m 0 is locally maximal

which we called the Greenberg-R-topology, and which we want to define on arbitraryE^d(K) now.

Definition 5.8. Let K/K denote a Z^dp-extension, d∈N. For every n ∈N0, we define

U(K, n) := {L∈ E(K, n)| P(L)⊆ P(K)}.

Remark 5.9. The U(K, n), together with∅, generate a topology on E^d(K).

Proof. The intersection of two sets U(K, n1) and U( ˜K, n2) is a finite union of sets of this type, or empty (compare the proof of Lemma 3.25, (i)):

Without loss of generality, we may assume that n1≥n2. Then U(K, n₁) ∩ U( ˜K, n₂) = {L∈ E(K, n₁)| P(L)⊆ P(K)∩ P( ˜K)}. This set might be empty. Otherwise, we choose setsI₁, . . . , I_r⊆ P(K)∩ P( ˜K) such that

• for every i = 1, . . . , r, there exists an element Li ∈ E(K, n1) such that P(Li) =I_i, and

• for everyM ∈ E(K, n₁) withP(M)⊆ P(K)∩ P( ˜K), we have P(M)⊆I_i for somei∈ {1, . . . , r}.

Then

U(K, n1) ∩ U( ˜K, n2) =

r

[

i=1

U(Li, n1).

We will see in Section 5.5 that, in contrast to the one-dimensional case, a full use of Fukuda theory for Z^dp-extensions requires a finer control on the ramification than is provided by the Greenberg-R-topology. In fact, it will not be enough to simply control which primes of K do ramify at all. We will moreover have to fix the rank of the maximal ‘torsion’ unramified subextension of ourZ^dp-extension (compare Definition 5.38 for details).

The Greenberg-R-topology, however, is fine enough in order to allow the application of the one-dimensional Fukuda method developed in Chapter 3 to suitableZp-extensions of K that are contained in ourZ^dp-extensions. This will be exploited in the next two sections, yielding the first results concerning the local behaviour of generalised Iwasawa invariants.

5.2 m

is locally maximal

We will now start to study the local behaviour of generalised Iwasawa invariants with respect to the topologies introduced above. Before formulating the first result, we prove a technical lemma.

Lemma 5.10. Let d ∈ N, d ≥ 2. Let K/K denote a Z^dp-extension, and let m0 := m0(K/K) ∈ N0. Then there exist only finitely many Z^d−1p -extensions M ⊆K of K such that

m0(M/K) > m0 .

If d= 2, then there exist only finitely many Zp-extensions M ⊆K of K such that µ(M/K)6=m₀(K/K).

Proof. We first note that there exist only finitely manyZ^d−1p -extensionsM ⊆K of K such that P(M) $ P(K). Indeed, each such M has to be contained in the inertia subfield of a prime of K ramifying in K whose inertia group is a subgroup of Gal(K/K) ofZp-rank 1.

Therefore, we will from now on assume thatP(M) =P(K).

Letf ∈Λd denote the characteristic power series of K/K, and let us write f = p^m0 ·g, with p - g. Consider the Fitting ideal (0) 6= F(X) = (p^m0g)·J ofX := Gal(H(K)/K), whereH(K) denotes the maximal unramifiedp-abelian extension ofK. Suppose that 06=h∈J is not divisible by p (such an element exists because J is not contained in the prime ideal (p)⊆Λ_dof height one).

By Lemma 4.20, the subsetC ⊂ε^d−2_d−1 of homomorphismsπ such that either π(g) ≡ 0 modporπ(h) ≡ 0 modpis finite. For everyπ ∈ε^d−2_d−1\C, the module X_π =X/(ker(π)·X) is a finitely generated torsion Λd−1-module (annihilated, for example, by p^m0 ·π(g·h)6= 0), and

p^m0 ·π(g·h) ∈ F(Xπ) (compare Lemma 4.35). Therefore

m0(Xπ) = m0.

If d = 2, then Lemma 4.3, (ii) implies that µ(M/K) = m₀(X_π) for the Zp-extensionM/K that corresponds toπ, provided that P(M) =P(K).

In order to handle the case d >2, we generalise Lemma 4.3, (ii) and show that m₀(M/K)≤m₀(X_π) ifM corresponds to some π ∈ε^d−2_d−1\C.

Proposition 5.11. Let j, r ∈ N, 2 ≤ j ≤ r −1. Let K/K denote a Z^rp -extension, and let M ∈ E^j,⊆K(K) denote aZ^jp-extension ofK contained in K. Let X:= Gal(H(K)/K), and suppose thatπ ∈ε^j−1_r−1 corresponds to the restric-tion map Gal(K/K) Gal(M/K). We assume thatXπ := X/(ker(π)·X) is a torsion Λj-module. Then

m₀(M/K) ≤ m₀(X_π) and l₀(M/K) ≤ l₀(X_π). If only finitely many primes of M ramify inK, then we have equalities.

Proof. We adapt the proof of Lemma 4.3, (ii). If H(M) denotes the maximal unramified p-abelian extension of M, then m0(M/K) = m0(Gal(H(M)/M)), by definition. The inclusionH(M)·K⊆H(K) implies that we have a surjective homomorphism

X = Gal(H(K)/K) ^//// Gal((H(M)·K)/K) . Note that kerπ={σ−1|σ ∈Gal(K/M)}. Since

(σ−1)·τ = ˜σ◦τ ◦σ˜⁻¹◦τ⁻¹ = τ ◦τ⁻¹ = 1

5.2. m0 IS LOCALLY MAXIMAL 179 for every τ ∈ Gal((H(M)·K)/K) and every σ ∈ Gal(K/M), it follows that ker(π)·Gal((H(M)·K)/K) ={1} (here ˜σ ∈Gal((H(M)·K)/M) denotes any lift ofσ, respectively). We therefore obtain a surjectiveZp[[Gal(M/K)]]∼= Λj -module homomorphism

X_π =X/(ker(π)·X) ^//// Gal((H(M)·K)/K)∼= Gal(H(M)/(H(M)∩K)). In particular,

m0(Gal(H(M)/(H(M)∩K))) ≤ m0(Xπ) and

l0(Gal(H(M)/(H(M)∩K))) ≤ l0(Xπ).

We will show that the Λ_j-module Gal(H(M)/(H(M)∩K)) is pseudo-iso-morphic to Gal(H(M)/M) and therefore

m0(Gal(H(M)/(H(M)∩K))) = m0(M/K) and

l₀(Gal(H(M)/(H(M)∩K))) = l₀(M/K). The reason for this is the fact that

Gal(H(M)/M)/Gal(H(M)/(H(M)∩K)) ∼= Gal((H(M)∩K)/M) is a finitely generatedZp-module and therefore is pseudo-null as a Λ_j-module.

Indeed, we may assume that Z := Gal((H(M) ∩K)/M) is in fact Zp-free, because the torsion subgroup of Z is finite. We write Λ_j = Zp[[T₁, . . . , T_j]].

Recall that j ≥ 2, by assumption. There exist distinguished polynomials in Zp[T1] as well as inZp[T2] that annihilate the finitely generatedZp-moduleZ, using the Weierstraß Preparation Theorem 1.14 and the assumption thatZ is torsion-free. In particular, these two polynomials are coprime when regarded as elements of Λj, and therefore Z is Λj-pseudo-null.

Now suppose that only finitely many primes of M ramify in K. Then the proof of the first inequality of Lemma 4.3, (ii) shows that

m₀(X_π) ≤ m₀(M/K) and l₀(X_π) ≤ l₀(M/K).

Indeed, if theZp-extension LofK in Greenberg’s original approach is replaced by the Z^jp-extension M/K, then the proof goes through without changes. In particular, the two groupsD and T remain finitely generated over Zp. By the above, Dand T therefore are pseudo-null as Λ_j-modules.

This also concludes the proof of Lemma 5.10.

We are now ready to state the main result of this section.

Theorem 5.12. Let K/K denote a Z^dp-extension. Then m0 := m0(K/K) is locally maximalwith respect to the Greenberg-R-topology, i.e., there exists an integer n∈N0 such that m0(L/K)≤m0(K/K) for every L∈U(K, n).

Proof. Ifd= 1, then the statement has been proved in Lemma 3.56 (recall that in this case,m₀(K/K) =µ(K/K), by Theorem 5.3).

Let us now assume that d = 2. Then there exist only finitely many Zp -extensionsM ⊆KofKwithµ(M/K)6=m0(K/K), by Lemma 5.10. In view of Lemma 5.7, (i), we may choose M ∈ E^⊆K(K) such that µ(M/K) =m₀(K/K) and P(M) = P(K). Let n ∈ N be large enough to ensure that in the one-dimensional neighbourhoodU(M, n) ofM ∈ E(K),µ(M/K) is locally maximal.

We may assume that every prime ramifying in the Zp-extension M/K has already started ramifying in the n-th intermediate field M_n.

Now consider the neighbourhood U := U(K, n) ofK∈ E^d(K). Let ˜K∈U. Then Mn ⊆Kn = ˜Kn ⊆ K˜, where ˜Kn denotes the subfield of ˜Kthat is fixed by Gal( ˜K/K)^pn. Since Gal( ˜K/K) ∼= Z^dp is torsion-free, we may choose a Zp -extension ˜M ⊆ K˜ of K containing M_n. In view of Lemma 5.7, (i), we may assume thatP( ˜M) =P( ˜K). Then ˜M ∈U(M, n)⊆ E(K), because

P( ˜M) = P( ˜K) ⊆ P(K) = P(M). Thereforeµ( ˜M /K)≤µ(M/K).

We let ˜X := Gal(H( ˜K)/K˜). Let ˜f ∈ Λ2 denote the characteristic power series of ˜K/K, and let ˜π ∈ ε⁰₁ denote the homomorphism corresponding to M˜ ⊆K˜ via Lemma 2.7. Then

µ( ˜M /K) = µ( ˜Xπ˜)<∞,

by Lemma 4.3, (ii), and µ( ˜Xπ˜)≥m0(˜π( ˜f)), by Proposition 4.34, (i). But then m0( ˜K/K) = m0( ˜f) ≤ µ( ˜M /K) ≤ µ(M/K) = m0(K/K).

Assume now that 3≤d is arbitrary. First note that all but finitely many Z^d−1p -extensions K^(d−1) ⊆ K of K satisfy P(K^(d−1)) = P(K) (compare the proof of Lemma 5.10). Moreover, Lemma 5.10 implies that we may choose K^(d−1) such thatm0(K^(d−1)/K)≤m0(K/K).

Inductively, we obtain aZp-extension M ⊆KofK such that µ(M/K) ≤ m₀(K/K) and P(M) = P(K).

Letn∈Nbe large enough to ensure that in the one-dimensional neighbourhood U(M, n) ofM ∈ E(K),µ(M/K) is locally maximal, and such that every prime ramifying in M/K has already started ramifying in the n-th intermediate field M_n of M/K.

LetU :=U(K, n)⊆ E^d(K). Suppose that ˜K∈U. As in the proof of d= 2, we can choose a Zp-extension

M˜ ∈ U(M, n) ∩ E^⊆K˜(K)

such that P( ˜M) =P( ˜K). Again, Lemma 4.3, (ii) implies that m0( ˜K/K) ≤ µ( ˜M /K) ≤ µ(M/K) ≤ m0(K/K).

5.3. l0 IS LOCALLY BOUNDED 181