The Inductive Argument

Also,

ord_Q(p^3eφ(v)Q) = 0⇔p^3e(ι◦φ(v)) =β(v) = 1⇔ b((v)^1/pℓ) (v)^1/pℓ = 1

⇔v is an p^ℓ-th power modulo Q.

On the other hand, we have

ord_Q(ϕ_q(v)) = l_Q(v) = 0⇔v is an p^ℓ-th power modulo Q.

It follows that there exists u ∈(Z/p^ℓZ)^× with ord_Q(ϕ_q(v)) = uord_Q(p^3eφ(v)Q), and the map sending

v 7→ϕ_q(v)−p^3euφ(v)Q

gives rise to a Gn-equivariant injective homomorphism from V to ⊕h∈Gn

h̸=1

(Z/p^ℓZ)Q^h. But the latter has no non-zero Gn-stable submodules, so

ϕ_q(v) =p^3euφ(v)Q, as required.

(iii) #(A(H_n)^χ)|p^ℓ and ¯v_Q(v) divides (p^ℓ/#(A(H_n)^χ))λ₂ in Λ^χ_n/p^ℓΛ^χ_n, and (iv) fΛ(G) is prime to I(H_n).

Then there exists a Gn-equivariant map φ:V →Λ_n/p^ℓΛ_n satisfying f φ(v) =λ₀λ₁λ₂v¯_Q(v).

Proof. This is a combination of [17, Lemma 8.2] and [11, Lemma 3.8.4].

Fix elements f₁, . . . , f_k ∈Λ(G) so that 0→ ⊕^k_i=1 Λ(G)

Λ(G)f_i →A(H∞)→Q→0 with Q finite. In particular,

char_Λ(A(H∞)) =

k

Y

i=1

f_i

!

Λ(G).

Theorem 5.3.2. (i) Ifp > 2, k is as above and χ∈G^∗, we have char_Λ(A(H_∞)^χ) divides I(D_p)^4k+4char_Λ( ¯E_H∞/C¯_H∞)^χ,

where D_p= ^Q_P|pD_P denotes the group generated by the decomposition groupsD_P of P in H∞/H.

(ii) If p= 2, k is as above and χ∈G^∗, we have

char_Λ(A(H∞)^χ) divides 2^6k+6char_Λ( ¯E_H∞/C¯_H∞)^χ.

Proof. We will prove this for p= 2. The case p > 2 is similar. Fix a generatorβ of char_Λ( ¯E_H∞/C¯_H∞)^χ. LetBbe an ideal of finite index in Λ(G) satisfying the conditions in Theorem 5.1.8 (ii) and Lemma 5.1.12. Take λ∈2I(G)B. By Lemma 5.1.1, λΛ_n/λ^2kΛ_n is finite. Also, by Corollary 5.1.10, βΛ(G) is prime to I(H_n), so Λ_n/βΛ_n is finite. It follows that λΛ_n/λ^2kβΛ_n is finite. Thus, for some ℓ>1, we have

2^ℓλΛ_n⊂(2^n+4k(#(A(H_n)^χ))λ^2kβ)Λ_n. (5.3.1) Now, by Corollary 5.1.11, there exists θ_λ,n: ¯E_Hn →Λ_n such that

λ²β ∈θ_λ,n( ¯C_Hn).

Thus, we may fix u∈C¯_Hn with θ_λ,n(u) = λ²β, and also we fix u₀ ∈ C_Hn with u≡u0 mod ( ¯CHn)^pℓ.

By Proposition 5.1.15, we have an Euler system αandσ∈Gal(H/K) withα^σ(n,1) = u₀. Let κ_α be the map defined in Proposition 5.1.16, and let c₁, . . . ,c_k ∈ A(H_n) be as given in Lemma 5.1.12. We will use induction to select primes Q₁, . . . ,Q_k+1 of H_n lying above primes q₁, . . . ,q_k+1 of K satisfying:

[Qi] =λc^χ_i inA(Hn)^χ, and qi ∈ Iℓ, (5.3.2)

¯

v_Q1(κ_α(q₁)^χ) = r₁2⁴λ²β and fi−1¯v_Qi(κ_α(a_i)^χ) =r_i2⁴λ²v¯_Qi−1(κ_α(ai−1)^χ), (5.3.3) wherea_i =q₁· · ·q_i and r_i ∈(Z/p^ℓZ)^×.

Fori= 1, we take c=λc^χ₁ ∈2I(G)A(H_n)^χ,W =E¯_Hn/E¯_Hn∩(H_n^×)^pℓχ,φ = 2θ_λ,n and apply Theorem 5.2.2 and Proposition 5.2.1. Then we obtain a prime Q₁ of H_n such that [Q₁] =λc^χ₁ inA(H_n)^χ and a primeq₁ ∈ I_ℓ lying below Q₁,

[(κ_α(q₁)^χ)]_Q1 =ϕ_q1(κ_α(1)^χ) = ϕ_q1(α^σ(n,1)^χ)

=r₁2³φ(u₀)Q₁ =r₁2⁴θ_λ,n(u₀)Q₁ =r₁2⁴λ²βQ₁. Thus, by the definition of [·]_Q1, we have

¯

v_Q1(κ_α(q₁)^χ) = r₁2⁴λ²β, which proves the first equality of (5.3.3).

Now, let 1 < i < k and suppose we have selected primes Q₁, . . .Q_i satisfying (5.3.2) and (5.3.3). We will define Q_i+1. Recall a_i = ^Q_j6iq_j.Let V_i be the Λ_n-

submodule of H_n^×/(H_n^×)^pℓ generated by κ_α(a_i)^χ. We will apply Lemma 5.3.1 with Q = Q_i, v =κ_α(a_i)^χ, λ₁ = λ₂ = λ and S = {q₁, . . . ,q_i−1}. This is possible because conditions (i), (ii) and (iv) of Lemma 5.3.1 are satisfied thanks to Proposition 5.2.1, Lemma 5.1.12 and Theorem 5.1.6, and (iii) is satisfied because by (5.3.3), ¯v_Qi(κ_α(a_i)^χ) divides 2⁴ⁱλ²ⁱβ in Λ^χ_n/2^ℓΛ^χ_n, so by the choice of ℓ made in (5.3.1), ¯v_Qi(κ_α(a_i)^χ) divides

p^ℓ/#(A(H_n)^χ)λ in Λ_n/2^ℓΛ_n. Thus, we obtain a map φ_i :V_i →Λ_n/p^ℓΛ_n such that f_iφ_i(κ_α(a_i)^χ) = 2λ²v¯_Qi(κ_α(a_i)^χ). (5.3.4)

Now, applying Theorem 5.2.2 by settingV =V_i, c=λc^χ_i+1, φ=φ_i, we obtain a prime q_i+1 ∈ I_ℓ and a prime Q_i+1 of H_n lying above it. Then (i) and (ii) of Theorem 5.2.2 gives (5.3.2) for i+ 1. Furthermore, by Proposition 5.2.1 (ii) and Theorem 5.2.2 (iii), for some r_i+1 ∈(Z/p^ℓZ)^× we have

fi[κα(ai+1)]Qi+1 =fiϕqi+1(κα(ai)^χ)

=r_i+12³f_iφ_i(κ_α(a_i)^χ)Q_i+1

=ri+12⁴λ²¯vQi(κ_α(a_i)^χ)Q_i+1,

where the last equation follows from (5.3.4). This proves (5.3.3) for i+ 1. Finally, combining (5.3.3) for 16i6k+ 1 gives

k

Y

i=1

f_iv¯_Qk+1(κ_α(a_k+1)^χ) =r2^4k+4λ^2k+2β in Λ_n/p^ℓΛ_n for some u∈(Z/p^ℓZ)^×. It follows that

char_Λ(A(H∞)) =

k

Y

i=1

f_i divides 2^4k+4λ^2k+2βΛ(G) = 2^4k+4λ^2k+2char_ΛE¯_H∞/C¯_H∞. This holds for everyλ∈2I(G)B, so in particular, holds forλbeing the greatest common divisor λ0 of all elements in 2I(G)B. It is easy to show that in this case we have λ0Λ(G) = 2I(G). This concludes the proof of Theorem 5.3.2, because char_Λ(A(H_∞)) is prime to I(G) by Theorem 5.1.6.

Corollary 5.3.3. Let p > 2. Then

char_Λ(A(H∞)) divides char_Λ( ¯E_H∞/C¯_H∞).

Proof. We have shown in Theorem 5.1.6 that char_Λ(A(H_∞)) is prime toI(D_p), so by Theorem 5.3.2, char_Λ(A(H_∞)) divides char_Λ( ¯EH∞/C¯H∞).

Recall that p-[H :K] by assumption.

Theorem 5.3.4. We havechar_Λ(X(H∞)) = char_ΛU_H∞/C¯_H∞if and only ifchar_Λ(A(H∞)) = char_ΛE¯_H∞/C¯_H∞, and

char_Λ(X(H∞))|2^e(6k+6)char_ΛU_H∞/C¯_H∞.

Proof. Recall from (4.4.2) that we have an exact sequence

0→E¯H∞/C¯H∞ →UH∞/C¯H∞ →X(H∞)→A(H∞)→0,

and therefore char_Λ(A(H∞))char_ΛU_H∞/C¯_H∞ = char_Λ(X(H∞))char_ΛE¯_H∞/C¯_H∞. The last assertion of the theorem follows from Theorem 5.3.2 and Corollary 5.3.3.

Proof of the main conjecture for H _∞ /H

6.1 The Iwasawa Invariants of X (H

_∞

)

Recall that

G ≃G×Γ,

where we identify G with Gal(H∞/K∞) and Γ with Gal(K∞/K). Recall that any Λ_I(G) =I[[G]]-moduleM can be decomposed into a direct sumM =⊕χ∈G^∗M^χ of its χ-components. Thus, let us consider I[[Γ]] as a Λ_I(G)-module via χ. Given a finitely generated torsion Λ_I(G)-module M, recall from Section 4.4 that char(M)⊂Λ_I(G) denotes the characteristic ideal of M. If X is a Λ(G)-module and χ ∈G^∗, we write X^χ for (X⊗b_ZpI)^χ. This is justified because we are only interested in char(X)^χ, and the characteristic ideals of a Γ-module behaves well under extension of scalars. This comes from the fact that we can identify I[[Γ]] with I[[T]].

Recall also that anyf(T)∈I[[T]] can be written uniquely, by thep-adic Weierstrass preparation theorem, in the form

f(T) = π^mP(T)U(T)

where π is a uniformiser of I, P(T) is a distinguished polynomial, i.e., a monic polynomial whose coefficients are divisible by π, and U(T) is a unit in I[[T]]. Let ϵ be the absolute ramification index of I. The invariants

µ(f) = m

ϵ and λ(f) = degP(T)

are called the Iwasawa µ-invariant and λ-invariant of f, respectively. The Iwasawa invariants of Λ(G)-modules are defined similarly, and ifM = X⊗b_ZpI is obtained from a Λ(G)-module X by extension of scalars to I, the invariants of M coincide with those of X.

Define f^χ = char (X(H∞))^χ and let g^χ = charUH∞/C¯_H∞^χ , and set f = ^Qf^χ and g =^Qg^χ. By Theorem 5.3.4, we have

Theorem 6.1.1. f^χ |π^ekg^χ for some integer k ≥0, e= 0 ifp > 2and e= 1 ifp= 2.

Thus, in order to show f^χ and g^χ define the same ideal, it remains to show that f and g the have the same Iwasawa invariants. We shall compute them separately, and show that they are equal. First, we compute at the invariants of X(H∞) using class field theory, and in Section 6.2 we compute the invariants of U_H∞/C¯_H∞ using the analytic class number formula.

Recall from the proof of Theorem 5.1.5 that X(H∞)/I(H_n)X(H∞) is equal to Gal(M(H_n)/H∞), where M(H_n) is the maximal abelian p-extension of H_n which is unramified outside the primes above p. Thus the asymptotic formula of Iwasawa [23, Theorem 13.13] gives:

Theorem 6.1.2. Let f be the characteristic power series for X(H∞) as a Zp[[Γ]]- module. For sufficiently large n, we have

ord_p(#(X(H_∞)/I(H_n)X(H_∞)) =µ(f)p^n−1−e+λ(f)(n−1−e) +c,

where µ(f) and λ(f) are the Iwasawa invariants of X(H∞) and c∈Z is independent of n.

We will now computep-adic valuation of the index [M(H_n) :H∞] using the methods of Coates and Wiles [7], and use it to find ord_p(#(X(H∞)/I(H_n)X(H∞)). We note that p is assumed to be an odd prime number in [7], but it can easily be extended to p= 2 in our case, because 2 splits in K and (p, h) = 1 by assumption.

Set [H_n:K] =d, which is equal to p^n−1−eh where e= 0 or 1 according as p is odd or even. Let ξ₁, . . . ξ_d denote the distinct embeddings ofH_n intoCp. SinceH_nis totally imaginary, rank_Z(E_Hn/(E_Hn)_tor) =d−1. We pick a basis ϵ₁, . . . , ϵd−1 forE_Hn/(E_Hn)_tor, and put ϵ_d = 1 +por 1 +p² according as pis odd or even.

Definition 6.1.3.

R_n= (dlogϵ_d)⁻¹det (log(ξ_i(ϵ_j)))_16i,j6d.

LetC_Hn denote the idele class group of H_n. For each n>1, let Yn =∩_m>nN_Hm/HnCHm

Let Φ_p =H_n⊗_KK_p.

LetP denote the set of primes ofH_n lying above p. LetU_Hn,P denote the group of units in the completion of H_n at P which are congruent to 1 moduloP, and lett >0 be such that p^−tO_P ⊂logU_Hn,P for each P∈P.

The p-adic logarithm gives a homomorphism log : U_Hn,P → H_n,P whose kernel has order w_P = #µ_p∞(H_n,P). Write logU_Hn = ^Q_P∈PlogU_Hn,P, so that we have log :U_Hn →Φ_p with kernelw_p =^Q_P∈Pw_P,

Lemma 6.1.4.

ord_p



[ ^Y

P∈P

p^−tO_P : logU_Hn]



= ord_p



w_p ^Y

P∈P

NP



+td.

Proof. See [6, Lemma 7].

LetVⁿ = 1 +pⁿO_p denote the local units of K_p which are congruent to 1 modulo pⁿ, and define D_n =V^1+eE¯_Hn ⊂ U_Hn, where e = 0 or 1 according as p > 2 or p = 2.

Furthermore, let ∆_Hn/K denote the discriminant ofH_n/K, and pick a generator ∆_n of the ideal ∆_Hn/KO_p.

Lemma 6.1.5.

ordp([logUHn : logDn]) = ordp







Rn

√∆_n



w_p ^Y

P∈P

NP





−1



+n+ 1.

Proof. Using methods analogous to [6, Lemma8], we can show that ord_p



[ ^Y

P∈P

p^−tO_P : logD_n]



= ord_p R_n

√∆_n

!

+td+n−e+ ord_plog(1 +p^1+e), (6.1.1) where e= 0 or 1 according as p >2 or p= 2. We have ord_p(log(1 +p^1+e)) = 1 +e, so the right hand side of (6.1.1) is equal to ord_p^√Rn

∆n

+td+n+ 1. The result now follows from Lemma 6.1.4.

Corollary 6.1.6.

ord_p([U_Hn :D_n]) = ord_p







R_n ω_Hn√

∆_n



 Y

P∈P

NP





−1



+n+ 1.

Proof. This is an immediate consequence of Lemma 6.1.5, obtained by applying the snake lemma to the following commutative diagram

0 D_n U_Hn U_Hn/D_n 0

0 logD_n logU_Hn logU_Hn/logD_n 0.

log log

with exact rows.

Lemma 6.1.7.

Y_n∩U_Hn = kerN_Φp/Kp |_UHn E¯_Hn = kerN_Φp/Kp |_Dn. Proof. See Lemma 5 and Lemma 6 of [6].

Lemma 6.1.8.

[Y_n∩U_Hn : ¯E_Hn] = ord_p





Rn

√∆_n

Y

P∈P

(1−(NP)⁻¹)



+n

Proof. By Lemma 6.1.7 and the definition of D_n, we have N_Φp/Kp(D_n) = (V^1+e)^d = (V^1+e)^n−e = Vⁿ⁺¹. Hence, applying Lemma 6.1.7 again, we obtain a commutative

diagram with exact rows

0 E¯_Hn D_n Vⁿ⁺¹ 0

0 Y_n∩U_Hn U_Hn Vⁿ 0.

NΦp/Kp

N_Φp/Kp

Lemma 6.1.8 now follows from Lemma 6.1.7 on noting that ord_p^Q_P∈P(1−(NP)⁻¹)= ord_p^Q_P∈P(NP)⁻¹) and [Vⁿ :Vⁿ⁺¹] =p.

Theorem 6.1.9. Let M(H_n) be the maximal abelian p-extension of H_n which is unramified outside the primes in P. Then

ord_p([M(H_n) :H∞]) = ord_p





h_HnR_n

√∆_n

Y

P∈P

(1−(NP)⁻¹)



+n, where h_Hn denotes the class number of H_n.

Proof. Let L(H_n) be the maximal unramified extension of H_n in M(H_n). Thus we may identify Gal(L(H_n)/H_n) withA(H_n), the p-primary part of the ideal class group of H_n. Class field theory gives an isomorphism

Y_n∩U_Hn/E¯_Hn ∼= Gal(M(H_n)/L(H_n)H∞).

Noting that L(H_n)∩H∞ =H_n because H∞/H_n is totally ramified atp, we obtain 0→Y_n∩U_Hn/E¯_Hn →Gal(M(H_n)/H_∞)→A(H_n)→0.

The theorem now follows from Lemma 6.1.8 and the fact that ord_p(#(A(H_n))) = ord_p(h_Hn).

Corollary 6.1.10. Let f be the characteristic power series for X(H∞)as a Γ-module.

Then for sufficiently large n,

µ(f)·p^n−1−e(p−1) +λ(f) = 1 + ord_p h_Hn+1R_n+1

√∆_n+1 /h_HnR_n

√∆_n

!

,

where µ(f) and λ(f) are the Iwasawa invariants of X(H∞).

Proof. By Theorem 6.1.9, it is clear that the right hand side of the above equation is equal to ord_p([M(H_n+1) :H∞]/[M(H_n) :H∞]). Recalling that Gal(M(H_n)/H_∞) = X(H_∞)/I(H_n)X(H_∞), Theorem 6.1.2 gives ord_p([M(H_n+1) :H∞]/[M(H_n) :H∞]) is equal to

(µ(f)p^n−e+λ(f)(n−e)+c)−(µ(f)p^n−1−e+λ(f)(n−1−e)+c) = µ(f)p^n−1−e(p−1)+λ(f).

This completes the proof of the corollary.

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