Introduction

LetKdenote the composite of allZp-extensions ofK, i.e., Gal(K/K)∼=Z^dp for somed∈N, and suppose thatd≥2. In the preceding section, we considered homomorphisms

πL: Zp[[Gal(K/K)]] ^//// Zp[[Gal(L/K)]]

for any fixedZp-extension Lof K. Baba˘ıcev more generally studied the set of all surjective homomorphisms

π: Λd //// Λ ,

where Λ = Λ1 =Zp[[T]] and Λ_d=Zp[[T1, . . . , T_d]] (see Definition 2.16).

Let Γ^d, respectively, Γ, denote free abelian pro-p-groups of rank d, respec-tively, of rank 1. Then we have topological isomorphisms Γ^d∼=Z^dp and Γ∼=Zp.

We will use some notation introduced in Section 2.1. Let ε(Γ^d) := {π: Γ^{d ////} Γ}

denote the set of all surjective Zp-module homomorphisms (i.e., continuous group homomorphisms) from Γ^d into Γ. In what follows, we will usually write the groups Γ^d and Γ multiplicatively, since in our applications, these groups will come up as Galois groups. Using the isomorphisms Γ^d ∼=Z^dp and Γ∼=Zp, we will identifyε(Γ^d) with the set

ε(Z^dp) := {π :Z^dp //// Zp} that has been studied in Section 2.1.

We will identify two homomorphismsπ1, π2:Z^dp −→Zp ifπ1 =π₂^u for some u ∈Z^∗p. This will be important for the application to Zp-extensions (compare Remarks 2.6, (1) and Lemma 2.7), and makes it possible to obtain an isomor-phism between ε(Z^dp) and the (d−1)-dimensional projective space P^d−1(Zp) overZpintroduced in Definition 2.1 (compare Proposition 2.5). Thereforeε(Γ^d) may be seen as a projective variety.

Using the isomorphism Gal(K/K)∼=Z^d_p, we may furthermore identifyε(Z^d_p) and

ε(Gal(K/K)) := {π: Gal(K/K) ^//// Zp} . This has been used in Lemma 2.7 in order to obtain a bijection

E(K) −→^∼ ε(Gal(K/K)) ;

roughly speaking, each L∈ E(K) corresponds to the restriction map π_L: Gal(K/K)Gal(L/K),

respectively.

Now let Γ and Γ^d be as above. Note that each homomorphism π ∈ ε(Γ^d) defines a homomorphism

π: Zp[[Γ^d]] ^//// Zp[[Γ]]

of the corresponding completed group rings. Let γ1, . . . , γd denote topological generators of Γ^d. Then Theorem 2.18 implies that there exists an isomorphism

ϕ: Zp[[Γ^d]] ^{∼ //} Λ_d = Zp[[T₁, . . . , T_d]]

induced by the mapγ_i7→1 +T_i, 1≤i≤d.

Ifγ₁⁰, . . . , γ_d⁰ is another system of topological generators of Γ^d, then γ_j⁰ =

d

Y

i=1

γ^a_i^i,j , 1≤j≤d ,

and A:= (a_i,j)_i,j ∈GL_d(Zp) is an invertible matrix overZp. The map induced by γ_j⁰ 7→1 +Tj, 1≤j≤d, yields another isomorphism

ϕ⁰ : Zp[[Γ^d]] ^{∼ //} Λ_d , again using Theorem 2.18. The commutative diagram

Λ_d

α

Zp[[Γ^d]]

ϕ ;;

ϕ⁰ ##

Λd

4.2. PROJECTIVE VARIETIES AND THEµ-INVARIANT 143 defines an automorphismα: Λd

−→∼ Λd, given by the substitution

Tj 7→ T_j⁰ :=

d

Y

i=1

(1 +Ti)^ai,j−1, 1≤j≤d .

Definition 4.6. A change of variables in Λd of the above shape is called admissible.

Lemma 4.7 (Baba˘ıcev). Let f(T₁, . . . , T_d) be a formal power series different from zero having coefficients in a commutative ring E of characteristic p6= 0.

Then there exists an admissible change of variables of the form X₁ = (1 +T₁)(1 +T_d)^a1 −1,

...

Xd−1 = (1 +Td−1)(1 +Td)^ad−1 −1, X_d = T_d,

witha₁, . . . , a_d−1 ∈N, under which f is carried to a series g(X₁, . . . , X_d) such thatg(0, . . . ,0, X_d)6= 0. Actually, a1, . . . , ad−1 may be chosen as

a1 = . . . = ad−1 = p^l, withl∈N sufficiently large.

Proof. See [Ba 76], Lemma 1. The proof given there in fact is an adaption of Lemmas 2 and 3 in [Bou 89], Chapter 7,§3, with the additional property that we want the changes of variables to be admissible.

Let π∈ε(Γ^d). If the topological generators of Γ^d are chosen such that the kernel ofπ : Γ^d−→Γ is generated by γ1, . . . , γd−1, and if δ :=π(γ_d), then δ is a topological generator of Γ. The induced homomorphismπ: Λ_d−→Λ is then given by

π(Ti) = π(γi−1) = π(γi)−π(1) = 1−1 = 0 for every 1≤i≤d−1, and

π(T_d) = π(γ_d)−1 = δ−1 = T .

Iff ∈Zp[[Γ^d]]∼=Zp[[T₁, . . . , T_d]], then we simply have π(f) =f(0, . . . ,0, T).

We will now see that for a given π, we may always choose topological gen-erators of Γ^d such thatπ obtains this canonical form.

Remark 4.8. For every π ∈ ε(Γ^d), we may choose topological generators γ1, . . . , γdof Γ^dsuch that the kernel ofπ: Γ^d−→Γ is generated byγ1, . . . , γd−1. Proof. The kernel of π : Γ^d −→ Γ is a Zp-submodule of Γ^d and therefore is Zp-free. Its rank has to be strictly smaller thand, sinceπ is surjective, and in fact, ker(π) hasZp-rank equal tod−1, sinceπ induces an exact sequence

0 ^// Zrank(ker(π))

p // Z^dp // Zp // 0 .

By the Principal Divisor Theorem (see [JS 06], Thm. VII.8.2), there exists a basisγ₁, . . . , γ_dof Γ^d(i.e., a set of topological generators of this multiplicatively written group) such that ker(π) is generated topologically by γ₁^a1, . . . , γ_d−1^ad−1, with a1, . . . , ad−1 ∈ Zp and a1 | a2 | . . . | ad−1. Let z := γ_d−1^ad−1 ∈ ker(π).

Now assume that p |ad−1 inZp. Then we can write z =y^p for some element y∈Γ^dthat does not lie in the kernel of π (since theγ_i^ai form a basis of ker(π), they are linearly independent). But then x := π(y) ∈ Zp is different from 1, and x^p = (π(y))^p = π(y^p) = π(z) = 1, which contradicts the fact that Γ is torsion-free (as being a freeZp-module). Thereforepdoes not divide ad−1, i.e., a₁, . . . , a_d−1 ∈Z^∗_p, and ker(π) is generated byγ₁, . . . , γ_d−1.

Definition 4.9. An element f ∈ Λ_d = Zp[[T₁, . . . , T_d]] is in Weierstraß normal form with respect to Td if

f = U·p^m·(T_d^k+ak−1T_d^k−1+. . .+a0),

wherem∈N0,k∈N,U ∈Λ^∗_dis a unit anda₀, . . . , ak−1 ∈(p, T₁, . . . , Td−1) are contained in the maximal ideal of the local ring Zp[[T₁, . . . , Td−1]]. f is called regular in Td iff is in Weierstraß normal form with respect toTd andm= 0 in the corresponding representation.

Remarks 4.10.

(1) Iff is in Weierstraß normal form with respect toT_d, thenf =U·p^m·f˜(T_d) with a distinguished polynomial

f˜(T_d) ∈ (Zp[[T₁, . . . , Td−1]])[T_d] in the sense of Definition 1.11.

(2) Ifπ∈ε(Γ^d) is a homomorphism such that ker(π) is generated topologically byγ₁, . . . , γd−1, thenδ :=π(γ_d) generates Γ. Iff ∈Λ_dis in Weierstraß nor-mal form with respect toT_din the variablesT₁, . . . , T_dinduced byγ₁, . . . , γ_d, then we can simply write

π(f) = π(U)·p^m·(T^k+ak−1·T^k−1+. . .+a₀),

with a_i = π(a_i) = a_i(0, . . . ,0) ∈ p·Zp, 0 ≤ i ≤ d−1. In particular, π(f)6= 0, and p|π(f) if and only if m >0, i.e., if and only if p|f.

(3) We may apply the Weierstraß Preparation Theorem 1.14 in the ring of power series

Zp[[T₁, . . . , T_d]] ∼= (Zp[[T₁, . . . , Td−1]])[[T_d]], sinceZp[[T1, . . . , Td−1]] is a local ring with maximal ideal

M_d−1 = (p, T1, . . . , Td−1)

which is complete with respect to theM_d−1-adic topology; compare Propo-sition 2.17, (i). This implies that an elementf ∈Λ_dis regular with respect toTd if and only iff 6∈(p, T1, . . . , Td−1)⊆Λd.

4.2. PROJECTIVE VARIETIES AND THEµ-INVARIANT 145 Lemma 4.11 (Baba˘ıcev). Let f ∈Zp[[Γ^d]] be non-zero. LetU ⊆ε(Γ^d) denote the set of homomorphismsπ: Γ^d−→Γ such that

(1) we can choose topological generators γ₁, . . . , γ_d of Γ^d such that ker(π) is generated byγ1, . . . , γd−1, and

(2) f is in Weierstraß normal form with respect to T_d in the variables induced by γ₁, . . . , γ_d via the map γ_i7→1 +T_i, 1≤i≤d.

Then U ⊆ ε(Γ^d) is open and dense in the topology defined on ε(Γ^d) via the bijectionε(Γ^d)−→^∼ ε(Z^dp)−→^∼ P^d−1(Zp) (compare Remarks 2.6, (2)).

Proof. This is basically an application of Lemma 4.7 and Remark 4.8, see Propo-sition 1 in [Ba 76] for details.

Definition 4.12. Let M denote a finitely generated Λ_d-module. For every surjective homomorphismπ: ΛdΛ, we defineMπ :=M/(ker(π)·M); this is a Λ-module, where we identify Λ = Λ_d/ker(π).

Note that this corresponds to the notion XL used by Greenberg (compare the preceding section).

Theorem 4.13 (Baba˘ıcev).

(i) Let M denote a finitely generated Λ_d-module, and let m := rank_Λd(M).

Then the subset

U := {π∈ε(Γ^d)|rankΛ(Mπ) =m} ⊆ ε(Γ^d) is open and dense in ε(Γ^d).

(ii) LetM denote a finitely generatedΛd-module, and assume that there exists a homomorphism π0 ∈ ε(Γ^d) such that Mπ0 is a finitely generated Zp -module. Then the setU ⊆ε(Γ^d) containing all π such that M_π is finitely generated overZp is open and dense in ε(Γ^d).

We recall that the Λd-rank of a finitely generated Λd-module N may be defined via

rank_Λd(N) := dim_Q(N⊗_ΛdQ),

where Q denotes the quotient field of Λd, and dimQ means the dimension as Q-vector space.

Proof. (i) This is Theorem 1 in [Ba 76]. Since Baba˘ıcev only gives a very brief proof, we will include here a proof giving full details.

Since M is a finitely generated Λ_d-module, there exists a surjection F ^// M ^// 0

for some free Λ_d-module F with basis f1, . . . , f_l. Let R ⊆ F denote the kernel of this map. Then R is finitely generated over Λ_d, since F is Noetherian as being finitely generated over the Noetherian ring Λd, and rankΛ_d(M) =m if and only if rankΛ_d(R) =l−m. Indeed, the sequence

0 ^// R ^// F ^// M ^// 0

of Λd-modules is exact by construction, and therefore the following se-quence of Q-vector spaces also is exact:

0 ^// (R⊗Q) ^// (F ⊗Q) ^// (M⊗Q) ^// 0 . (?) Note that in general, tensoring a sequence of Λd-modules with a Λd-module N will be only right-exact (see [JS 06], p. 184 for an example over the ringZ). A Λ_d-moduleN is calledflat if tensoring withN is exact on both sides. In our situation,N =Q= Quot(Λ_d) is equal to the quotient field of Λ_d, and therefore flat by Corollary 3.6 in [AM 69], proving the exactness of the sequence (?). But the dimension of vector spaces is additive on exact sequences, and therefore

rank_Λd(R) + rank_Λd(M) = dim_Q(R⊗Q) + dim_Q(M⊗Q)

= dim_Q(F ⊗Q)

= rankΛ_d(F) = l ,

proving that rankΛd(M) =m if and only if rankΛd(R) =l−m.

Letr1, . . . , rq denote generators of R⊆F. There exist elementsai,j ∈Λ_d such that

r_i =

l

X

j=1

a_ijf_j , 1≤i≤q .

Now the condition rankΛd(R) =l−m is equivalent to the fact that there exists a non-vanishing minor of the matrix (a_ij)_i,j of orderl−m, whereas every minor of order greater thanl−mis zero. Letf ∈Λ_ddenote the non-trivial minor of (aij)i,j. By Lemma 4.11, there exists an open and dense subsetU ⊆ε(Γ^d) such that for everyπ ∈U,π(f) is in Weierstraß normal form, and in particular non-zero. We will show that rank_Λd(M_π) =mfor π∈U, proving (i).

We have a surjection

F_π =F/(ker(π)·F) ^{ψ //} M_π =M/(ker(π)·M) ^// 0

induced by the surjective Λ_d-module homomorphism F −→^ψ M −→ 0, which maps ker(π)·F into ker(π)·M. The map R_π −→^ϕ F_π induced by 0 −→ R −→^ϕ F perhaps is not injective, so we divide out the kernel Xπ

and define a Λ-module ˜R_π :=R_π/X_π. Then the sequence

0 ^// R˜_π ^{ϕ˜ //} F_π ^{ψ //} M_π ^// 0 (??) is exact, where the induced injective map ˜ϕ : ˜R_π −→ F_π is defined via r+ ker(ϕ) 7→ ϕ(r).

Indeed, it remains to show that ker(ψ)⊆im( ˜ϕ). Letf ∈F be such that ψ(f + ker(π)·F)∈ker(π)·M. Write

ψ(f) = X

i

αi·mi

4.2. PROJECTIVE VARIETIES AND THEµ-INVARIANT 147 We know from the first part of the proof that there exists a non-vanishing minorf ∈Λ_dof the matrix (aij)i,j of orderr. The setU ⊆ε(Γ^d) has been π(f)6= 0, this matrix has a non-vanishing minor of order r, proving that rank_Λ(R_π) ≥ rank_Λ(ϕ(R_π)) ≥ r. Let J ⊆ {1, . . . , q} denote the set of indices corresponding to the submatrix of (aij) whose determinant is the minor f.

withβj ∈ker(π)⊆Λd,j= 1, . . . , l. But then 0 =

l

X

j=1

(X

i∈I

λiaij−βj)·fj

and thereforeP

i∈Iλiaij =βj ∈ker(π) for all 1≤j ≤l, recalling that the fj are Λ_d-linear independent.

If I was equal to J, then we would obtain a non-trivial vanishing linear combination of the rows of (aij)i,j∈J in Λ = Λd/ker(π). But this would contradict the fact that

π(f) = det((aij)i,j∈J) 6= 0. Therefore, rank_Λ( ˜R_π)≥ |J|=r.

On the other hand, if rankΛ( ˜Rπ) was strictly larger than r, then there would exist a non-vanishing minor g of the matrix (a_ij) of order greater thanr, since ˜R_π is generated by the cosets ofr₁, . . . , r_q. Sinceπ: Λ_d−→Λ is surjective, we could lift g to a non-vanishing minor of (aij) of order greater than r, in contradiction to the fact that rank_Λd(R) =r.

(ii) Now suppose thatM_π0 is a finitely generatedZp-module. ThenM_π0 is a torsion Λ-module. Greenberg has shown that this happens only if there exists an annihilator f ∈Λ_dof M such that f 6∈ker(π0) (see Lemma 4.2, (i)). Furthermore, we may assume that f is not divisible by p. Indeed, µ(Mπ0) = 0 by Proposition 1.31, (iii). Therefore the characteristic poly-nomial g(T) ∈ Zp[T]⊆ Λ of Mπ0 is not divisible by p. g(T) annihilates the elementary Λ-module E_Mπ

0. Since the finite kernel of the pseudo-isomorphismMπ0

−→∼ EMπ0 may be annihilated by an appropriate power ofT, by Nakayama’s Lemma (compare Remark 3.49), we may augmentg in order to obtain an annihilatorg of Mπ0 that is still not divisible by p.

Using the arguments from the proof of Lemma 4.2, (i), it follows thatM is a torsion Λ_d-module, and that there exists an annihilator f ∈Λ_dof M such that

f ≡ g^l mod (ker(π0)),

where l denotes the number of generators of the finitely generated Λ_d -module M. In particular, p-f.

Since there exists a surjective homomorphism (Λ_d/(f))^l −→ M, it will suffice to prove assertion (ii) for the moduleN := Λ_d/(f). By Lemma 4.11 and Remarks 4.10, (2), there exists an open and dense subsetU ⊆ε(Γ^d) such that for every π∈U, the imageπ(f) =u·f˜is the product of a unit u∈Zp[[T]]^∗= Λ^∗ and a distinguished polynomial ˜f ∈Zp[T]. Therefore

Nπ = Λ/(π(f)) = Λ/( ˜f) ∼= Z^{deg( ˜}p ^f)

is finitely generated overZp for everyπ ∈U.

Using this theorem, Baba˘ıcev proved his first result concerning the Iwasawa µ-invariant in Zp-extensions of K. In order to apply the theory developed so

4.2. PROJECTIVE VARIETIES AND THEµ-INVARIANT 149 far, we let Γ = Γ¹ := Zp and Γ^d := Gal(K/K) ∼= Z^dp, where K denotes the composite of allZp-extensions ofK, as usual. Then the study of surjectiveZp -module homomorphisms π ∈ε(Γ^d) corresponds to the study of Zp-extensions ofK (compare Lemma 2.7).

Definition 4.14. Let c∈N0. Define E^µ>c(K) to be the set ofZp-extensions L/K satisfying µ(L/K) > c. Furthermore, let E⁰(K) denote the set of Zp -extensionsL/K such that µ(L/K) = 0.

Theorem 4.15(Baba˘ıcev). If there exists a Zp-extensionL∈ E⁰(K)such that only finitely many primes ofLlie overp, then the subsetE⁰(K)ofE(K)is open and dense.

Proof. This is Theorem 4 in [Ba 76]. Whereas the proof given there uses coho-mology theory, we will use more elementary arguments. LetY := Gal(H(K)/K) denote the Galois group of the maximal p-abelian unramified extension of K. Then Y is a finitely generated torsion Λd-module (compare Theorem 1 in [Gr 73]). Furthermore, if π denotes the surjective homomorphism corre-sponding to L/K via Lemma 2.7, then our assumptions on L imply that Yπ := Y /(ker(π)·Y) is a finitely generated Zp-module (compare Lemma 4.3, (i)).

Theorem 4.13, (ii) implies that there exists an open and dense subsetU of ε(Gal(K/K)) such that Yπ is a finitely generated Zp-module for every π∈U.

We will now make use of the following fact.

Lemma 4.16.For everyZp-extensionL/K, and corresponding homomorphism π∈ε(Gal(K/K)), we have an exact sequence

Y_π ^// X_π ^// Gal((H(L)∩K)/L)

of Λ-modules, where X_π := Gal(H(L)/L) denotes the Galois group of the max-imalp-abelian unramified extensionH(L) of L, and Yπ is defined as above.

Proof. Let F denote the subfield of H(K) fixed by ker(π) ·Y ⊆ Y. Thus, Gal(H(K)/F) = ker(π)·Y and Gal(F/K)∼=Y_π =Y /(ker(π)·Y). Assume that we have chosen a set of topological generatorsγ1, . . . , γ_dof Gal(K/K) such that the kernel ofπ∈ε(Gal(K/K)) is generated by γ₁, . . . , γd−1.

Claim 4.17. The maximal p-abelian unramified extension H(L) of L is con-tained in F.

Proof. Since ker(π)⊆Λ_dis generated by T₁, . . . , Td−1, the subfield F of H(K) is fixed by< T1, . . . , Td−1>·Y.

Note thatH(K) actually is Galois over L. Since

< T1, . . . , Td−1>·Y ⊆ Gal(H(K)/L)

is a closed subgroup, it follows thatF ⊆H(K) is also Galois overL. We claim thatF is the maximal subextension that is abelian overL; this relies on the fact that< T₁, . . . , Td−1>·Y corresponds to the topological commutator subgroup of Gal(H(K)/L), as we will see in Chapter 5 (compare Lemma 5.19).

Indeed, Gal(K/L) acts on Gal(H(K)/K) =Y via conjugation. Since γ_i·σ·γ_i⁻¹·σ⁻¹(x) = ((γ_i−1)·σ)(x) = (T_i·σ)(x) = x

for everyσ ∈Y,x∈F,i= 1, . . . , d−1, it follows thatγ_i·σ·γ_i⁻¹(x) =σ(x) for each σ ∈Y,x∈F, i.e., γi·σ·γ_i⁻¹ =σ for every σ ∈Gal(F/K), proving that F/L is abelian.

Conversely, if M ⊆H(K) is abelian overL, then γ_i·σ·γ⁻¹_i =σ for every σ ∈ Gal(M/K) and every i ∈ {1, . . . , d−1}, since Gal(K/L) is generated by γ1, . . . , γd−1. But then M ⊆H(K)^{<T1,...,Td−1>·Y} =F.

Since H(L) is abelian over L, and H(L) ⊆ H(K) (compare Proposition 1.34), it is now immediate that H(L)⊆F.

Now let X_π := Gal(H(L)/L). Then H(L) := H(L) ·K ⊆ F. We let X_π := Gal(H(L)/K) and summarise our situation in the following diagram:

H(K)

F

ker(π)·Y

K

Y

Yπ

Xπ

H(L)

L Xπ H(L)

K

Since it is possible that K∩H(L) %L, we may not conclude that X_π ∼=X_π. However, since H(L)⊆F, we have a surjective map

Yπ //// Xπ , σ ^// σ+ Gal(F/H(L)) =: σ .

Furthermore, since Xπ = Gal(H(L)/K) ∼= Gal(H(L)/(K∩H(L))), we obtain a map Xπ −→ Xπ, induced by restriction to H(L). Note that this latter map will not be surjective wheneverH(L)∩K6=L. However, the cokernel will yield us the desired exact sequence:

First note that the composite

i:Yπ // Xπ // Xπ , σ ^// σ ^// σ|_H(L) ,

is well-defined since every element in Gal(F/H(L)) fixes H(L)⊆H(L). Let j:X_π = Gal(H(L)/L) ^// Gal((H(L)∩K)/L), τ ^// τ|_(H(L)∩K) ,

4.2. PROJECTIVE VARIETIES AND THEµ-INVARIANT 151 denote the canonical restriction map. Then the sequence

Yπ i // Xπ

j // Gal((H(L)∩K)/L)

is exact. First of all, it is clear that the image of i is contained in ker(j), sinceσ|_(H(L)∩K) = id∈ Gal((H(L)∩K)/L) for every σ ∈ Gal(H(L)/K). On the other hand, if τ|_(H(L)∩K) = id for some τ ∈ Xπ = Gal(H(L)/L), then τ ∈Gal(H(L)/(H(L)∩K)), and therefore τ is the restriction toH(L) of some element in Gal(H(L)/K) = X_π. Since Y_π X_π is surjective, we obtain that τ =i(σ) for a suitable σ.

This shows that Xπ is a finitely generated Zp-module if Yπ is finitely gen-erated over Zp, since rank_Zp(Gal((K ∩ H(L))/L)) ≤ d−1. In particular, X_π is finitely generated over Zp and therefore Λ-torsion for every π ∈U, i.e., U ⊆ E(X) in the notation of Section 4.1. The assertion of Theorem 4.15 now follows from Corollary 4.5, (ii) and Proposition 1.31, (iii).

Im Dokument A new approach to the investigation of Iwasawa invariants (Seite 153-163)