′ A A′′ 0in . → → → → A

13 Remark 3.1.7. The admissibility of a morphism is closed under composi- tion in an exact category, i.e. the composition of admissible epimorphisms (resp. admissible monomorphisms) is again an admissible epimorphism (resp. admissible monomorphism)

Example 3.1.8. The category (R)isanexactcategory.Indeed (R)is afullsubcategoryoftheabeliancategoryP (R)offinitelygeneratedP R- modules. M

Definition 3.1.9 (Q-construction for an exact category). Let be an exact category. Define a category Q as follows: Q has the sameA objects as . AmorphismfromA to B in QA is an equivalenceA class of diagrams A A q i A o o B′ / / B, where i is an admissible monomorphism and q is an admissible epimorphism q i q i in .TwosuchdiagramsA B′ B and A ′ B′′ ′ B are equivalent A ! " ! " if there is an isomorphism B′ B′′ such that the diagram → q i A o o B′ / / B ≃

q′  i′ A o o B′′ / / B commutes. The composition of two morphisms A ! B′ " B and B ! C′ " C is the equivalence class of A ! B′ B C′ " C,whichisdefinedby the diagram × A O

B′ / / B O O q′′

i′′ B′ B C′ / / C / / C. × ′

Remark 3.1.10. The morphism q′′ (resp. i′′)intheabovelastdiagramis certainly an admissible epimorphism (resp. an admissible monomorphism). Indeed, we have Ker(q′′) = Ker(C′ B) ,andthusB′ C′ since ∼ # ∈A ×B ∈A is closed under extension. Therefore q′′ is an admissible epimorphism. The A rest of our claim follows from Coker(i′′) ∼= Coker(B′ " B). We can also check that the equivalence class of the above diagram depends only on the equivalence classes of A ! B′ " B and B ! C′ " C.

14 Definition 3.1.11 (Higher K-groups of an essentially small exact category). For a small exact category ,wedefinethei-th K-group of by the i +1st homotopy group of BQ ; A A A K ( ):=π (BQ ) i A i+1 A for i 0. For≥ an exact category which has a set of isomorphism classes of objects, we define A

K ( ):=K ( ′) i A i A for i 0, where ′ is a small subcategory equivalent to . ≥ A A Remark 3.1.12. The above definition is independent of the choice of ′. A Indeed, if two exact categories and ′ are equivalent then Q and Q ′ are A A A A equivalent. Therefore, if both and ′ are small categories, then K ( ) = A A i A ∼ K ( ′)foralli. i A Definition 3.1.13 (Higher K-groups of rings). For a ring R,wedenote (R) the exact category of finitely generated projective R-modules, and defineP the i-th K-group Ki(R)by K (R):=π (BQ (R)) i i+1 P for i 0. ≥ Example 3.1.14 (Quillen [25]). Consider a finite field Fq with q elements. We have Z if i =0, Ki(Fq)=⎧ 0 if i :even> 0, ⎨⎪ Z/(qi 1)Z if i :odd. − ⎪ 3.2 Lemmas ⎩

For a totally real field L, wm(L)denotesthelargestintegern so that the Galois group of L(µn)/L is killed by m.

Lemma 3.2.1. Let p be a prime number and Bn the n-th layer of the cyclo- tomic Zp-extension of Q. For each even integer m 2, we have ≥ 2pn ℓ1+vℓ(m) if p 1 m, − | ℓ 1 m wm(Bn)=⎧ −#| ⎪ 2 ℓ1+vℓ(m) otherwise. ⎨⎪ ℓ 1 m −#| ⎪ ⎩⎪ 15 Proof. It suffices to consider the Galois groups Gal(Bn(µℓt )/Bn)foreach prime ℓ. (i) The case when ℓ = p and ℓ =2.Foranyintegert 1, we have ̸ ̸ ≥ t 1 Gal(Bn(µℓt )/Bn) = (Z/ℓZ)× Z/ℓ − Z. ∼ × Thus we have

1+vℓ(m)ifℓ 1 m, vℓ (wm(Bn)) = − | ( 0 otherwise.

(ii) The case when ℓ = p and ℓ =2.Foranyintegert 2, we have ̸ ≥ t 2 Gal(Bn(µ2t )/Bn) = (Z/4Z)× Z/2 − Z. ∼ × Thus we have

v2 (wm(Bn)) = 2 + v2(m). (iii) The case when ℓ = p =2.Foranyintegert 1, we have ̸ ≥ t 1 Gal(Bn(µpn+t )/Bn) = (Z/pZ)× Z/p − Z. ∼ × Thus we have

n +1+vp(m)ifp 1 m, vp (wm(Bn)) = − | ( 0 otherwise.

(iv) The case when ℓ = p =2.Foranyintegert 2, we have ≥ t 2 Gal(Bn(µ2n+t )/Bn) = (Z/4Z)× Z/2 − Z. ∼ × Thus we have

v2 (wm(B2,n)) = n +2+v2(m).

Let ψn be a generator of the group of Dirichlet characters associated to n+1 n+2 Bn.Letfn be the conductor of ψn,i.e.fn = p if p 3, and fn =2 if p =2.Forallintegersm 1weknowthat ≥ ≥ B ζ (1 m)= L(1 m, ψ)= m,ψ , (3.2.1) Bn − − − m ψ ψn ψ ψn 1 2 ∈⟨# ⟩ ∈⟨# ⟩

16 where L(s, ψ)istheDirichletL-functionofψ and Bm,ψ is the mth generalized Bernoulli number attached to ψ.TheseBernoullinumbersaredefinedby

fψ ψ(a)XeaX ∞ Xm = Bm,ψ Q[[X]], efψX 1 m! ∈ a=1 m=0 * − * where fψ is the conductor of ψ.WehavethefunctionalequationonDirichlet L-functions s π(s δψ) τ(ψ) 2π Γ(s)cos − L(s, ψ)= δ L(1 s, ψ), 2 2√ 1 ψ fψ − 1 2 − 1 2 where Γ(s)istheGammafunction,ψ is a nontrivial Dirichlet character of f ψ 2π√ 1a/fψ conductor fψ, τ(ψ)= a=1 ψ(a)e − is the Gauss sum,

, 0ifψ is an even character, δψ = ( 1ifψ is an odd character.

Now we give an evaluation on the complex absolute value of Bm,ψn /m. Lemma 3.2.2. Let m be an even integer 2. For all n 1 we have ≥ ≥

2 (m 1)! m 1 Bm,ψn 10 (m 1)! m 1 − f − f < < − f − f . 3 (2π)m n n m 3 (2π)m n n 4 4 3 4 4 3 4 4 Proof. By the functional equation4 on Dirichlet4 L-functions, we have B m,ψn = L(1 m, ψ ) m | − n | 4 4 4 4 (m 1)! m 1 4 4 =2 − f − f L m, ψ . 4 4 (2π)m n n n 3 4 & '4 1 4 5 4 Thus it is enough to prove that 3 < L(m, ψn) < 3 .Thisfollowsfromthe following evaluation; | |

∞ ψ (a) L(m, ψ ) 1 = n n − am 4a=2 4 4* 4 4 4 4 42 4 4 4∞ 1 π4 2 4 = 4 1 < . ≤ a2 6 − 3 a=2 *

17 1 Next we give an evaluation on the ℓ-adic valuation of 2 Bm,ψn /m.Weuse acongruenceonℓ-adic L-functions, which follows from the Taylor expansion of Lℓ(s, χ); for a nontrivial Dirichlet character χ whose conductor fχ is not divisible by ℓ2 if ℓ 3, by 23 if ℓ =2,wehave ≥ 1 1 L (1 m, χ) L (1 m′,χ)(modℓ) 2 ℓ − ≡ 2 ℓ − for all integers m, m′ 1andbothsidesareℓ-integral ([37, Corollary 5.13]). ≥ Lemma 3.2.3. Let m be an even integer 2. For each prime number ℓ = p, ≥ ̸ 1 B v m,ψn =0 for large enough n>0. ℓ 2 m 1 2 Proof. We know that

m m 1 Bm,ψn L (1 m, ψ ω )= (1 ψ (ℓ)ℓ − ) , ℓ − n ℓ − − n m m m 1 Lℓ(0,ψnωℓ )= (1 ψnωℓ − (ℓ))B1,ψ ωm 1 . − − n ℓ − (i) the case when p =2. 2 n+2 Since ℓ =2,ℓ does not divide f m =2 ℓ which is the conductor of ψnωℓ m ̸ ψnωℓ . Hence we have a congruence on ℓ-adic L-functions 1 1 L (1 m, ψ ωm) L (0,ψ ωm)(modℓ). 2 ℓ − n ℓ ≡ 2 ℓ n ℓ Here we note that ℓ 1doesnotdividem 1. In fact, ℓ 1isaneveninteger, − − − m 1 on the other hand, m 1isanoddinteger.Thismeansthatψnωℓ − (ℓ)=0. Thus we have −

1 Bm,ψn 1 B1,ψ ωm 1 (mod ℓ). 2 m ≡ 2 n ℓ − Therefore

1 Bm,ψn 1 vℓ =0 vℓ B1,ψ ωm 1 =0. 2 m ⇐⇒ 2 n ℓ − 1 2 1 2 1 By Washington’s result [39], we have vℓ( B m 1 )=0forn large enough, 2 1,ψnωℓ − and hence we get the conclusion.

(ii) the case when p 3. ≥

18 2 n+1 For an odd prime ℓ, ℓ does not divide f m = ℓp ,andfortheprime ψnωℓ 3 m n+1 2(=ℓ), 2 does not divide fψnω2 = fψn = p since m is an even integer. Hence we have a congruence on ℓ-adic L-functions 1 1 L (1 m, ψ ωm) L (0,ψ ωm)(modℓ). 2 ℓ − n ℓ ≡ 2 ℓ n ℓ Here we note that ℓ 1doesnotdividem 1asinthecase(i)ifℓ is an odd − m 1 − m 1 prime. If ℓ =2,thenω2 − = ω2.Inanycase,wehaveψnωℓ − (ℓ)=0.Thus we have

1 Bm,ψn 1 B1,ψ ωm 1 (mod ℓ). 2 m ≡ 2 n ℓ − Therefore

1 Bm,ψn 1 vℓ =0 vℓ B1,ψ ωm 1 =0. 2 m ⇐⇒ 2 n ℓ − 1 2 1 2 1 By Washington’s result [39], we have vℓ( B m 1 )=0forn large enough, 2 1,ψnωℓ − and we get the conclusion.

1 Remark 3.2.4. The above result on vℓ( 2 Bm,ψn /m)canalsobeprovedby using gamma transforms of appropriate rational function measures in [30]. Indeed, let m be an even integer 2, g any integer and put ≥ m 1 g a d − a=1 X Rm(X)= X Qℓ(X), dX 1 Xg ∈ 1 2 1,− 2 where the right-hand side is independent of the choice of g.Theintegerg will be taken sufficiently large in the proof for technical reason. Let αm be a Qℓ-valued rational function measure on Zp whose associated rational function is Rm.TheΓ-transformofαm is the function of characters ψ on Zp× of the second kind for p defined by

Γαm (ψ)= ψ(x)dαm(x). 5Zp× By the same method as in the proof of [30, Proposition 4.1], we have

Γαm (ψ)=L p (1 m, ψ) { } − for all but finitely many Dirichlet characters ψ of the second kind for p.Then [30, Remark after Theorem 3.1] implies that 1 1 vℓ L(1 m, ψ) = vℓ L p (1 m, ψ) =0 2 − 2 { } − 1 2 1 2

19 for all but finitely many ψ of the second kind for p. Here the first equality holds since ψ(p)=0forallbutfinitelymanyψ and thus the Euler factor 1+m 1 ψ(p)p− =1. −

Lemma 3.2.5. For n>0 large enough, we have v2(Bm,ψn /m) < 1.

Proof. At first, we show that for any odd integers a, b Z, ∈ ψ (b)=ψ (a) b a, 2n+2 a (mod 2n+2). n n ⇐⇒ ≡ − n+2 n+2 Assume that ψn(b)=ψn(a)andb /a(mod 2 ). As a, b (Z/2 Z)× = ≡ ∈ 1, 5 ,wecanwritethemuniquelyasa ( 1)xa 5ya , b ( 1)xb 5yb (mod ⟨−n+2 ⟩ ≡ − ≡ − 2 )forsomeintegersxa,xb,ya,yb satisfying 0 xa,xb 1, 0 ya,yb n yb y≤a ≤ ≤ ≤ 2 1. Then we have ζ n = ψ (b)=ψ (a)=ζ n and thus we get y = y . − 2 n n 2 b a By our assumption, we may assume xa =0andxb =1.Thereforeweget b 5yb a (mod 2n+2). ≡− ≡− Next, we show that for any odd integers a, b Z, ∈ ψ (b)= ψ (a) b 2n+1 + a, 2n+1 a (mod 2n+2). n − n ⇐⇒ ≡ −

Assume that ψn(b)= ψn(a). Let xa,xb,ya,yb be integers as above. Then n 1 n 1 − n xa xb 2 − we see yb 2 − + ya (mod 2 ). Thus we have b ( 1) − 5 a (mod ≡ n 1 ≡ − 2n+2). Note that we have 52 − 2n+1 +1 (mod2n+2). Since a is an n+1 ≡ n+1 n+2 odd integer, we get b 2 a + a 2 + a (mod 2 )ifxa = xb,and n+1 n+1≡ n≡+2 b 2 a a 2 a (mod 2 )ifxa = xb. ≡−Put x =2−n+1≡.Thenwehave− ̸

n fn 2 aiψ (a)= ai (x a)i (x + a)i +(2x a)i ψ (a). n − − − − n a=1 a=1 * * & ' Note that ψ (a) 1 a 2n,a :odd is a basis of the ring of integers { n | ≤ ≤ }

20 Z[ζ2n ]. For each even integer i>0, we have ai (x a)i (x + a)i +(2x a)i − − − − i i i i j i j i j i j = a x ( a) − x a − − i j − − i j j=0 j=0 * 1 − 2 * 1 − 2 i i j i j + (2x) ( a) − i j − j=0 * 1 − 2 i i i j i j i j i j = x ( a) − x a − − i j − − i j " j=1 j=1 $ * 1 − 2 * 1 − 2 i i j i j + (2x) ( a) − i j − j=1 * 1 − 2 i i j i j = 2 x a − − i j j=1 1 − 2 j*:even i i i j i j i j i j + (2x) a − (2x) a − i j − i j j=1 1 − 2 j=1 1 − 2 j*:even j*:odd

i i i j j i j i j i j i 1 = 2+2 x a − (2x) a − i 2x a − i j − − i j − · · j=2 1 − 2 j=2 1 − 2 j*:even & ' j*:odd

i i 2 i j 1 j 2 i j j 2 i i j =2x ⎛ ( 1+2 − )x − a − 2(2x) − a − ⎞ i j − − i j j=2 1 − 2 j=2 1 − 2 ⎜j*:even j*:odd ⎟ ⎜ ⎟ ⎝ ⎠ i 1 2xia − . − Thus we have 2n (ai (x a)i (x + a)i +(2x a)i)ψ (a) − − − − n a=1 * 2n i 1 n+3+v2(i) ( 2xia − )ψ (a) / 0(mod2 ) ≡ − n ≡ a=1 * 21 for every n such that n +1=v2(x) 1+v2(i). Finally, we claim that for n large≥ enough we have

f 1 n v (B )=v amψ (a) < 1+v (m) (3.2.2) 2 m,ψn 2 f m 2 " n a=1 $ * and thus v (B /m) < 1. In fact, for i =2, 4,...,m 2wehave 2 m,ψn − fn m i m i v2 fn − Bm i a ψn(a) 2(n +2)+0 1+(n +2+v2(i)) i − ≥ − " a=1 $ 1 2 * =3n +5+v2(i). For i = m 1, we note that − m 1 fn − m 1 j m 1 j 0=fnBm 1,ψn = fn − − − Bm 1 j a ψn(a) − j − − j=1 " a=1 $ * 1 2 * since ψn is an even character. Thus we have

fn m m 1 v f B a − ψ (a) 2 n m 1 1 n " a=1 $ 1 − 2 * m 2 fn − m m 1 j m 1 j = v2 fn B1 fn − − − Bm 1 j a ψn(a) m 1 − j − − " " j=1 " a=1 $$$ 1 − 2 * 1 2 * (n +2)+1 1+((n +2)+0 1+0) ≥ − − =2n +3. For i = m,wehave

fn m v2 a ψn(a)

3.3 The orders of even K-groups

Let ψn be as in the previous section. Let hn =#K2m 2( Bn ). The work of − Voevodsky and Rost on the Bloch–Kato Conjecture in [36]O implies that there are isomorphisms up to finite 2-torsion:

2 K2m 2( Bn ) = Hmot( Bn , Z(m)), − O ∼ O

22 where the right hand side denotes the motivic cohomology of Spec( Bn ). The deviation between K-theory and motivic cohomology depends onOm mod 8 and is known by work of Rognes–Weibel (cf. [28]). Since Bn is totally real, the Iwasawa Main Conjecture, which was proved by Wiles [41] for odd p and for p =2incaseofanabelianfield,impliesthatforanyevennumberm 2 ≥ 2 #Hmot( Bn , Z(m)) ζBn (1 m)= O . − ± wm(Bn)

bn Thus we have hn = 2 wm(Bn)ζBn (1 m)forsomeintegerbn.ByLemma 3.2.1 and (3.2.1), we± have −

hn Bm,ψσ =2bn′ pδp n , (3.3.1) hn 1 × × − m − σ Gal(Q(ζpn )/Q) 4 4 ∈ # 4 4 4 4 where bn′ = bn bn 1, 4 4 − − 1ifp 1 m, δp = − | ( 0otherwise. We define c = v ( hn ). n p hn 1 − Now we give an asymptotic formula for cn by Iwasawa theory when p is odd. Let A be the p-Sylow subgroup of the ideal class group of the Q(µpn+1 ) n+1 CM field Q(µp )and =limA (µ n+1 ).Letω be the Teichm¨uller character X Q p for ∆= Gal(Q(µp)/Q), i.e.←− for a Zp pZp, ω(a)isdefinedtobea(p 1)th ∈ \ i − root of unity such that a ω(a)(modp). Let ω be the submodule of on which each σ ∆actsby≡ ωi(σ). X X ∈ Proposition 3.3.1. If p is odd, then we have cn = λ for n large enough, 1 m where λ is the Iwasawa λ-invariant λ( ω − ). X Proof. We show that there exist constants λ and ν so that vp(hn)=λn + ν for n large enough. Let − be the submodule of on which complex conjugation acts by 1. TheX Quillen–Lichtenbaum Conjecture,X which now is a theorem due to − Voevodsky and Rost [36], implies that K2m 2( Bn ) Zp is isomorphic to 2 − O ⊗ H (Spec( n [1/p]), Zp(m)) for each n (cf. [31]), and the ´etale cohomology is ´e t OB isomorphic to the Pontryagin dual of the Galois coinvariant of − Zp(m 1) (cf. [3], [20]). Therefore we have isomorphisms of groups X ⊗ −

K2m 2( Bn ) Zp = ( − Zp(m 1))Gal(B (µp)/Bn) − O ⊗ ∼ X ⊗ − ∞ 1 m ω − and we know that ( − Zp(m 1))Gal(B (µp)/Bn) =( )Gal(B /Bn). Note X ⊗ − ∞ X ∞ that the Iwasawa µ-invariant µ(Q(µp)) is 0. Thus we get the conclusion from the Iwasawa theory on ideal class groups.

23 Next we will give an upper bound on cn when p =2.Bydefinition,we have

m fn 1 m i m i Bm,ψn = fn − Bm i a ψn(a) , f · i · − n i=1 " a=1 $ * 1 2 * where Bm i are the Bernoulli numbers. − n Proposition 3.3.2. cn

hn n Z 0 is unbounded. { | ∈ ≥ } Furthermore, for any even number m 2 and each prime p, there are ≥ infinitely many prime numbers which divide hn for some n Z 0, i.e. ∈ ≥ # ℓ : prime number there exists n Z>0 such that ℓ hn = . { | ∈ | } ∞ Proof. To prove the former statement, we show that the quotient hn/hn 1 − tends to infinity as n . By (3.3.1) and Lemma→∞ 3.2.2, we have

hn Bm,ψσ =2bn′ pδp n hn 1 × × − m − σ Gal(Q(ζpn )/Q) 4 4 ∈ # 4 4 4 φ(pn)4 b δp 2 (m 1)! m 1 4 4 2 n′ p − f − f (n ), ≥ 3 (2π)m n n −→ ∞ →∞ 1 2 3 where δ =1ifp 1 m and otherwise δ =0. p − | p We show the latter statement, namely the existence of infinitely many prime numbers which divide hn for some n. We first consider the case when p>2. Assume that there are only finitely many prime numbers which divide hn for some n,i.e.

ℓ :primenumber there exists n Z>0 such that ℓ hn = ℓ1,...,ℓr { | ∈ | } { } for some integer r.Foreachℓ =2,p,thereexistsanintegerN(ℓ )suchthat i ̸ i ℓi does not divide hn/hn 1 for every n>N(ℓi) by Lemma 3.2.3. Hence by − 24 the finiteness of ℓi’s, there exists an integer N0 such that ℓi except for 2 and p does not divide hn/hn 1 for every n>N0 and for every i.Fortheprime2, − n there exists an integer N1 such that v2(hn/hn 1)=bn′ + φ(p )v2(Bm,ψn /m)= n − bn′ + φ(p )foreveryn>N1 by Lemma 3.2.3. Furthermore, we know that there exist a constant λ and an integer N2 such that cn = vp(hn/hn 1)=λ − for every n>N2 by Proposition 3.3.1. Thus we have

hn n n =2bn′ 2φ(p )pcn =2bn′ 2φ(p )pλ hn 1 − for every n>max(N0,N1,N2). On the other hand we have

φ(pn) hn b δp 2 (m 1)! m 1 n′ − 2 p − m fn fn hn 1 ≥ 3 (2π) − 1 n 2 bn′ (m 1)φ(p ) 3 > 2 fn −

n+1 for n large enough as in the proof of the former statement. Since fn = p , we have

φ(pn) (m 1)(n+1)φ(pn) λ 2 >p − − for n large enough. This is a contradiction.

Next we study the case when p =2.Bythesamemethodasabove,we see that hn/hn 1 is a power of 2 for n large enough. Thus, we have −

hn n =2cn < 2bn′ 21+φ(2 ) hn 1 · − for n large enough by Proposition 3.3.2. On the other hand we have

φ(2n) hn b 2 (m 1)! m 1 n′ − 2 2 − m fn fn hn 1 ≥ · 3 (2π) − 1 2 3 b (m 1)φ(2n) > 2 n′ 2 f − · · n n+2 for n large enough as in the proof of the former statement. Since fn =2 , we have

1+φ(2n) 1+(m 1)(n+2)φ(2n) 2 > 2 − for n large enough. This is a contradiction.

25 Chapter 4

The plus and the minus Selmer groups

In this chapter, we study the Λ-module structure of the plus and the minus Selmer groups Sel±(F ,E[p∞])∨.In 4.1, we define the plus and the minus ∞ Selmer groups following Shin-ichi Kobayashi,§ and fix a global setting. In 4.2, we study the Λ-module structure of the plus and the minus local con- § ditions (E±(k ) Qp/Zp)∨ in a local setting. In 4.3, we first prove that ∞ ⊗ § the classical p-primary Selmer group Sel(F ,E[p∞])∨ has no nontrivial finite ∞ Λ-submodules (Theorem 4.3.5) under our setting. Finally, we prove that the plus and the minus Selmer groups Sel±(F ,E[p∞])∨ also have no nontrivial ∞ finite Λ-submodules (Theorem 4.3.8).

4.1 Definitions

Let p be a prime number, F afiniteextensionofQ,andE an elliptic curve defined over F .ForafiniteextensionL/F ,thep-primary Selmer group for E over L is defined by

1 1 H (Lv,E[p∞]) Sel(L, E[p∞]) := Ker H (L, E[p∞]) , −→ E(L ) / " v v Qp Zp $ # ⊗ where v runs through all the places of L, Lv is the completion of L at the 1 place v,andE(Lv) Qp/Zp is regarded as a subgroup of H (Lv,E[p∞]) by the Kummer map. For⊗ a number field L which is an infinite extension of F , we define the p-primary Selmer group for E over L by

Sel(L, E[p∞]) := lim Sel(L′,E[p∞]), −→L′

26 where L′ runs through all the subfields of L that are finite extensions of F ,andthetransitionmapsarerestrictionmapsbetweenthecohomology groups.

We denote Fn = F (µpn+1 ), F 1 = F and F = n Fn,whereµpn denotes n − ∞ the group of p th roots of unity. We fix a generator (ζpn )ofZp(1), namely, n < p for each n 0, ζpn is a primitive p th root of unity such that ζpn+1 = ζpn . Then by≥ definition, we have

Sel(F ,E[p∞]) = lim Sel(Fn,E[p∞]). ∞ −→n Throughout this thesis, we fix the following notations:

p is an odd prime number, • F is a finite extension of Q, •

E is an elliptic curve defined over a subfield F ′ of F . • ss Denote Sp the set of all primes of F ′ lying above p where E has super- singular reduction. Throughout this thesis, we assume the following:

any prime w Sss is unramified in F , • ∈ p ss Fw′ = Qp for any prime w Sp ,whereFw′ is the completion of F ′ at • the prime w, ∈

E has good reduction at any prime w p of F ′, • | Sss is nonempty, and • p ss aw =1+NF / (w) #Ew(Fw)=0foranyprimew Sp ,whereNF / • ′ Q − ∈ ′ Q is the norm for F ′/Q, Ew is the reduction of E at w,andFw is the residue field of F ′ at w.% % When p 5, the condition a =0isautomaticallysatisfiedsincewehave ≥ w p aw and aw 2√p. | | |≤ ss ss Denote Sp,F the set of all primes of F lying above Sp . + Following Shin-ichi Kobayashi [17] we define subgroups E (Fn,v)and ss E−(F )ofE(F )foreachprimev S ,anddefineplusandminus n,v n,v ∈ p,F Selmer groups Sel±(Fn,E[p∞]), Sel±(F ,E[p∞]) as the following (see also ∞ [11] and [14]).

27 Definition 4.1.1. (1) For a prime v Sss and n 1, let F be the ∈ p,F ≥− n,v completion of Fn at the unique prime of Fn lying above v.Wedefine

+ Trn/m+1 P E(Fm,v) E (Fn,v)= P E(Fn,v) ∈ , ∈ for all even m, 1 m n 1 = 4 − ≤ ≤ − > 4 4 Trn/m+1 P E(Fm,v) E−(Fn,v)= P E(Fn,v) ∈ , ∈ 4 for all odd m, 1 m n 1 = 4 − ≤ ≤ − > 4 4 where Trn/m+1 : E(Fn,v) E(Fm4+1,v)isthetracemap. (2) The plus and the→ minus Selmer groups are defined by

1 H (Fn,v,E[p∞]) Sel±(Fn,E[p∞]) := Ker Sel(Fn,E[p∞]) , ⎛ −→ E (Fn,v) p/ p ⎞ v Sss ± Q Z ∈!p,F ⊗ ⎝ ⎠ Sel±(F ,E[p∞]) := lim Sel±(Fn,E[p∞]). ∞ −→n

We denote the Pontryagin dual of a module M by M ∨.Let = Gn Gal(Fn/F )and =Gal(F /F ). Then the group ring Zp[ n]actsnat- G∞ ∞ G urally on Sel±(Fn,E[p∞])∨ and the completed group ring Λ( ):=Zp[[ ]] G∞ G∞ on Sel±(F ,E[p∞])∨. ∞ The Pontryagin dual of the usual p-primary Selmer group Sel(F ,E[p∞])∨ ∞ is not a torsion Λ( )-module, however, Sel±(F ,E[p∞])∨ are known to be G∞ ∞ Λ( )-torsion in the case when F = Q (cf. [17] Theorem 2.2). G∞ In this thesis, we study the Λ-module Sel±(F ,E[p∞])∨ in our setting, ∞ where Λ= Zp[[Gal(F /F0)]] In the following we also assume that ∞

both Sel±(F ,E[p∞])∨ are Λ-torsion. • ∞ 4.2 The formal groups and the norm sub- groups

In this section we study the local conditions of the plus and the minus Selmer groups defined in the previous section. The problem is completely in local setting, and thus we use the following local setting unless otherwise specified. Let E/Qp be an elliptic curve with ap =0,andE the formal group defined over Zp associated with the minimal model of E over Qp.Letk be a finite unramified extension of Qp of degree d =[k : Qp], and) k the ring of integers O of k.Foreachn 1, let k = k(µ n+1 ), and m be the maximal ideal ≥− n p n of kn.Letk = n 1 kn,andm = n 1 mn.LetGn =Gal(kn/Qp), ∞ ≥− ∞ ≥− G =Gal(k /Qp),Γn =Gal(k /kn),Γ=Γ0(= Gal(k /k0)), and ∆= ∞ ∞ < ∞ < ∞ 28 Gal(k(µp)/k)=Gal(k0/k 1). Let ϕ be the Frobenius map of Gal(k/Qp)= ϕ − p G 1 characterized by x x (mod p k). We denote Λ= Zp[[Γ]]. We fix − ≡ O atopologicalgeneratorγ Γ. Then we identify Zp[[Γ]] with Zp[[X]], and ∈ Zp[[G ]] with Zp[G0][[X]] by identifying γ with 1 + X. ∞

4.2.1 Honda’s theory In this subsection we recall Honda’s theory of formal groups. We restrict ourselves to one-dimensional commutative formal groups defined over . Ok Let kϕ[[T ]] (resp. k,ϕ[[T ]]) be the non-commutative power series ring on O ϕ T with the multiplication rule: Tα = α T for α k (resp. α k). We define a -submodule of the (commutative) power∈ series ring∈kO[[X]] by Ok P

∞ i = f(X)= aiX k[[X]] iai k for i 0,f(0) p k . P ( ∈ 4 ∈O ≥ ∈ O ? i=0 4 * 4 i 4 For u = ∞ b T [[T ]] and f ,wedefineanelementu f by i=0 i ∈Ok,ϕ ∈4 P ∗ ∈P , ∞ i i (u f)(X)= b f ϕ (Xp ), ∗ i i=0 * where the supersucript ϕi of f means that we apply ϕi to the coefficients of the power series f.Wecancheckthat is a left k,ϕ[[T ]]-module by the action . P O ∗ Definition 4.2.1. An element u = u(T ) [[T ]] is said to be special if ∈Ok,ϕ u p (mod deg 1). Let u be a special element of k,ϕ[[T ]]. An element f is≡ said to be of Honda type u,iff satisfies the followingO two conditions:∈P

(i) f(X) αX (mod deg 2) for some α ×, ≡ ∈Ok (ii) (u f)(X) 0(modp [[X]]). ∗ ≡ Ok For a one-dimensional formal group F defined over Zp,wedenotethe formal logarithm by logF and the formal exponential by expF .Weremark that log (X) . F ∈P Theorem 4.2.2 (Honda).

(i) Let F be a formal group of dimension 1 defined over . Then there Ok exists a unique Eisenstein polynomial u = u(T ) k[T ] such that log (X) is of the Honda type u. ∈O F ∈P

29 (ii) Let F and G be formal groups defined over k of dimension 1 whose logarithms are of the same Honda type u. ThenO exp log (X) is an G ◦ F element of [[X]] and gives an isomorphism F ∼ G . Ok → (iii) Let u be an Eisenstein polynomial. Then there exist an element of of the Honda type u.Iff is of the Honda type u, there existsP a formal group defined over ∈Pwith the logarithm f. Ok Proof. See [9] Theorem 2, 4, and Proposition 2.6 and 3.5.

4.2.2 The formal groups associated to E

In this subsection we introduce a system of local points (dn)n which is in- trinsically same with Byoung Du Kim’s one, and study E(mn)foralln.The goal of this subsection is to describe the Zp[Gn]-modules E(mn)intermsof the system of local points (dn)n. ) ) Proposition 4.2.3. For any n, E(mn) is Zp-torsion-free.

Proof. We can prove this by the same) method as the proof of [17] Proposition 8.7.

The above proposition implies that the formal logarithm logE(X)induces an injective homomorphism log : E(mn) kn for all n,sincethekernelof E ! the logarithm of a formal group precisely→ consists of the elements of finite order. ! ) For such a one-dimensional formal group F defined over Zp with height 2, the formal logarithm logF induces isomorphisms as in the following propo- sition (cf. The proof of Proposition 2.1 in [19], and Lemma 2.4 in [8]).

Proposition 4.2.4. Let F be a one-dimensional formal group defined over Zp with height 2. For a finite extension L/Qp, denote by mL its maximal ideal. Then the logarithm log : F (m ) L induces isomorphisms F L → j j log : F (m ) ≃ m F L −→ L for all j>v (p)/(p2 1), where v is the normalized valuation of L so that L − L vL(πL)=1for a uniformizer πL of L.

Following [11] and [14] we construct a system of local points (dn)n. Fix a generator ζ of the group of roots of unity in k.Thenζ is a primitive d (p 1)th root of unity, and we have k = Qp(ζ). −

30 m 1 m 2 p p (m) ϕ − ϕ − Let g(X)=(X +ζ) ζ k[X], g (X)=g g g(X)= (X +ζ)pm ζpm for m −1and∈gO(0)(X)=X.Wedefineaformalpowerseries◦ ◦···◦ − ≥ logG (X)by ∞ g(2m)(X) log (X)= ( 1)m . G − pm m=0 * We can check that

(n+1) 2 2 (n+1) (logϕ− )ϕ (Xp )+p logϕ− (X) 0(modp) G G ≡ (n+1) (n+1) ϕ− ϕ− and (logG )′(X) k[[X]] for each n.ThismeansthatlogG (X)is of the Honda type T∈2 +Op for each n. Theorem 4.2.5 (Honda [9] Theorem 5). Let E be an elliptic curve defined 2 over Qp with good reduction at p. Then log (X) is of the Honda type T E − apT + p. ! Proposition 4.2.6. Over the ring of the unramified quadratic extension of Qp, the formal group E is isomorphic to the Lubin-Tate formal group of height 2 with the parameter p. − ) Hence by Theorem 4.2.2 we see that

(i) there is a formal group Gn defined over k whose formal logarithm (n+1) ϕ− O logGn is given by logG for each n,and (ii) the power series exp log is contained in [[X]] and gives an iso- E ◦ Gn Ok morphism Gn E over k for each n. → ! O We fix a generator (ζpn )ofZp(1), namely, for each n 0, ζpn is a primitive ) ≥(n+1) n p ϕ− p th root of unity such that ζpn+1 = ζpn .Letπn = ζ (ζpn+1 1) mn for n 1andπ =0forn< 1. For each n,wecaneasilyshowthat− ∈ ≥− n − (n+1) (m),ϕ− g (πn)=πn m (4.2.1) − for any m 0bydirectcalculation. Put ≥

(n+3) (n+5) (n+7) ε = ζϕ− p ζϕ− p2 + ζϕ− p3 n − −··· ∞ i 1 ϕ (n+1+2i) i = ( 1) − ζ − p m − ∈ k i=1 * for n 1. Since log : G (m ) m is an isomorphism for all n (cf. ≥− Gn n k → k Proposition 4.2.4), there is ϵ G (m )suchthatlog (ϵ )=ε for n 1. n ∈ n k Gn n n ≥− 31 Definition 4.2.7. We define d =exp log (ϵ [+] π ) n E ◦ Gn n Gn n for n 1, where [+] is the addition! of G . ≥− Gn n For n m,wedenotebyTr : E(m ) E(m )thetrace(norm)with ≥ n/m n → m respect to the group-law E(X, Y ). ) ) Proposition 4.2.8. The system of local points (d ) E(m ) satis- ) n n n 1 n fies ∈ ≥− @ (1) Trn/n 1(dn)= dn 2 for each n 1, ) − − − 1 ≥ (2) Tr0/ 1(d0)= (ϕ + ϕ− )d 1. − − − Proof. We prove this by the same method as the proof of Lemma 8.9 in [17].

Since logE is injective (cf. Proposition 4.2.3) and commute with the action of Gn on E(mn), it is enough to show that the relation holds after applying ! logE to both sides of the equality. We have) !

logE(dn)=logGn (ϵn[+]Gn πn) =log (ϵ )+log (π ) ! Gn n Gn n n+1 [ 2 ] m πn 2m = ε + ( 1) − . n − pm m=0 * Here the last equality follows from (4.2.1) and πn =0forn 1. For n 1, we have ≤− ≥ n+1 [ 2 ] (n+1) π ϕ− m n 2m Trn/n 1 log (dn)=pεn ζ p + ( 1) − − E − − pm 1 m=1 − ! n 1 * [ −2 ] m πn 2 2m = εn 2 ( 1) − − − − − − pm m=0 * = log (dn 2). − E − For n =0,wehave !

1 ϕ− Tr0/ 1 log (d0)=(p 1)ε0 ζ p − E − −1 = (ϕ + ϕ− )ε 1 ! − − 1 = (ϕ + ϕ− )log (d 1). − E − !

32 Remark 4.2.9. As long as we define local points as values of certain power 1 series at certain points, the factor ϕ+ϕ− in the condition (2) always appears (cf. [18] Proposition 3.10, (3.3)). Although Byoung Du Kim did not mention 1 explicitly in [11], [12], this factor ϕ + ϕ− was an obstruction. He assumed in [11] and [12] that k = Qp when he considered the plus Selmer groups in 1 order to make the situation simpler, i.e. ϕ + ϕ− =2inZp[G 1]=Zp.In − this thesis, we consider general unramified extension k/Qp,carefullytaking 1 into account this factor ϕ + ϕ− in Zp[G 1]. − 1 Lemma 4.2.10. ϕ + ϕ− is a unit in Zp[G 1] if and only if d / 0 (mod 4). − ≡ 1 2 Proof. First we note that ϕ+ϕ− Zp[G 1]× if and only if 1+ϕ Zp[G 1]×. − − If d is odd, we have ∈ ∈

2d 2 2 2 4 − 2d 2 d 1 (1 + ϕ )(1 ϕ + ϕ +( 1) 2 ϕ − )=1+( 1) − − −··· − − =2.

If d is even, we have

d 2 d 2 2 2 4 − d 2 − (1 + ϕ )(1 ϕ + ϕ +( 1) 2 ϕ − )=1+( 1) 2 − −··· − − 2ifd / 0(mod4), = ≡ ( 0ifd 0(mod4), ≡ d 2 2 4 − d 2 and 1 ϕ + ϕ +( 1) 2 ϕ − =0inZp[G 1]. Since 2 is invertible in − −··· − ̸ − Zp[G 1], we get the conclusion of Lemma 4.2.10. − 1 Remark 4.2.11. What we really proved is the following; (1) ϕ + ϕ− is a 1 unit if d / 0(mod4),(2)ϕ + ϕ− is a zero-divisor if d 0(mod4). ≡ ≡ Moreover, we can easily check the following lemma.

Lemma 4.2.12. We have

1 AnnZp[G 1](ϕ + ϕ− ) − 0 if d / 0(mod 4), = ≡ 2 4 d 2 ( 1 ϕ + ϕ ϕ − Zp[G 1] if d 0(mod 4). ⟨ − −···− ⟩ − ≡ 1 and rankZp AnnZp[G 1](ϕ + ϕ− )=2if d 0 (mod 4). − ≡

To describe the quotient modules E(mn)/E(mn 1)usingthelocalpoints − (dn)n,wepreparethefollowinglemma. ) ) 33 Lemma 4.2.13. We have mn/mn 1 = πn Zp[Gn] for n 0. − ⟨ ⟩ ≥ Proof. It is enough to show that ζ(ζpn+1 1) generates mn/mn 1 as a Zp[Gn]- − module. − τ We first observe the ring of integers of k .LetP = (ζ m 1) Okn n m { p − | τ Gal(k /k) for m 1andP = 1 .Since = [ζ n+1 ], we have ∈ m } ≥ 0 { } Okn Ok p

kn = P0 P1 Pn+1 . O ⟨ ∪ ∪···∪ ⟩Ok n+1 τ x x a a ζ m Thus, for kn ,wecanwrite = 0 + m=1 τ Gal(km/k) m,τ ( p 1) ∈O ∈ τ − with a ,a . With this notation, since each (ζ m 1) already has 0 m,τ ∈Ok , , p − positive valuation, we see that x m if and only if a m . ∈ n 0 ∈ k Take any class in mn/mn 1 with a representative x mn.Writex = n+1 − τ ∈ a a ζ n+1 0 + m=1 τ Gal(km/k) m,τ ( p 1) .Inthissummation,thesummands ∈ τ − a0 and am,τ (ζpm 1) with 1 m n are contained in mn 1.Since k = , , − ≤ ≤ − d 1 O ϕi − Zp[ζ]= ζ Zp[G 1],eachan+1,τ k can be written as an+1,τ = i=0 bτ,iζ ⟨ ⟩ − ∈O with bτ,i Zp.Thereforewehave ∈ , τ x a (ζ n+1 1) ≡ n+1,τ p − τ Gal(kn+1/k) ∈ * d 1 − ϕi τ bτ,iζ (ζpn+1 1) (mod mn 1). ≡ − − τ Gal(kn+1/k) i=0 ∈ * * ϕi τ Here, ζ (ζ n+1 1) with 0 i d 1, τ Gal(k /k)areexactlythe p − ≤ ≤ − ∈ n+1 Galois conjugates of ζ(ζpn+1 1) by Gn = G 1 Gal(kn+1/k). This completes − the proof. − ×

Proposition 4.2.14. For n 0, we have log (E(mn)) mn + kn 1 and the ≥ E ⊆ − formal logarithm logE induces canonical isomorphisms of Zp[Gn]-modules, ! ) ! E(mn)/E(mn 1) ≃ mn/mn 1. − −→ − By these isomorphisms, d is sent to π . In particular, we have ) n ) n

E(mn)/E(mn 1)= dn Zp[Gn]. − ⟨ ⟩ Proof. We prove this by the same method as the proof of Proposition 8.11 ) ) in [17] or Proposition 4.9 in [10]. For the first statement, we only note that by the commutative diagram

log E / k , E(mn) ; n ! ww exp log ww G 1 E = ww − ◦ ∼ wlog ) ww G 1 !  w − G 1(mn) − 34 it is enough to consider logG 1 (= logG )onG 1(mn)insteadoflogE on E(mn). − − Then we can show that logG 1 (G 1(mn)) mn + kn 1 as in [17], [10]. − − ⊆ − ! Since we have log (mn) kn 1 =log (mn 1), the natural map ) E ∩ − E − ! ! E(mn)/E(mn 1) (mn + kn 1)/kn 1 = mn/mn 1 − −→ − − ∼ − is injective. Since) εn ) m 1 and πn 2m kn 2 for m 1, we have ∈ − − ∈ − ≥ n+1 [ 2 ] m πn 2m log (d )=ε + π + ( 1) − E n n n − pm m=1 ! * πn (mod kn 1). ≡ −

Since πn generates mn/mn 1 as a Zp[Gn]-module (cf. Lemma 4.2.13), the − above injection is in fact a bijection and dn generates E(mn)/E(mn 1)asa − Zp[Gn]-module. ) ) Corollary 4.2.15. We have

d 1 Zp[G 1] if n = 1, − − E(mn)= ⟨ ⟩ − ( dn,dn 1 Zp[Gn] if n 0. ⟨ − ⟩ ≥ ) Proof. The case when n = 1followsfromE(m 1) = m 1 (see Proposition − ∼ − 4.2.4) and Nakayama’s lemma.− The case when n 0followseasilyfrom ≥ Proposition 4.2.14 and the trace relations satisfied) by the dn (see Proposition 4.2.8).

Remark 4.2.16. We defined the system of local points (dn)n following By- oung Du Kim [11] and M. Kim [14] in the above. We can take another system of local points instead of (dn)n.Indeed,whatweneedforthefollowingdis- cussion is a system of local points (dn)n which satisfies the following three conditions

1. Trn/n 1(dn)= dn 2 for each n 1(Proposition4.2.8(1)), − − − ≥ 1 2. Tr0/ 1(d0)= (ϕ + ϕ− )d 1 (Proposition 4.2.8 (2)), − − −

3. E(mn)/E(mn 1)= dn Zp[Gn] (Proposition 4.2.14). − ⟨ ⟩ Such a) system) (dn)n obviously admits at least a difference of multiplication by a unit in Zp[G 1]×.Shin-ichiKobayashiconstructedsuchasystemof − local points also in [18] Proof of Proposition 3.12 by using another formal power series ℓ (X)andasystem(ζ n+1 1) instead of log (X)andasystem ϵ p − n G 35 (ϵn[+]Gn πn)n.Inoursetting,theformalpowerseriesℓϵ(X)isdefinedforeach ϵ E(m )by ∈ k ∞ f (2m)(ε X) ) ℓ (X)=ε + ( 1)m ′ k[[X]], ϵ − pm ∈ m=0 * 2 1 p where ε =logE(ϵ) mk, ε′ =(ϕ + p)εp− k, f(X)=(X +1) 1 and f (m)(X)isthem∈-iterated composition of f∈.ByusingthisformalpowerO − series, Kobayashi! defined d for each n 1by ϵ,n ≥− (n+1) ϕ− d =exp ℓ (ζ n+1 1) E(m ). ϵ,n E ◦ ϵ p − ∈ n Then the first and the second! conditions, which are listed above, are satisfied. ) If we take ε m such that m = ε ,thenthethirdconditionisalso ∈ k k ⟨ ⟩Zp[Gn] satisfied. We also note that we can take such a ε m ,sincem is known to ∈ k k be a cyclic Zp[Gn]-module.

4.2.3 The norm subgroups Following Shin-ichi Kobayashi [17] (and M. Kim [14]), we define the n-th + plus subgroup E (mn), the n-th minus subgroup E−(mn)andthen-th norm subgroup C (mn)ofE(mn); ) ) Definition 4.2.17. We define ) + Trn/m+1 P E(mm) E (mn)= P E(mn) ∈ , ∈ for all even m, 1 m n 1 = 4 − ≤ ≤ − > 4 ) ) ) 4 Trn/m+1 P E(mm) E−(mn)= P E(mn) 4 ∈ ∈ for all odd m, 1 m n 1 = 4 − ≤ ≤ − > 4 ) ) ) 4 for n 0. We denote E±(m )=4 n E±(mn). We also define ≥ ∞

Trn/m<+1 P E(mm) C (mn)= P E()mn) ) ∈ ∈ for all m n (mod 2), 1 m n 1 = 4 ≡ − ≤ ≤ − > 4 ) ) 4 for n 0andC (m 1)=E(4m 1). ≥ − − By the following lemma, it is enough to study E (m )insteadofE (k ) ) ± n ± n in our purpose. ) Lemma 4.2.18. The natural maps E±(m ) E±(k ) induce isomorphisms n → n E±(mn) Qp/Zp ≃ E±(kn) Qp/Zp for all n, and thus we have ⊗ → ⊗ ) E±(m ) Qp/Zp ≃ E±(k ) Qp/Zp. ) ∞ ⊗ −→ ∞ ⊗

) 36 Proof. We consider the following commutative diagrams

0 / E(mn) / E(kn) / E(Fk) / 0 O O O ι± ) %n ? ? 0 / E±(mn) / E±(kn) / An± / 0, where E is the reduction) of E modulo p, Fk is the residue field of k and An± is the cokernel of E±(m ) E±(k ). Since E±(m )=E(m ) E±(k ), n → n n n ∩ n we see% that the right vertical arrows ιn± are injective. Thus An± are finite as E(Fk)isfinite.Wealsonotethat) An±[p∞]=0,since) E/Qp has) supersingular reduction. From the above, our claim will follow immediately. % By comparing two definitions, we get the following relations between the plus subgroups (the minus subgroups) and the norm subgroups.

Lemma 4.2.19. We have

C (mn) if n is even, E+(m )= n ⎧ C (mn 1) if n is odd, ⎨ − ) ⎩ C (mn) if n is odd, E−(m )= n ⎧ C (mn 1) if n is even. ⎨ − ) We now describe C (mn)intermsofthesystemoflocalpoints(⎩ dn)n,and thus we get a description of plus and minus subgroups E±(mn). Proposition 4.2.20. (1) For each n 1, the n-th norm subgroup is gen- ≥− ) erated by dn and d 1 as a Zp[Gn]-module; −

C (mn)= dn,d 1 Zp[Gn]. ⟨ − ⟩ (2) For each n 0, we have an exact sequence ≥

0 E(m 1) C (mn) C (mn 1) E(mn) 0, (4.2.2) −→ − −→ ⊕ − −→ −→ where the first) map is diagonal embedding by inclusions,) and the second map is (a, b) a b. 1→ −

37 Proof. We will prove this by the same method as the proof of Proposition 8.12 in [17]. The main difference is the element d 1 in the first statement. − We first show that C (mn) C (mn 1)=E(m 1)forn 0. It is clear ∩ − − ≥ that C (mn) C (mn 1) E(m 1)bydefinition.LetP C (mn) C (mn 1). ∩ − ⊇ − ∈ ∩ − We show that, if P E(mm)forsomem ) 0, then P E(mm 1). In ∈ n≥ m ∈ − the case when m n (mod) 2), we have p − P =Trn/m P E(mm 1) ≡ ∈ − by the definition of C (m)n 1). Therefore for σ Gal(km/km) 1), we have n m σ − ∈ − p − (P P ) = 0. Hence, by Proposition 4.2.3, P E(mm 1). In) the case − ∈ − when m n 1(mod2),ourclaimisshownsimilarlybyconsideringC (m ) ≡ − n instead of C (mn 1)intheaboveargument.Theotherinclusionisclearly) − true. For the moment, let C ′(mn)betheZp[Gn]-submodule of E(mn)generated by dn and d 1.Bythetracerelationsondn,clearlywehaveC (mn) C ′(mn). − ⊇ We now prove )

C (mn)=C ′(mn), C (mn)+C (mn 1)=E(mn) − for n 0, simultaneously by induction. ) In≥ the case when n =0,wehave

C (m0)=E(m0)= d0,d 1 Zp[G0] = C ′(m0), ⟨ − ⟩ C (m0)+C (m 1)=E(m0)+E(m 1)=E(m0) ) − − by Corollary 4.2.15. ) ) ) In the case when n 1, by the induction hypothesis we have ≥

E(mn 1)=C (mn 1)+C (mn 2), C (mn 2)=C ′(mn 2) (4.2.3) − − − − − and by the trace relation we have C ′(mn 2) C ′(mn). Therefore, by Propo- ) − sition 4.2.14 and (4.2.3), we have ⊆

E(mn)= dn Zp[Gn] + E(mn 1) ⟨ ⟩ − =(dn Zp[Gn] + C ′(mn 2)) + C (mn 1) ) ⟨ ⟩ ) − − C ′(mn)+C (mn 1). ⊆ −

In particular, we have C (mn) C ′(mn)+C (mn 1). This implies that ⊆ − C (m )=C ′(m ). Indeed, if P C (m ), then there exist Q C ′(m )and n n ∈ n ∈ n R C (mn 1)suchthatP = Q+R.ThenweseethatR = P Q C (mn) ∈ − − ∈ ∩ C (mn 1)=E(m 1). Note that E(m 1) C ′(mn)sinced 1 C ′(mn). So − − − ⊆ − ∈ we get P = Q + R C ′(m )andthusC (m )=C ′(m ). It is now clear that ∈ n n n C (mn)+C (m)n 1)=C ′(mn)+C )′(mn 1)=E(mn). − −

38 ) Remark 4.2.21. We check here that the norm subgroup C (mn)isnota cyclic Zp[Gn]-module generated by dn if and only if d =[k : Qp] 0(mod4) and n is even. ≡ (1) When n is odd or d / 0(mod4),weseethatd 1 is automatically − contained in d .Thusinthesecasesweseethatthenormsubgroup≡ ⟨ n⟩Zp[Gn] C (mn)isacyclicZp[Gn]-module generated by dn for each n; C (m )= d . n ⟨ n⟩Zp[Gn] Indeed, when n is odd, we have

n+1 2 d 1 =( 1) Tr1/0 Trn 2/n 3 Trn/n 1 dn dn Zp[Gn]. − − ··· − − − ∈⟨ ⟩ When d / 0(mod4)andn is even, we have ≡ n+2 2 1 1 d 1 =( 1) (ϕ + ϕ− )− Tr0/ 1 Trn 2/n 3 Trn/n 1 dn dn Zp[Gn], − − − ··· − − − ∈⟨ ⟩ 1 since ϕ + ϕ− Zp[G 1]× by Lemma 4.2.10. ∈ − (2) When d 0(mod4),d 1 cannot be contained in dn Zp[Gn] for any ≡ − ⟨ ⟩ even n.ThusinthiscasethenormsubgroupC (mn)isnotacyclicZp[Gn]- module generated by d for each even n.Indeed,ifE(m )= d ,then n 0 ⟨ 0⟩Zp[G0] E(m 1)= Tr0/ 1(d0) Zp[G 1] since Tr0/ 1 : E(m0) E(m 1)issurjective. − − − − − ⟨ ⟩ →) 1 Since E(m 1) = Zp[G 1], this means that Zp[G 1]=(ϕ+ϕ− )Zp[G 1], which − ∼ − − − is) impossible by Lemma 4.2.10. ) ) ) Definition 4.2.22. Define dn± by

n+2 n+1 ( 1) 2 dn if n is even, ( 1) 2 dn if n is odd, + d = − d− = − n ⎧ n+1 n n ( 1) 2 dn 1 if n is odd, ( ( 1) 2 dn 1 if n is even. ⎨ − − − −

Remark⎩ 4.2.23. By the relation between C (mn)andE±(mn)(cf.Lemma 4.2.19), we can translate Proposition 4.2.20 in terms of the plus and the + + + minus systems of points (d ) and (d−) such that E ()m )= d ,d− n n n n n ⟨ n 0 ⟩Zp[Gn] and E−(mn)= dn−,d0− Zp[Gn]. As in Remark 4.2.21, we see that the the plus + ⟨ ⟩ ) + subgroups E (mn)arecyclicZp[Gn]-modules generated by dn for all n if and only) if d / 0(mod4),ontheotherhandtheminussubgroupsE−(m )are ≡ n always cyclic) Zp[Gn]-modules generated by dn− for all n. ) Let χ :∆ Zp× be a character of ∆= Gal(k(µp)/k). We denote 1 → 1 εχ = p 1 σ ∆ χ(σ)σ− Zp[∆]. If M is a Zp[∆]-module, then M can be decomposed− ∈ into ∈ , M = εχM. χ !

39 χ We denote by M the χ-component εχM. Since we have Gn = G 1 ∆ Gal(kn/k0), we can regard a Zp[Gn]-module ∼ − × × as a Zp[∆]-module.

n i n i Corollary 4.2.24. Let χ :∆ Zp× be a character and qn = i=0( 1) p − . Then we have → − , d(q +1) if n : odd and χ = 1, χ n rankZp C (mn) = ( dqn otherwise, for each n 0 and ≥

χ d if χ = 1, rankZp C (m 1) = − 0 if χ = 1. = ̸

Proof. Since C (m 1)=E(m 1) = Zp[G 1], we obtain the latter statement. − − ∼ − Moreover, from the exact sequence (4.2.2) we get a recurrence sequence ) χ χ rankZp C (mn) +rankZp C (mn 1) − χ χ =rankZp E(m 1) +rankZp E(mn) . − By the theory of formal groups,) we have )

r χ r χ E(mn) ∼= (mn) (as Zp[Gn]-modules), and ⎧#E(m )χ/E(mr )χ =#mχ/(mr )χ < ⎨ ) n n n n ∞ for r sufficiently⎩ large.) Thus) we have

χ r χ rankZp E(mn) =rankZp E(mn) r χ χ n =rankZp (mn) =rankZp mn = dp ) ) for each n 0. Therefore we obtain the former statement. ≥ We introduce here some notation that will be used throughout the remain- n n 1 p p 1 ip − der of the thesis. Let ωn(X):=(1+X) 1andΦn(X):= i=0− X be n − the p th cyclotomic polynomial. We define ω±(X):=1and 0 , + + + ωn (X)= Φm(1 + X),% ωn (X)=Xωn (X), 1 m n,m:even ≤ ≤# ω%n−(X)= Φm(1 + X),ωn−(X)=Xω%n−(X). 1 m n,m:odd ≤ ≤# % % 40 Note that ω (X)=ω∓(X)ω±(X)foralln 0. We write ω (X), ω±(X)and n n n ≥ n n ωn±(X)simplybyωn, ωn± and ωn± respectively.

We identify Zp[Gn]withZp[G0][X]/ ωn p[G0][X] by sending γn to 1 + X, % ⟨ ⟩Z % where γn is the image% of γ in Zp[Gn]. n i n i Set qn = ( 1) p − as in Corollary 4.2.24 and q 1 := 0. Put i=0 − −

, qn if n is even, qn if n is odd, + qn := qn− := ( qn 1 if n is odd, ( qn 1 if n is even. − − + n Note that qn + qn− = p for all n 0. ≥ χ For later use, we rephrase Corollary 4.2.24 in terms of E±(mn) and qn± as in the following corollary. ) Corollary 4.2.25. Let χ :∆ Zp× be a character. Then we have → + χ + rankZp (E (mn) )=dqn ,

d(q− +1) if χ = 1, ) χ n rankZp (E−(mn) )= ( dq− if χ = 1. n ̸ ) For a character χ of ∆, we define Zp[χ]tobetheZp[∆]-module which is Zp as a Zp-module, and on which ∆acts via χ,namelyσ x = χ(σ)x for · σ ∆andx Zp[χ]. ∈ ∈

Proposition 4.2.26. Let χ :∆ Zp× be a character. We have →

Zp[G 1][X] Zp[G 1] − − + ⊕1 if χ = 1, (ωn , (ϕ + ϕ− )) Zp[G 1][X] + χ ⎧ ⟨ − ⟩ − E (mn) ∼= ⎪ Zp[G 1][X] ⎪ [χ] − if χ = 1, ⎨ Zp% Zp + ) ⊗ ωn Zp[G 1][X] ̸ ⟨ ⟩ − ⎪ ⎪ ⎩ Zp[G 1][X]/ ωn− Zp[G 1][X] if χ = 1, − ⟨ ⟩ − χ E−(m ) n ∼= ⎧ Zp[G 1][X] − ⎪ Zp[χ] Zp if χ = 1, ⎨ ⊗ ωn− Zp[G 1][X] ̸ ) ⟨ ⟩ − as [G ]-modules. ⎩⎪ Zp n % Proof. There is a surjective homomorphism

Zp[G0][X] + + ψ : Zp[G 1] dn ,d0− Zp[Gn] = E (mn) ω ⊕ − −→ ⟨ ⟩ ⟨ n⟩Zp[G0][X] ) 41 + d0− d 1 − obtained by sending (1, 0) to dn and (0, 1) to p 1 (= p 1 ). We have a relation − − + + + + + + + ωn dn = ωn 2dn 2 = = ω0 d0 = Xd0 =0. − − ··· 1 1 1 Since ε1 = p 1 σ ∆ σ− = p 1 Tr0/ 1 Zp[∆], we also have another relation − ∈ − − ∈ , + + 1 + + 1 + 1 d0− ε1ωn dn = Tr0/ 1 ωn dn = Tr0/ 1 d0 =(ϕ + ϕ− ) . p 1 − p 1 − p 1 − − − Thus the% map ψ induces a surjective% homomorphism

Zp[G0][X] [G ] ω+ Zp 1 n Zp[G0][X] ⊕ − + + ψ : ⟨ ⟩ d ,d− = E (m ). (ε ω+, (ϕ + ϕ 1)) −→ ⟨ n 0 ⟩Zp[Gn] n ⟨ 1 n − − ⟩Zp[G0][X] ) This map ψ is% injective since the source and the target of ψ are free Zp- + modules of the same Zp-rank d(p 1)qn (cf. Corollary 4.2.25). Thus we have −

Zp[G0][X] [G ] ω+ Zp 1 + ⟨ n ⟩Zp[G0][X] ⊕ − E (mn) = + 1 ∼ (ε1ω , (ϕ + ϕ )) ⟨ n − − ⟩Zp[G0][X] ) Zp[G0][X] Zp[G 1] = % ⊕ − ∼ (ω+, 0), (ε ω+, (ϕ + ϕ 1)) ⟨ n 1 n − − ⟩Zp[G0][X]

εχZp[G0][X] εχZp[G 1] = % ⊕ − ∼ (ε ω+, 0), (ε ε ω+, ε (ϕ + ϕ 1)) χ χ n χ 1 n χ − Zp[G0][X] ! ⟨ − ⟩ as Zp[Gn]-modules, where the last isomorphism% is obtained by the character decomposition. Since we have

Zp[G 1][X] Zp[G 1]ifχ = 1, − ⊕ − εχZp[G0][X] εχZp[G 1] = ⊕ − ∼ ( Zp[G 1][X] if χ = 1, − ̸ and

+ + 1 (ε ω , 0), (ε ε ω , ε (ϕ + ϕ− )) ⟨ χ n χ 1 n − χ ⟩Zp[G0][X] + + 1 (ωn , 0), (ωn , (ϕ + ϕ− )) Zp[G 1][X] if χ = 1, = ⟨ % − ⟩ − ∼ + ( ωn Zp[G 1][X] if χ = 1 ⟨ ⟩ − % ̸

42 as Zp[G 1][X]-modules, we have −

+ χ εχZp[G0][X] εχZp[G 1] E (m ) = ⊕ − n ∼ (ε ω+, 0), (ε ε ω+, ε (ϕ + ϕ 1)) ⟨ χ n χ 1 n − χ − ⟩Zp[G0][X] ) Zp[G 1][X] Zp[G 1] −% − + + ⊕ 1 if χ = 1, (ωn , 0), (ωn , (ϕ + ϕ− )) Zp[G 1][X] ⎧ ⟨ − ⟩ − ∼= ⎪ Zp[G 1][X] ⎪ [χ] − if χ = 1 ⎨ Zp Zp % + ⊗ ωn Zp[G 1][X] ̸ ⟨ ⟩ − ⎪ ⎪ + + 1 as Zp[Gn]-modules.⎩ Since (ωn , 0) = X(ωn , (ϕ+ϕ− )), we get the conclusion + χ − for E (mn) . % Similarly) to the above, we have

E−(m )= d− n ⟨ n ⟩Zp[Gn]

) = Zp[G0][X]/ ωn−, (σ 1)ωn− σ ∆ p[G0][X] ∼ ⟨ − | ∈ ⟩Z

Zp[G 1][X] Zp[G 1][X] − A[χ] − ∼= Zp Zp ωn− Zp[G 1][X] ⊕ ⊗ ωn− Zp[G 1][X] − χ=1 1 − 2 ⟨ ⟩ !̸ ⟨ ⟩ χ as Zp[Gn]-modules. So we get the conclusion for E−(m% n) .

Remark 4.2.27. When d / 0(mod4)andχ =) 1,thedescriptionofthe + χ ≡ Galois module E (mn) in Proposition 4.2.26 can be made more simpler. Explicitly, we claim that the homomorphism ) p[G 1][X] p[G 1] + ≃ Z Z Zp[G 1][X]/ ωn − ⊕ − − ⟨ ⟩ −→ (ω+, (ϕ + ϕ 1)) ⟨ n − − ⟩ 1 given by x (x, 0) is an isomorphism. Indeed, since ϕ + ϕ− Zp[G 1]× 1→ % ∈ − in this case (see Lemma 4.2.10), (x, y) Zp[G 1][X] Zp[G 1]isequivalent 1 1 + ∈ − ⊕ − to (x + y(ϕ + ϕ− )− ωn , 0) and thus the map is surjective. On the other + 1 hand, if (x, 0) (ωn , (ϕ + ϕ− )) for x Zp[G 1][X], then there exists ∈⟨ − ⟩ + ∈ − 1 a(X) Zp[G 1][X]suchthat% x = a(X)ωn and 0 = a(0)(ϕ + ϕ− ). Again ∈ − − a(X) by Lemma 4.2.10, we% see that a(0) = 0. So we get x = ω+ ω+ and X n ∈⟨ n ⟩ thus the map is injective. %

In the rest of this thesis, we abbreviate Zp[G 1][X]-modules S Zp[G 1][X] − − generated by some set S to S as in the above remark. ⟨ ⟩ ⟨ ⟩

43 4.2.4 The plus and the minus local conditions

χ In this subsection, we study the Λ-module (E±(m ) Qp/Zp)∨ and prove ∞ Proposition 4.2.31. We also study the Λ-module ⊗ ) 1 ∨ H (k ,E[p∞]) ∞ "E±(m ) Qp/Zp $ ∞ ⊗ and prove Proposition 4.2.35.)

χ We first study (E±(m ) Qp/Zp)∨. ∞ ⊗ Since E±(m )areZp-torsion-free, we have an exact sequence ∞ ) χ χ χ 0 )E±(m ) E±(m ) Qp E±(m ) Qp/Zp 0. −→ ∞ −→ ∞ ⊗ −→ ∞ ⊗ −→

From this exact) sequence,) we get the Γn-invariant-coinvariant) exact sequence

Γn χ χ 0 E±(mn) Qp/Zp E±(m ) Qp/Zp −→ ⊗ −→ ∞ ⊗ B χ C ) E)±(m ) [p∞] 0 (4.2.4) −→ ∞ Γn −→ B C ) χ for each n 0. We will compute the rightmost modules E±(m ) [p∞] ≥ ∞ Γn χ B C for all n 0tostudytheΛ-module(E±(m ) Qp/Zp)∨. ∞ ) Define≥δ by ⊗ ) 0ifd / 0(mod4)orχ = 1, δ = ≡ ̸ ( 2otherwise. .

χ ∨ Proposition 4.2.28. Let χ :∆ Zp× be a character. Then (E±(m ) )Γn [p∞] → ∞ are free Zp-modules for all n, and we have B C ) + χ ∨ rankZp (E (m ) )Γn [p∞] = dqn− + δ, ∞ B C ) d(q+ 1) if χ = 1, χ ∨ n − rankZp (E−(m ) )Γn [p∞] = ∞ ( dq+ if χ = 1. B C n ̸ )

44 More precisely, we have

+ χ E (m ) [p∞] ∞ Γn B C Zp[G 1][X] 1 ) − Ann [G ](ϕ + ϕ− ) p/ p if χ = 1, ω− Zp 1 Q Z ⟨ n ⟩ ⊕ − ⊗ ∼= ⎧ B C Zp[G" 1][X] ⎪ − p/ p if χ = 1, ⎨ ω− Q Z ⟨ n ⟩ ⊗ ̸ ⎪ B C ⎩ χ " E−(m ) [p∞] ∞ Γn B C Zp[G 1][X] ) − + Qp/Zp if χ = 1, ωn ⊗ = ⎧ ⟨ ⟩ ∼ ⎪ ⎪ Zp[G 1][X] ⎨ − Qp/Zp if χ = 1 ω%+ ⊗ ̸ ⟨ n ⟩ as -modules.⎪ Zp ⎩⎪ + χ Proof. We prove the claim for E (m ) [p∞]inthecasewhenχ = 1. ∞ Γn We can prove the rest of the claimsB similarly.C Since we have ) + χ + χ + χ E (m ) [p∞]= E (m ) /ωnE (m ) [p∞] ∞ Γn ∞ ∞ B C B C ) ) + χ ) + χ ∼= lim E (mm) /ωnE (mm) [p∞], (4.2.5) m n −→≥ B C where the transition maps ) )

+ χ + χ + χ + χ E (m ) /ω E (m ) [p∞] E (m ) /ω E (m ) [p∞] m n m −→ m+1 n m+1 areB multiplication-by-p mapsC when mBis odd and identity maps whenC m is ) ) ) ) + χ + χ even, we will calculate E (mm) /ωnE (mm) [p∞]foreachn and m.Since + + E (mm)=E (mm 1)ifB m is odd, we may assumeC m is even. − We consider the case) when n is even.) By Proposition 4.2.26 we have ) ) + χ + χ Zp[G 1][X] Zp[G 1] E (m ) /ω E (m ) = − ⊕ − . (4.2.6) m n m ∼ (ω+, (ϕ + ϕ 1)), (ω , 0) ⟨ m − − n ⟩ We can) show that ) Zp[G 1][X] Z%p[G 1] − ⊕ − [p∞] (ω+, (ϕ + ϕ 1)), (ω , 0) ⟨ m − − n ⟩ 1 Zp[G 1][X] AnnZp[G 1](ϕ + ϕ− ) − − . (4.2.7) ∼=% m n m n p −2 , ω ⊕ p −2 ⟨ n−⟩ ⟨ ⟩

% 45 m n + − + + Indeed, since we have ωm p 2 ωn (mod ωn)andωn = ωn−ωn ,thereisan exact sequence ≡

1 % Zp[G 1][X] AnnZp[G 1](ϕ + ϕ− ) − 0 m n− m n −→ p −2 , ω ⊕ p −2 ⟨ n−⟩ ⟨ ⟩ Zp[G 1][X] Zp[G 1] − ⊕ − −→ (ω+, %(ϕ + ϕ 1)), (ω , 0) ⟨ m − − n ⟩ Zp[G 1][X] Zp[G 1] − ⊕ − 0, (4.2.8) −→ (ω%+, (ϕ + ϕ 1)), (ω+, 0), (αω+, 0) −→ ⟨ m − − n n ⟩ + + where the first map is (x, y) (xωn + yωn , 0) and α is a generator of the % 1→ 1 % Zp[G 1]-module AnnZp[G 1](ϕ+ϕ− )(cf.Lemma4.2.12).Thereisananother − − exact sequence %

Zp[G 1][X] 0 − −→ ω+,αω+ ⟨ n n ⟩ Zp[G 1][X] Zp[G 1] − ⊕ − −→ (ω+, %(ϕ + ϕ 1)), (ω+, 0), (αω+, 0) ⟨ m − − n n ⟩ Zp[G 1] − 0, (4.2.9) −→ ϕ%+ ϕ 1 −→ % ⟨ − ⟩ whose leftmost and rightmost modules are both Zp-free, where the first map is x (x, 0) and the second map is (x, y) y.Thustherightmostmodule 1→ 1→ in (4.2.8), which is the same as the middle module in (4.2.9), is Zp-free. Our claim (4.2.7) follows from this. By (4.2.5), (4.2.6), and (4.2.7), we get

+ χ χ χ E (m ) [p∞] = lim E±(mm) /ωnE±(mm) [p∞] ∞ Γ ∼ n m n B C −→≥ B C ) )Zp[G 1][X] ) Zp[G 1] lim − ⊕ − [p∞] ∼= + 1 m n (ωm, (ϕ + ϕ− )), (ωn, 0) −→≥ ⟨ − ⟩ 1 Zp[G 1][X] AnnZp[G 1](ϕ + ϕ− ) − = lim %m n− m n ∼ − − m n p 2 , ωn− ⊕ p 2 −→≥ ⟨ ⟩ ⟨ ⟩ Zp[G 1][X] 1 − = % AnnZp[G 1](ϕ + ϕ− ) Qp/Zp ∼ ω ⊕ − ⊗ 1 ⟨ n−⟩ 2 m n m (n 1) − − when n is even. By replacing p %−2 with p 2 in the above discussion, we get the statement also in the case when n is odd.

46 + χ ∨ From this description, we see that (E (m ) )Γn [p∞] is Zp-free and ∞ B C + χ ∨ ) rankZp (E (m ) )Γn [p∞] ∞ B C Zp[G 1][X] 1 − =rankZp ) +rankZp AnnZp[G 1](ϕ + ϕ− ) ω − 1 ⟨ n−⟩ 2 & ' = dqn− + δ. %

Corollary 4.2.29. Let χ :∆ Zp× be a character. Then the Γn-coinvariants χ → ((E±(m ) Qp/Zp)∨)Γn are free Zp-modules for all n, and we have ∞ ⊗ ) + χ ∨ n rankZp E (m ) Qp/Zp = dp + δ, ∞ ⊗ Γ 1B C 2 n ) χ ∨ n rankZp E−(m ) Qp/Zp = dp . ∞ ⊗ Γ 1B C 2 n Proof. It follows from Corollary) 4.2.24, Proposition 4.2.28 and the exact sequence (4.2.4).

Proposition 4.2.30. Let χ :∆ Zp× be a character. There exist injective homomorphisms of Λ-modules →

+ χ ∨ d δ E (m ) Qp/Zp Λ⊕ (Λ/X)⊕ , ∞ ⊗ −→ ⊕ B χ C∨ d E)−(m ) Qp/Zp Λ⊕ ∞ ⊗ −→ B C with finite cokernels.)

+ χ Proof. We prove the claim for (E (m ) Qp/Zp)∨.Wecanprovetherest ∞ of the claims similarly. ⊗ + χ We note that (E (m ) Q)p/Zp)∨ has no nontrivial finite Λ-submodule ∞ ⊗ since its Γn-coinvariants are free Zp-modules for all n 0(seeCorollary ≥ 4.2.29). Thus by the) structure theorem for Λ-modules, there exist irreducible distinguished polynomials fj,nonnegativeintegersr, s, t, mi, nj,andan injective homomorphism

s t + χ ∨ r mi nj f : E (m ) Qp/Zp Λ⊕ Λ/p Λ/fj =: ∞ ⊗ −→ ⊕ ⊕ E i=1 j=1 B C ! ! ) with finite cokernel Z.

47 We show that r = d, ⎧ s =0(inotherwordsmi =0foralli), ⎪ 0ifd / 0(mod4)orχ = 1, ⎪ t = ≡ ̸ ⎨⎪ 2otherwise, and = (f n1 ,...,fnt )=(2, X, X )ift =2. ⎪ 1 t ⎪ { } ⎪ ⎩ + χ ∨ f From the exact sequence 0 E (m ) Qp/Zp Z 0, we → ∞ ⊗ →E→ → get the Γn-invariant-coinvariant exactB sequences C ) Γn + χ Z (E (m ) Qp/Zp)∨ −→ ∞ ⊗ Γn B C /ω) Z/ω Z 0 (4.2.10) −→ E nE−→ n −→ for all n. Note that, the first maps in (4.2.10) are 0-maps for all n,since + χ nj ((E (m ) Qp/Zp)∨)Γn is Zp-free. Then we see that mi =0andfj ωn ∞ (and n 1)⊗ for all n sufficiently large since Z/ω Z is bounded as n | . j ≤ n →∞ Thus) we get s =0here.Wenowhave

n + χ dp + δ =rankZp (E (m ) Qp/Zp)∨ ∞ ⊗ Γn B t C =rank ( /)ω )=rpn + n deg f (4.2.11) Zp E nE j j j=1 * for all n sufficiently large. Thus we get r = d. In the case when d / 0(mod4)orχ = 1,weget0= t n deg f from ≡ ̸ j=1 j j the above discussion. Thus we get nj =0whichisthedesiredresult,i.e. t =0. , We finally consider the case when d 0(mod4)andχ = 1.Wemay ≡ assume nj =1forallj.Inthiscase,wehave

t

2= deg fj, (4.2.12) j=1 * n f ω (= (1 + X)p 1) (4.2.13) j| n − for all n sufficiently large. We narrow down the possible combinations of (t, f ,...,f )satisfyingthesetwoconditions(4.2.12)and(4.2.13).Ifp 5, { 1 t} ≥ there is a unique combination (t, f1,...,ft )=(2, X, X ), since deg fj 2. 2 { } { } ≤ If p =3,sinceω1 = X(X +3X +3),therearetwopossiblecombinations

48 2 (t, f1,...,ft )=(2, X, X ), (1, X +3X +3 ). By showing that the last{ combination} is impossible,{ } we{ complete the} proof. Indeed, we have 2 rankZp /ω0 = d +1 withthecombination(t, f1,...,ft )=(1, X + 3X +3E). OnE the other hand, from the exact sequence{ (4.2.10)} for n =0,we{ must have} rank /ω = d +2andthuswegetthedesiredconclusion. Zp E 0E We now get the following proposition which is an important ingredient for the proof of Proposition 4.2.35.

χ Proposition 4.2.31. Let χ :∆ Zp× be a character. Then (E±(m ) → ∞ ⊗ Qp/Zp)∨ has no nontrivial finite Λ-submodule and its Λ-rank is d. ) Proof. This follows from Proposition 4.2.30. In the rest of this subsection, we study the Λ-module

1 ∨ H (k ,E[p∞]) ∞ . "E±(m ) Qp/Zp $ ∞ ⊗ We consider the exact sequence)

1 ∨ H (k ,E[p∞]) 1 0 ∞ H (k ,E[p∞]) ∨ ∞ → "E±(m ) Qp/Zp $ −→ ∞ ⊗ & ' ∨ E±(m ) Qp/Zp 0. (4.2.14) ) −→ ∞ ⊗ → B C We studied the Λ-module structure of the) rightmost module. We also know the Λ-module structure of the middle module by the following fact (Propo- sition 4.2.32).

Proposition 4.2.32 (Greenberg [5] 3Corollary2). Let L be a finite exten- § sion of Qp and L some Zp-extension of L. Put ΛL = Zp[[Gal(L /L)]].If ∞ 1 ∞ E(L )[p∞]=0, then H (L ,E[p∞])∨ is a free ΛL-module and its ΛL-rank ∞ ∞ is 2[L : Qp];

1 2[L:Qp] H (L ,E[p∞])∨ = ΛL⊕ . ∞ ∼

We can apply Proposition 4.2.32 in our setting as L = k0, L = k . ∞ ∞ Indeed we see that E(k )[p∞]=0byProposition4.2.3. ∞ Here we recall the following useful lemma on equivalent conditions on freeness of Λ-modules and on triviality of finite Λ-submodules.

49 Lemma 4.2.33. Let M be a finitely generated Λ-module. Γ (1) M is a free Λ-module if and only if M =0and MΓ is a free Zp- module. (2) M has no nontrivial finite Λ-submodule if and only if M Γ is a free Zp-module. Proof. See for example [24] Proposition 5.3.19. Applying the following lemma to the exact sequence (4.2.14), we can now 1 determine the Λ-module structure of (H (k ,E[p∞])/(E±(k ) Qp/Zp))∨. ∞ ∞ ⊗ Lemma 4.2.34. Let f : M N be a surjective homomorphism) of Λ- modules. Suppose that M is a→ free Λ-module of Λ-rank r, and that N is Λ-module of Λ-rank s which has no non-trivial finite Λ-submodule. Then its kernel Ker f is a free Λ-module of rank r s. −

Proof. We put M0 := Ker f.Thenbytakingtheinvariant-coinvariantexact sequence, we have

0 M Γ M Γ N Γ M M . −→ 0 −→ −→ −→ 0,Γ −→ Γ Γ Since M is a free Λ-module, we have M =0andMΓ is a free Zp-module by Lemma 4.2.33. Since N has no non-trivial finite Λ-submodule, we see that Γ Γ N is a free Zp-module by Lemma 4.2.33. Hence we have M0 =0andM0,Γ is a free Zp-module. Thus M0 is a free Λ-module again by Lemma 4.2.33. It is easy to see that the Λ-rank of M is r s. 0 − Proposition 4.2.35. We have

H1(k ,E[p ]) ∨ ∞ [k0:Qp] ∞ ∼= Λ⊕ . E±(k ) Qp/Zp 1 ∞ ⊗ 2 Proof. It follows from Proposition 4.2.31, Proposition 4.2.32, and Lemma 4.2.34 for the exact sequence (4.2.14).

4.2.5 More on the plus and the minus local conditions The discussion in the previous subsections is enough to prove our main the- orem. In this subsection, we further determine the explicit structure of the χ ∨ Λ-module E±(m ) Qp/Zp .Forthatpurpose,weshowthefollowing ∞ ⊗ lemma. B C )

50 Lemma 4.2.36. Let f : M N be an injective homomorphism of Λ- → modules with finite cokernel. Suppose that M/ωnM is Zp-free and NΛ-tors = x N ωnx =0 for all n sufficiently large. Then f induces an isomor- phism{ ∈ | }

f : M/M ≃ N/N . Λ-tors −→ Λ-tors Proof. We regard M as a Λ-submodule of N by f.SinceN/M is finite, we see that ω N M for all n sufficiently large. We thus have n ⊂

Coker(f)=N/(M + NΛ-tors)

ωn × ≃ ω N/ω M −→ n n " M/ω M → n for all n sufficiently large. Since M/ωnM is Zp-free for all n sufficiently large, we get M/MΛ-tors ∼= N/NΛ-tors for such n.

Theorem 4.2.37. Let χ :∆ Zp× be a character. We have →

+ χ ∨ d δ E (m ) Qp/Zp = Λ⊕ (Λ/X)⊕ , ∞ ⊗ ∼ ⊕ B χ C∨ d E)−(m ) Qp/Zp = Λ⊕ . ∞ ⊗ ∼ B C + χ Proof. We prove this) theorem for (E (m ) Qp/Zp)∨.Wecanprovethe ∞ rest of the claim similarly. ⊗ + χ ∨) d δ Let M = E (m ) Qp/Zp , N =Λ⊕ (Λ/X)⊕ , f be the map ob- ∞ ⊗ ⊕ tained in PropositionB 4.2.30. ThenC the assumptions in Lemma 4.2.36 are sat- isfied (see Proposition) 4.2.30, Corollary 4.2.29). Thus we have M/MΛ-tors = d ∼ Λ⊕ by Lemma 4.2.36. We now consider the commutative diagram

/ M / / M/M / 0 Λ-tors _ M _ Λ-tors 0

f0 f f ∼=    δ d δ d 0 / (Λ/X)⊕ / Λ⊕ (Λ/X)⊕ / Λ / 0. ⊕ ⊕ δ We see that MΛ-tors ∼= (Λ/X)⊕ ,sinceCoker(f0)isfinite.Therefore,the above horizontal exact sequence splits and thus we get

d δ M = M/M M = Λ⊕ (Λ/X)⊕ . ∼ Λ-tors ⊕ Λ-tors ∼ ⊕

51 4.3 Finite Λ-submodules

We use the notations and the assumptions which are described in Section 4.1. Let Γ= Gal(F /F0)andΛ=Zp[[Γ]]. We fix a topological generator ∞ γ Γ. Then we identify the completed group ring Zp[[Γ]] with the ring of ∈ power series Zp[[X]] by identifying γ with 1 + X.

4.3.1 Finite Λ-submodules of Sel(F ,E[p∞])∨ ∞ In this subsection, we study finite Λ-submodules of the Pontryagin dual of the p-primary Selmer group. The aim of this subsection is to prove Theorem 4.3.5. The following proposition was essentially proved by Hachimori and Mat- suno in [7].

Proposition 4.3.1. Let L be a finite extension of Q, L /L a Zp-extension, ∞ Ln its n-th layer, and E an elliptic curve defined over L. Put ΓL =Gal(L /L) ∞ and ΛL = Zp[[ΓL]]. Let Xn be the kernel of the restriction map

Sel(Ln,E[p∞]) Sel(L ,E[p∞]) −→ ∞ and X := lim Xn where the projective limit is taken with respect to the ∞ corestriction←− maps. Assume that the Zp-rank of Sel(Ln,E[p∞])∨ is bounded as n . Then →∞ the maximal finite ΛL-submodule of Sel(L ,E[p∞])∨ is isomorphic to X . ∞ ∞ In particular if we further assume that E(L)[p]=0, then Sel(L ,E[p∞])∨ ∞ has no nontrivial finite ΛL-submodule.

Proof. The proof in [7] works as well if the assumption on the ΛL-torsionness of Sel(L ,E[p∞])∨ is replaced by the assumption on the boundedness of the ∞ Zp-ranks of Sel(Ln,E[p∞])∨,asTakejipointedoutin[34].

We check that we can apply the above proposition in our setting L = F0 and L = F . ∞ ∞

Lemma 4.3.2. The morphisms Sel±(Fn,E[p∞]) Sel±(F ,E[p∞]) are in- ∞ jective for all n 0. → ≥ Proof. We can prove this by the same method as the proof of Lemma 9.1 in [17].

52 We assume from here that both Sel±(F ,E[p∞])∨ are Λ-torsion. We ∞ denote the Iwasawa λ-invariant of Sel±(F ,E[p∞])∨ by λ±. ∞ Let

1 1 H (Fn,v,E[p∞]) Sel (Fn,E[p∞]) := Ker Sel(Fn,E[p∞]) , ⎛ −→ E(Fv) p/ p ⎞ v Sss Q Z ∈#p,F ⊗ ⎝ ⎠ ss where Sp,F is the set of all primes of F lying above p where E has supersin- gular reduction. By the exact sequence (4.2.2), we have an exact sequence

1 1 1 H (Fn,v,E[p∞]) H (Fn,v,E[p∞]) H (Fn,v,E[p∞]) 0 + −→ E(Fv) Qp/Zp −→ E (Fn,v) Qp/Zp ⊕ E−(Fn,v) Qp/Zp ⊗ ⊗ ⊗ H1(F ,E[p ]) n,v ∞ 0 −→ E(Fn,v) Qp/Zp −→ ⊗ for each n and for each prime v of F lying above p.Thus,foreachn,weget the following exact sequence

1 ι + 0 Sel (F ,E[p∞]) Sel (F ,E[p∞]) Sel−(F ,E[p∞]) −→ n −→ n ⊕ n η Sel(F ,E[p∞]), (4.3.1) −→ n where ι is the diagonal embedding by inclusions and the map η is (x, y) x y. 1→ − Proposition 4.3.3. The cokernel of η in the exact sequence (4.3.1) is finite.

Proof. We can prove this by the same method as the proof of Lemma 10.1 in [17].

Proposition 4.3.4. The Zp-rank of Sel(Fn,E[p∞])∨ is bounded as n . More precisely, we have →∞

+ rank Sel(F, E[p∞])∨ +rank Sel(F ,E[p∞])∨ λ + λ− Zp Zp n ≤ for every n.

1 Proof. Since the restriction map Sel(F, E[p∞]) Sel (Fn,E[p∞]) is injective, we have →

1 rank Sel(F, E[p∞])∨ rank Sel (F ,E[p∞])∨. Zp ≤ Zp n

53 Hence by Lemma 4.3.2 and Proposition 4.3.3, we get

rankZp Sel(F, E[p∞])∨ +rankZp Sel(Fn,E[p∞])∨ + rankZp Sel (Fn,E[p∞])∨ +rankZp Sel−(Fn,E[p∞])∨ ≤ + λ + λ− ≤ for every n from (4.3.1). The boundedness of the Zp-ranks follows from this immediately. From the above argument, we can prove the following theorem.

+ Theorem 4.3.5. Assume that both Sel (F ,E[p∞])∨ and Sel−(F ,E[p∞])∨ ∞ ∞ are Λ-torsion. Then Sel(F ,E[p∞])∨ has no nontrivial finite Λ-submodule. ∞

Proof. The Zp-rank of Sel(Fn,E[p∞])∨ is bounded as n (cf. Proposition →∞ 4.3.4). Further, we have E(F0)[p]=0byProposition4.2.3.Thuswecan apply Proposition 4.3.1 and get the desired result.

4.3.2 Finite Λ-submodules of Sel±(F ,E[p∞])∨ ∞ Finally, we study finite Λ-submodules of the Pontryagin duals of the plus and the minus Selmer groups. The aim of this subsection is to prove our main theorem (Theorem 4.3.8). We prove that the triviality of finite Λ-submodules of Sel(F ,E[p∞])∨ is ∞ inherited to that of Sel±(F ,E[p∞])∨. ∞ Let us consider the following exact sequence of Λ-modules coming from the definition of the Selmer groups;

1 H (F ,v,E[p∞]) ∨ ι± ∞ Sel(F ,E[p∞])∨ ∞ ss E±(F ,v) Qp/Zp −→ v Sp,F 1 ∞ 2 ∈! ⊗ Sel±(F ,E[p∞])∨ 0. (4.3.2) −→ ∞ −→

Proposition 4.3.6. Assume that Sel±(F ,E[p∞])∨ is Λ-torsion. Then the ∞ map ι± in (4.3.2) is injective.

Proof. We have rankΛ(Sel(F ,E[p∞])∨) v Sss [F0,v : Qp](cf.[6]Theo- ∞ ≥ ∈ p,F rem 1.7). By Proposition 4.2.35, we have ,

H1(F ,E[p ]) ∨ ,v ∞ [F0,v:Qp] ∞ ∼= Λ⊕ E±(F ,v) Qp/Zp 1 ∞ ⊗ 2

54 ss for each prime v Sp,F .Fromthisandtheexactsequence(4.3.2),wesee that ∈

1 H (F ,v,E[p∞]) ∨ [F0,v : Qp]=rankΛ ∞ ss ⎛ ss E±(F ,v) Qp/Zp ⎞ v Sp,F v Sp,F 1 ∞ 2 ∈* ∈! ⊗ ⎝ ⎠ rankΛ (Sel(F ,E[p∞])∨) . ≥ ∞ Thus we get

1 H (F ,v,E[p∞]) ∨ rankΛ ∞ =rankΛ (Sel(F ,E[p∞])∨) . ∞ ⎛ ss E±(F ,v) Qp/Zp ⎞ v Sp,F 1 ∞ 2 ∈! ⊗ ⎝ ⎠ From this, we see that the kernel Ker ι± is Λ-torsion. Therefore we get the conclusion since the leftmost direct sum in the exact sequence (4.3.2) is a torsion-free Λ-module. The following proposition is a key tool for the proof of our main theorem. Proposition 4.3.7 (Greenberg [6] p.104–105). Let f : M N be an in- jective homomorphism of Λ-modules. Suppose that N is a finitely→ generated Λ-module which has no nontrivial finite Λ-submodule, and that M is a free Λ-module. Then the cokernel Coker(f) has no nontrivial finite Λ-submodule.

Proof. We put N1 := Coker(f). Then by taking the invariant-coinvariant exact sequence of 0 M N N 0, we get an exact sequence → → → 1 → M Γ N Γ N Γ M . (4.3.3) −→ −→ 1 −→ Γ Γ Since M is a free Λ-module, we see that M =0andMΓ is a free Zp- module by Lemma 4.2.33. Since N has no non-trivial finite Λ-submodule, Γ we see that N is a free Zp-module by Lemma 4.2.33. Thus from the exact Γ sequence (4.3.3), we see that N1 is a free Zp-module and therefore N1 has no nontrivial finite Λ-submodule again by Lemma 4.2.33. + Theorem 4.3.8. Assume that both Sel (F ,E[p∞])∨ and Sel−(F ,E[p∞])∨ + ∞ ∞ are Λ-torsion. Then both Sel (F ,E[p∞])∨ and Sel−(F ,E[p∞])∨ have no ∞ ∞ nontrivial finite Λ-submodule. Proof. By Proposition 4.2.35, we have

H1(F ,E[p ]) ∨ ,v ∞ [F0,v:Qp] ∞ ∼= Λ⊕ E±(F ,v) Qp/Zp 1 ∞ ⊗ 2 for each prime v Sss .ThuswecanapplyProposition4.3.6andProposition ∈ p,F 4.3.7 for f = ι±.Thus,byTheorem4.3.5,wegetthedesiredresult.

55 Finally, we consider the following special setting. Let E be an elliptic curve defined over Q such that E has good supersingular reduction at a prime number p with ap =0.LetF = Q(µm)suchthatGCD(p, m)=1, F0 = F (µp)andF the cyclotomic Zp-extension of F0.Inthissetting,weget ∞ unconditionally the following non-existence of nontrivial finite Λ-submodules.

+ Corollary 4.3.9. Both Sel (F ,E[p∞])∨ and Sel−(F ,E[p∞])∨ have no ∞ ∞ nontrivial finite Λ-submodule.

+ Proof. It is actually proved that both Sel (F ,E[p∞])∨ and Sel−(F ,E[p∞])∨ ∞ ∞ are Λ-torsion, due to Kobayashi and Kato. Therefore, the claim follows from Theorem 4.3.8.

56 Acknowledgements

The author would like to express his sincere gratitude to his supervisor, Professor Masato Kurihara, for successive encouragement, helpful discus- sions, many suggestions, and comments. The author is deeply grateful to Dr. Rei Otsuki for his collaboration on the study of the non-existence of nontrivial finite Λ-submodules of Selmer groups. The author is also deeply grateful to Professors Hiroyasu Izeki, Kenichi Bannai, and Taka-aki Tanaka for their helpful comments on the manuscript of this thesis. The author would like to thank Dr. Chan-Ho Kim, Dr. Takashi Miura, Dr. Kazuaki Murakami, Dr. Jiro Nomura, and the other members of number theory for discussion, encouragements, and fruitful time in Keio.

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