The χ-components on the class group and on E/C

Recall that we fixed a decomposition Gal(L∞/K)∼= Γ⁰×H withH = Gal(L∞/K∞).

Letγ⁰ be a topological generator of Γ⁰. To simplify notation we will use the notation γ_n⁰ for the elementγ⁰²ⁿ. Let Γ = Gal(L∞/L). There exists a power of 2 such that Γ^02m is contained in Gal(L∞/L). In particularL∞/L^Γ

02m

∞ is totally ramified at all primes above p and Γ^02m+n = Γ^2m

Lemma 3.3.12. The kernel of the multiplication of(1−γ_n⁰) onA∞ is finite for every n.

Proof. This follows directly from the fact that all finite subextensions of L∞/K are abelian overKand that thep-adic Leopoldt conjecture holds for any abelian extension ofK. In particular, thep-adic Leopoldt conjecture holds for every fieldLn(see [Ru 1, page 705] for more details).

Lemma 3.3.13. Let χ be a character of H and n ≥ m⁰. Then there is a Λχ,n

homomorphism

Aχ,n→

k

M

i=1

Λχ,m+n−m⁰/(g_i),

with uniformly bounded cokernel. Here, g_i is the restriction of g_i to level n.

Proof. This proof is very similar to [Gr, Lemma 3.10]: By [Wash, page 281] the module A_n is isomorphic to A∞/νm+n−m⁰,mY for some submodule Y. Consider the map

φ_n:A∞/(1−γ_m+n−m⁰ 0)A∞→A_n.

By definition, the kernel is isomorphic to νm+n−m⁰,mY /(1−γ_m+n−m⁰ 0)A∞, which is bounded by the size of Y /(1−γ_m⁰ )A∞ ≤ A∞/(1−γ⁰_m)A∞. By Lemma 3.3.12 this quotient is finite and the kernel ofφn is uniformly bounded. Thus, the kernel of the natural projections

A∞,χ/(1−γ_m+n−m^{0 0})A∞,χ →Aχ,n

is uniformly bounded and we can deduce the claim from (3.2).

Let Γ⁰_n2,n1 = Γ⁰²ⁿ¹/Γ⁰²ⁿ² forn2 > n1. Recall that Γ⁰²ⁿ fixes the field Lm⁰+n−m for n > m. Hence Gal(Ln2−m+m⁰/Ln1−m+m⁰) = Γ⁰_n2,n1.

Lemma 3.3.14. Let E_m⁰ 0+n2−m be the p-units in Lm⁰+n2−m. Then we have that

|H¹(Γ⁰_n2,n1, E_m⁰ 0+n2−m)|is uniformly bounded for n₂ ≥n₁ ≥m.

Iwasawa [Iw 2, page 267] proves the same result for the group of thep-units instead of thep-units.

Proof. Let A⁰_n = An/Bn where Bn is the group generated by the ideal classes of the ideals above p for all n. Then the capitulation kernel C_n2,n1 of the natural homomorphism

A⁰_n1+m0−m→A⁰_n2+m0−m

is uniformly bounded for alln2≥n1≥m. This can be seen as in [Iw 2, Theorem 10]

using the corresponding definition ofA⁰_n. If we can show that Cn2,n1 ∼=H¹(Γ⁰_n2,n1, E_m^{0 0}_+n2−m)

we are done. Leta∈Cn2,n1 and a=cB_n1+m⁰−m. Leta be an ideal inc. Then there is someγ ∈Ln2−m+m⁰ such that (γ) =AB, whereB is only divisible by ideals above p. Recall that τ = γ⁰²ⁿ¹ is a generator for Γ⁰_n2,n1. Thus, γ^τ−1 ∈ E_n⁰

2+m⁰−m. Note that the image of γ^τ−1 in H¹(Γ⁰_n2,n1, E_m⁰ 0+n2−m) is independent of the choice ofc,A

3.3. ELLIPTIC UNITS AND EULER SYSTEMS 67 and γ. If γ^τ−1 ∈ E^0τ−1_n2+m⁰_−m, then γ =ηα with α ∈Ln1+m⁰−m and η ∈ E_n⁰

2+m⁰−m. We obtain thatais trivial. Hence, we have an injective homomorphism

Cn2,n1 →H¹(Γ⁰_n2,n1, E_m^{0 0}_+n2−m).

It remains to show that it is surjective. Lete∈E_m⁰ 0+n2−mlie in the kernel of the norm N: Ln2+m⁰−m → Ln1+m⁰−m. Then there is some element γ ∈ Ln2+m⁰−m such that e=γ^1−τ. As ideal we see that (γ) =ABfor some idealAthat is a lift fromLn1+m⁰−m

and some ramified idealB. Letc= [A] and a=cB_n1+m⁰_−m. Then a∈C_n2,n1 and a is a preimage of the image ofe inH¹(Γ⁰_n2,n1, E_m⁰ 0+n2−m).

As a consequence we obtain:

Lemma 3.3.15. There is a constant k such that

|(1−γ_m⁰ )H¹(Γ⁰_n2,n1, E_m⁰_+n2−m)| ≤2^k and |(1−γ_m⁰ )Hb⁰(Γ⁰_n2,n1, E_m⁰_+n2−m)| ≤2^k for any pair (n1, n2) with n2> n1≥m.

Proof. The proof follows the ideas of [Ru 1, Lemma 1.2]. But it is restated here as we use weaker assumptions. LetE_m⁰_+n2−m be the units of Lm⁰+n2−m and R_m⁰_+n2−m

be theZ-free group defined by the exact sequence

0→ E_m⁰_+n2−m →E_m⁰ 0+n2−m→R_m⁰_+n2−m→0.

AsL∞/Lm⁰ is totally ramified we see that Γ^0m acts trivially onR_m⁰_+n2−m. We know from Lemma 3.3.14 that|H¹(Γ⁰_n2,n1, E_m⁰ 0+n2−m)| is uniformly bounded. Further, we have the exact sequence

Hb⁰(Γ⁰_n2,n1, R_m⁰_+n2−m)→H¹(Γ⁰_n2,n1,E_m⁰_+n2−m)→H¹(Γ⁰_n2,n1, E_m⁰ 0+n2−m).

The first term is annihilated by 1−γ_m⁰ and the last term is uniformly bounded. It follows that (1−γ_m⁰ )H¹(Γ⁰_n2,n1,E_m⁰_+n2−m) is uniformly bounded.

It is an immediate consequence from [Ja, V Theorem 2.5] that q(E_m⁰ 0+n2−m) = 2^{(n2−n1)(1−s)}, wheres is the number of primes abovep. Thus,

2^{(n2−n1)(s−1)}|H¹(Γ⁰_n2,n1, E_m^{0 0}_+n2−m)|=|Hb⁰(Γ⁰_n2,n1, E_m^{0 0}_+n2−m)|.

Consider the surjective mapHb⁰(Γ⁰_n2,n1, E_m⁰ 0+n2−m)→N_n1,mE_m⁰ 0+n1−m/N_n2,mE_m⁰ 0+n2−m

induced byNn1,m = (γ_n⁰₁ −1)/(γ_m⁰ −1). Using thatNn1,m(1−γ_m⁰ ) = (1−γ_n⁰₁) and that Γ⁰²ⁿ¹ is precisely the group fixingLm⁰+n1−m we see that the subgroup

((1−γ_m⁰ )E_m^{0 0}_+n1−m+Nn2,n1E_m^{0 0}_+n2−m)/Nn2,n1E_m^{0 0}_+n2−m

is certainly contained in the kernel. Note that N_n1,mE_m⁰ 0+n1−m/N_n2,mE_m⁰ 0+n2−m is the kernel of the natural mapHb⁰(Γ⁰_n2,m, E_m⁰ 0+n2−m)→Hb⁰(Γ⁰_n1,m, E_m⁰ 0+n1−m). Thus, we obtain:

|(1−γ_m⁰ )Hb⁰(Γ⁰_n2,n1, E_m^{0 0}_+n2−m)| ≤ |Hb⁰(Γ⁰_n2,n1, E_m⁰ 0+n2−m)||Hb⁰(Γ⁰_n1,m, E_m⁰ 0+n1−m)|

|Hb⁰(Γ⁰_n2,m, E_m⁰ 0+n2−m)|

≤ 2^{(n2−n1)(s−1)+k}2^{(n1−m)(s−1)+k}

2^{(n2−m)(s−1)} = 2^2k,

where 2^k is the uniform bound on H¹(Γ⁰_n2,n1, E_m⁰ 0+n2−m). It is easy to verify that the natural map Hb⁰(Γ⁰_n2,n1,E_m⁰_+n2−m) → Hb⁰(Γ⁰_n2,n1, E_m⁰ 0+n2−m) is an injection. As

|(O(Ln)/P)^×|is coprime to 2 for every prime ideal Pabovep, the claim follows.

Lemma 3.3.16. Let n≥m⁰ and consider the projection πn:E∞/(1−γ_m+n−m^{0 0})E∞→En.

There exists an integerksuch that2^k(1−γ_m⁰ )annihilates the kernel and the cokernel of π_n for all n≥m⁰.

Proof. We have an exact sequence

∞←nlim⁰H¹(Γ⁰_m+n0−m⁰,m+n−m⁰, E_n⁰)→

→E_∞/(1−γ_m+n−m⁰ 0)E∞→E_n→ lim

∞←n⁰Hb⁰(Γ⁰_m+n0−m⁰,m+n−m⁰, E_n).

Then the first and the last term of the above sequence are annihilated by 2^k(1−γ_m⁰ ) due to Lemma 3.3.15.

Lemma 3.3.17. Let U∞ be defined as in the introduction. Then we have i) U∞⊗_ZpQp ∼= Λ⊗_ZpQp and

ii) U∞,χ⊗_ZpQp ∼= Λχ⊗_ZpQp.

Proof. Claim i) follows as in [Bl, Lemma 3.5 Claim 2]. Bley gives two references for this proof. Note that the second one is only stated forp >2 but the proof works for p= 2 as well (we will actually give the details in Lemma 3.4.1).

Claim ii) can be proved as follows:

U∞⊗_ZpZp(χ)⊗_ZpQp∼= Λ⊗_ZpZp(χ)⊗_ZpQp.

Let I_χ ⊂ Z(χ)[H] be the module generated by σ −χ(σ) for σ ∈ H. It is an easy verification that

U∞⊗_ZpZp(χ)⊗_ZpQp/I_χ(U∞⊗_ZpZp(χ)⊗_ZpQp)

∼= Λ⊗_ZpZp(χ)⊗_ZpQp/Iχ(Λ⊗_ZpZp(χ)⊗_ZpQp).

It is proved in [Ts, Lemma 2.1] thatMχ∼= (M⊗_ZpZp(χ))/Iχ(M⊗_ZpZp(χ)). Further, for any moduleM we see that

M⊗_ZpZp(χ)⊗_ZpQp/I_χ(M⊗_ZpZp(χ)⊗_ZpQp)

= (M⊗_ZpZp(χ)/I(χ)(M⊗_ZpZp(χ)))⊗_ZpQp =M_χ⊗_ZpQp. Using this forU∞ and Λ the second claim follows.

3.3. ELLIPTIC UNITS AND EULER SYSTEMS 69 Lemma 3.3.18. Let hχ be the characteristic ideal of (E∞/C∞)χ and n≥m. Then there exist constants n₀, c₁ and c₂ independent of n, a divisor h⁰_χ of h_χ and a Gal(Lm⁰+n−m/K)-homomorphism

ϑ_m⁰+n−m:E_m⁰+n−m,χ→Λn,χ

such that

i) h⁰_χ is prime to 1−γ_v⁰ for allv,

ii) (γ_n⁰₀ −1)^c12^c2h⁰_χΛ_n,χ ⊂ ϑ_m⁰+n−m(im(C_m⁰+n−m,χ)), where im(C_m⁰+n−m,χ) de-notes the image of Cm⁰+n−m,χ in Em⁰+n−m,χ.

Proof. From the second claim of Lemma 3.3.17 and the fact that Λχ ⊗_Zp Qp is a principal ideal domain we obtain that the submoduleE∞,χ⊗_ZpQp is free cyclic over the ring Λ_χ⊗_ZpQp. We obtain a pseudo isomorphismf:E∞,χ →Λ_χ⊕ M, where M is an elementary Λχ-module of finite exponent. Let p denote the natural projection of Λ_χ⊕ M →Λ_χ. Letα =p◦f. Consider the following diagram

0 E∞,χ E∞,χ 0

0 M ΛχL

M Λχ 0

f α

p

It is immediate that the kernel ofα is annihilated by some power of 2 and that the cokernel ofα is finite.

Consider the map

π_n⁰ :E∞/(1−γ_n⁰)→E_m⁰+n−m.

Note that π_n⁰ = π_n+m⁰−m. The rest of the proof is exactly the same as [Bl, Lemma 3.5]; we just have to substituteEn by Em⁰+n−m due to the index shift we defined at the beginning of the present section. Further, we have to write (1−γ⁰_m) instead of (1−γ⁰) in all computations due to the fact that Lemma 3.3.16 is weaker than the corresponding claim in Bley’s case, meaning thatm= 0 in his setting. We still give a complete proof below for the convenience of the reader.

LetW_m⁰+n−m be the image ofπ⁰_n inE_m⁰+n−m and define T = Tor_Zp[H]( Coker (π⁰_n),Zp(χ)).

Then we obtain the following two exact sequences:

T W_m0+n−m,χ E_m0+n−m,χ Coker (π_n⁰)_χ 0

ker(π_n⁰)_χ E_∞,χ/(1−γ_n⁰)E_∞,χ ^π W_m0+n−m,χ 0 .

0n

=

Letπ_n,χ⁰ be the compositum ofπ_n⁰ andβ:Wm⁰+n−m,χ→Em⁰+n−m,χ. Then we obtain 0→ker(π⁰_n,χ)→E∞,χ/(1−γ_n⁰)E∞,χ →E_m⁰+n−m,χ→ Coker (π⁰_n)_χ →0 It is immediate from Lemma 3.3.16 that the cokernel ofπ_n,χ⁰ is annihilated by 2^k(1− γ_m⁰ ). In the next step we also want to bound the size of ker(π_n,χ⁰ ). Let therefore e∈ ker(π_n,χ⁰ ). Then π_n⁰(e) lies in the image of T and π_n⁰((1−γ_m⁰ )2^ke) = 0. Hence, (1−γ_m⁰ )2^kelies in the image of ker(π⁰_n)_χ inE∞,χ/(1−γ_n⁰)E∞,χ. Thus, Lemma 3.3.16 implies that (1−γ_m⁰ )²2^2ke= 0. Therefore, (1−γ_m⁰ )²2^2k annihilates the kernel and the cokernel ofπ_n,χ⁰ .

We are no able to construct the homomorphism ϑ_m⁰+n−m: Let e ∈ E_m⁰+n−m,χ. Then (1−γ_m⁰ )²2^2ke has a preimage z in E∞,χ/(1−γ_n⁰)E∞,χ and therefore also in E∞,χ. By abuse of notation we denote both preimages by z. Define

ϑ_m⁰_+n−m(e) = (1−γ_m⁰ )²2^2kα(z) mod (1−γ_n⁰)Λ_χ.

As the definition of ϑ_m⁰+n−m depends a priori on the choice of z, we first have to check that ϑ_m⁰+n−m is well defined. Assume that z⁰ is another preimage, then the image of z−z⁰ inE∞,χ/(1−γ_n⁰)E∞,χ lies in the kernel of π⁰_n,χ. In particular,

(1−γ⁰_m)²2^2k(z−z⁰)∈(1−γ_n⁰)E∞,χ

andϑ_m⁰_+n−m is indeed well defined.

From the exact sequence 0→ C∞ →E∞ → E∞/C∞ →0 we obtain an embed-ding

E∞,χ/im(C∞,χ),→(E∞/C∞)_χ.

Ash_χ(E∞/C∞)_χ is finite the same holds for the quotientsh_χ(E∞,χ/im(C∞,χ)) and hχ(α(E∞,χ)/α(im(C∞,χ))). Due to the definition of α we can find an integer ssuch that 2^s∈α(E∞,χ) and such that 2^shχα(E∞,χ) ⊂α(im(C∞,χ)). Hence 2^2shχ lies in α(im(C∞,χ)). Choosez∈im(C∞,χ) such that α(z) = 2^2sh_χ. Then we obtain

2^2s+4k(1−γ_m⁰ )⁴hχ= 2^4k(1−γ_m⁰ )⁴α(z).

Letz_m⁰+n−m =π⁰_n,χ(z)∈im(C_m⁰+n−m,χ). Clearly, a preimage of 2^2k(1−γ_m⁰ )²z_m⁰+n−m

is 2^2k(1−γ_m⁰ )²z. Then we obtain

ϑ_m⁰+n−m(z_m⁰+n−m) = 2^4k(1−γ_m⁰ )⁴α(z) = 2^4k(1−γ_m⁰ )2^2sh_χ.

Note that (1−γ_n⁰)|(1−γ_n⁰0) for all n⁰ ≥n. In particular, there is ann≥m and an c1 such that hχ(1−γ_m⁰ )⁴ |h⁰_χ(1−γ_n⁰₀)^c1. From this the claim is immediate.

Lemma 3.3.19. Let M=Ln for somenand let ∆be a subgroup ofG=Gal(M/K).

Let χ be a character of ∆, M = 2^l and A=Qs

i=1q_i ∈Sn,l. Let Q be a divisor of q_s in M and let C = [Q]the ideal class of Q. Let B ⊂A(M) be the subgroup generated by ideals dividing q₁, . . . ,q_s−1. Let x ∈ M^×/(M^×)^M be such that [x]_r = 0 for all (r,A) = 1. Let W ⊂ M^×/(M^×)^M be the Zp[G]-submodule generated by x. Assume that there are elementsE, η, g∈Zp[G]such that

3.3. ELLIPTIC UNITS AND EULER SYSTEMS 71 i) E·ann_Zp[G](Cχ)⊂gZp[G]χ, where Cχ is the image of C in (A/B)χ.

ii) Zp[G]_χ/g(Zp[G])_χ is finite.

iii) M ≥ |A_χ(M)||η((I_qs/M Iqs)/[W]qs)χ|.

Then there exists aG-homomorphism

ψ:W_χ→(Z/MZ)[G]_χ such thatgψ(x)Q_χ= (Eη[x]_qs)_χ in (I_qs/M I_qs)_χ.

Proof. This is [Bl, Lemma 3.8]. The proof is the same as [Gr, Lemma 3.12].

To prove the central theorem of this section we need the following lemma.

Lemma 3.3.20. [Gr, Lemma 3.13] Let∆be any finite group andN aZp[∆]-module.

Let χ be a character of ∆ and n:N →N_χ the natural projection. Then there exists a Zp[∆]-homomorphism

εχ:Nχ→N such thatn◦ε_χ=|∆|.

Let q be an element in S_n,l and A in I_q. Then there is an element v_Q(A) in Z/2^lZ[Gal(Ln/K)] such that A= v_Q(A)Q. We will use this notation in the follow-ing theorem, which allows us to relate the characteristic ideal of Aχ to the one of (E∞/C∞)_χ. The proof follows the ideas of [Bl].

Theorem 3.3.21. Let M = Ln and G = Gal(M/K) for n large enough. Let χ be a character of H ⊂ Gal(M/K). For 1 ≤ i ≤ k let C_i ∈ A_χ(M) = A_χ be such that t(Ci) = (0,0,· · · ,2^c3,0· · ·,0)in Lk

i=1Λχ,m+n−m⁰/(g_i) where t is the map defined in Lemma 3.3.13 and 2^c3 annihilates the cokernel. Let C_k+1 in Aχ be arbitrary. Let d= 3c+ 4, where c is defined in Theorem 3.3.8. Then there are prime ideals Q_i in M such that

i) [Q_i]_χ= 2^dC_i.

ii) q_i=Q_i∩K is inSn,n. iii) one has

(v_Q1(κ(q₁))_χ=u₁|H|(γ⁰_n

0 −1)^c12^d+c2h⁰_χ mod 2ⁿ (3.3) (gi−1vQi(κ(q1q₂. . .q_i))χ (3.4)

=ui|H|(γ_n⁰

0 −1)^ci−11 2^2d+c3(v_Qi−1(κ(q1. . .q_i−1))χ mod 2ⁿ for 2≤i≤k+ 1.

Proof. By Lemma 3.3.18 there exists an element ξ⁰ in im(Cn,χ) with the property ϑ_n(ξ⁰) = (1−γ_n⁰₀)^c12^c2h⁰_χ. By approximating ξ⁰ with a global elliptic unit we can find ξ ∈ Cn such that ϑn(ξ) = (1−γ_n⁰₀)^c12^c2h⁰_χ mod MΛχ,m+n−m⁰. We can apply Lemma 3.3.5 to find an Euler system α^σ(n,r) such that α^σ(n,(1)) = ξ. Let i = 1

and C be a preimage of Ci under the map An → Aχ,n. Choose M = 2ⁿ and W = O(M)^×/(O(M)^×)^M. Consider

ψ:W →Z/MZ[G] x7→(ε_χ◦ϑ_n)(x^v),

where v is such that x^v ∈En for all x and εχ is defined as in Lemma 3.3.20. Then Theorem 3.3.8 implies that we can find an ideal Q₁ satisfying i) and ii). We know further from Theorem 3.3.8 that ϕ_q1(w) = 2^duψ(w)Q₁. As α^σ(n,(1)) is a unit we can apply Proposition 3.3.7² and obtain

v_Q1(κ(q1))Q1 = [κ(q1)]q1 =ϕq1(κ(1))

=ϕq1(ξ) = 2^duψ(ξ)Q1 = 2^duvεχ((γ_n⁰₀ −1)^c12^c2h⁰_χ)Q1 mod 2ⁿ. Projecting to theχcomponent and using the definition of ε_χ we get (3.3).

We will now define the idealsQ_i inductively. Assume that we have already found the idealsQ₁, . . . Q_i−1 and let a_i−1 =Qi−1

j=1q_j. Using point iii) recursively we see Y

j≤i−2

gj(vQi−1(κ(ai−1))χ=|H|ⁱ⁻¹u2^{(i−2)(2d+c3)+d+c2}(γ⁰_n0 −1)^c1+

Pi−2 j=1c^j₁

h⁰_χ.

LetDi=|H|ⁱ⁻¹u2^{(i−2)(2d+c3)+d+c2}. By enlargingc1 we can assume thatc1+Pi−2 j=1c^j₁ is bounded bycⁱ⁻¹₁ and sett_i =cⁱ⁻¹₁ . It follows thatv_Qi−1(κ(ai−1))_χ|D_ih⁰_χ(γ_n⁰₀−1)^ti. Define N = (γ⁰_n0 −1)^ti(I_qi−1/(M I_qi−1 +Zp[G][κ(ai−1)]_qi−1)_χ). As h⁰_χ is coprime to everyγ_n⁰ −1 we see that Λχ,m+n−m⁰/h⁰_χ is finite and further

|N| ≤ |Λ_χ,m+n−m⁰/D_i||Λ_χ,m+n−m⁰/h⁰_χ|.

Choose now 2^l = M >max(|A_χ(M)||Λ_χ,m+n−m⁰/D_i||Λ_χ,m+n−m⁰/h⁰_χ|,2ⁿ). We want to apply Lemma 3.3.19 with E = 2^c3+d, η = (γ_n⁰₀ −1)^ti, g = gi−1, A = a_i−1, and x = κ(ai−1). To do so we have to check the assumptions. It follows directly from Proposition 3.3.7 i) that [x]r= 0 for allrcoprime to a_i−1. We now have to check the conditions i)-iii) from Lemma 3.3.19.

i) By definition Cχ = [Qi−1]χ = 2^dCi−1. The annihilator of t(C) is given by gi−1/(2^c3+d, gi−1). As property i) holds for allQ_j withj≤i−2 we obtain that E·ann_Zp[G](C_χ)⊂gi−1Zp[G]_χ.

ii) It is immediate from Lemma 3.3.13 that Zp[G]χ/gZp[G]χ is finite.

iii) M >|A_χ||N|=|A_χ||η( ^{Iqi−1/(M Iqi−1)}

Zp[G][κ(ai−1)]qi−1)χ|.

Thus, we obtain a homomorphism

ψ_i:W_χ→Z/MZ[G]_χ

2We have to ensure that q1 lies in the domain ofκ, i.e. q1 has to satisfy a certain coprimality condition. As Theorem 3.3.8 gives us infinitely many primes we can just assume thatq1 lies in the domain ofκ.

3.3. ELLIPTIC UNITS AND EULER SYSTEMS 73 with gi−1ψi(κ(ai−1)) = (2^c3+d(γ_n⁰₀ −1)^tivQi−1(κ(ai−1)))χ. Let Πχ be the projection W →W_χ and defineψ=ε_χ◦ψ_i◦Π_χ. LetM be as in the previous paragraph and C a preimage ofC_i. Then Theorem 3.3.8 gives us a prime ideal Q_i satisfying i) and ii) (recall that 2ⁿ|M). Further,ϕqi(κ(ai−1)) = 2^duψ(κ(ai−1))Qi. Then we obtain

v_Qi(κ(q1. . .q_i))Qi= [κ(q1. . .q_i)]qi =ϕqi(κ(q1. . .q_i−1))

= 2^duψ(κ(ai−1))Q_i. Projecting to theχ-component and using the definition ofψ_i we obtain

(gi−1v_Qi(κ(q1q₂. . .q_i))χ=ui|H|(γ_n⁰

0−1)^ci−11 2^2d+c3(v_Qi−1(κ(q1. . .q_i−1))χ, which finishes the proof.

To derive a relation betweenhχ and Qs

i=1gi we need the following result:

Corollary 3.3.22. Let M be a finite abelian extension of K and consider H/K∞M (the maximal p-abelian unramified extension of K∞M). Then A∞ = Gal(H/K∞M) is finitely generated as aZp-module

Proof. Gal(H/K∞M) is a quotient ofGal(M∞/K∞M). The latter is finitely generated due to Theorem 2.1.1.

Theorem 3.3.23. Char(A∞,χ)|Char((E∞/C∞)_χ).

Proof. The main argument of this proof is analogous to [Wash, page 371]. From (3.3) and (3.4) we obtain that Qk

i=1g_iv_Qk+1(κ(q₁. . .q_k+1))_χ = ηh⁰_χ mod 2ⁿ, where η= ˜u|H|^k+12^{k(2d+c3)+d+c2}(γ⁰_n0−1)^c1+Pkj=1cj1 for some unit ˜u. It follows thatQ_k

i=1g_i divides ηh⁰_χ in Λχ,m+n−m⁰/2ⁿΛχ,m+n−m⁰. For every n we can find an element z_n such that Q_k

i=1g_iz_n=ηh⁰_χ in Λχ,m+n−m⁰/2ⁿΛχ,m+n−m⁰. Thez_n’s have a convergent subsequence and we obtain thatQk

i=1gi |ηh⁰_χin Λχ. By Lemma 3.3.12 and Corollary 3.3.22 Char(A∞,χ) is coprime toη and the claim follows.

Remark 3.3.24. Vigui´e proves in [Vi-2] as well that there is a power of 2 such thatChar(A∞,χ)|2^vChar((E∞/C∞)_χ). Using Corollary 3.3.22 this implies Theorem 3.3.23. He follows the proofs of Bley very closely as well but he uses a slightly different version of Theorem 3.3.8 and does not use the relation betweenΓand Γ⁰ in the same manner as we did here in Section 3.3.2 to construct the homorphismϑ_n.

Corollary 3.3.25. Char(A∞)|Char(E∞/C∞)

Proof. As Theorem 3.3.23 holds for all characters and Char(A∞) is coprime to 2 this is immediate.

Im Dokument Classical Conjectures in Iwasawa Theory for the split prime Z_p-extension and the cyclotomic Z_p-extension (Seite 65-74)