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On the Iwasawa [mu]-invariants of branched [Z.sub.p]-covers.

1. Introduction. In this article, following the analogies between knots and primes ([Mor12]), we establish relative genus theory for a branched cover of rational homology 3-spheres ([QHS.sup.3]). Then we formulate analogues of Iwasawa's theorems on [mu]-invariants ([Iwa73]) in 3-dimensional topology by using relative genus theory.

Genus theory for number fields was first studied for quadratic, abelian, and Galois extensions over Q by Hasse, Iyanaga Tamagawa and Leopoldt, and Frohlich. The case over a general number field k was formulated by Furuta in [Fur67] and is called relative genus theory. A role of the co-invariant group was also discussed in [Yok67]. Genus theory for 3-manifolds, on the other hand, was formulated in [Mor01] and [Mor12] for the cyclic case over an integral homology 3-sphere ([ZHS.sup.3]), and was also discussed in [Uek14]. In this paper, we generalize these results and establish relative genus theory for a branched Galois cover of oriented, connected, and closed 3-manifolds. In addition, by employing Niibo's idele ([Nii14], [NU]), we give an alternative proof which is parallel to the one for number fields.

Next, we recall Iwasawa theory. Let p be a prime number and let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] denote the ring of p-adic integers. A field [k.sub.[infinity]] obtained as a [Z.sub.p]-extension of a number field is called a [Z.sub.p]-field. Such [k.sub.[infinity]] is a limit of cyclic extensions [k.sub.n]/k of degree [p.sup.n]. As an analogue of [Z.sub.p]-extension, we consider an inverse system of cyclic branched p-covers [h.sub.n]: [M.sub.n] [right arrow] M of [QHS.sup.3] which are branched over a link L in M, and call it a branched [Z.sub.p]-cover. For these objects, the Iwasawa invariants [lambda], [mu], v are defined and studied ([Iwa59], [HMM06], [KM08], [KM13], [Uek]), and they describe the behaviors of the orders of p-parts of the ideal class groups Cl([k.sub.n]) and [H.sub.1]([M.sub.n]). Moreover, as an analogue of an extension of [Z.sub.p]-fields, the notion of a morphism (branched Galois cover) of branched [Z.sub.p]-covers was introduced in [Uek]. It is a compatible system of branched covers on each layer.

In [Iwa73], Iwasawa studied the behavior of the [mu]-invariants in a p-extension of [Z.sub.p]-fields, by employing relative genus theory. He gave a construction of a [Z.sub.p]-extension with arbitrary large p, and also proved that there are infinitely many [Z.sub.p]-fields with [mu] = 0. We formulate their analogues.

Notation. For a group G and a G-module A, let [A.sup.G] and [A.sub.g] = A/[I.sub.G]A denote the G-invariant subgroup and the G-co-invariant quotient, where [I.sub.G] = (g - 1 | g [member of] G) < Z[G] is the augmentation ideal. If G is a finite cyclic group, then #[A.sup.G] = #[A.sub.G] holds.

2. Relative genus theory for number fields. First, we recall the case for number fields. We assume that algebraic extensions of Q are contained in C, and number fields are finite over Q.

Definition 2.1. Let k'/k be an abelian extension of number fields. The relative genus field [k'.sup.g] of k'/k is the maximal unramified extension of k' abelian over k. The degree [g.sub.k']/k = ([k'.sup.g] : k') is called the relative genus number.

In addition, for a Galois extension k'/k, we define the same notions by considering the maximal unramified extension [k'.sup.g] of k' which is obtained as a composite of k' and an abelian extension of k instead.

Theorem 2.2 ([Fur67]). Let k'/k be a finite Galois extension of a number field, and let [k'.sub.0]/k denote its maximal abelian subextension. Then,

[g.sub.k']/k = #Cl(k) [[product].sub.p] [e'.sub.p]

where p runs through all the primes of k, [e'.sub.p] denotes the ramification index of the maximal abelian subextension of [K.sub.B]/[k.sub.p] for a prime B of K dividing p, [epsilon] the unit group of k, and [eta] the group of elements in [epsilon] everywhere locally norm.

Theorem 2.3 ([Yok67, Proposition 1]). Let k'/k be a cyclic extension of number fields with G = Gal (k'/k) = <[sigma]>. Then [g.sub.k']/k = # [Cl(k').sub.G] = #([Cl(k')/Cl(k').sup.1-[sigma]]) = #[Cl(k').sup.G] holds.

By combining these two theorems, we can estimate the increase of class numbers in extensions.

3. Relative genus theory for rational homology 3-spheres. In this section, we formulate analogues of the two theorems in [section]2. They generalize the results of [Mor01] and [Mor12] originally for a branched cyclic cover over a [ZHS.sup.3]. We also give an alternative proof by employing Niibo's idele.

In the following, we assume that 3-manifolds are oriented, connected, and closed, and that branched covers of 3-manifolds are branched over links and are equipped with base points. In order to discuss analogues of class numbers, we sometimes assume that 3-manifolds are [QHS.sup.3]'s. A 3-manifold M is a [QHS.sup.3] if and only if [H.sub.1](M) < [infinity].

Definition 3.1. For a finite branched abelian cover h : N [right arrow] M of 3-manifolds, the relative genus cover of h is the maximal unbranched cover [N.sup.g] [right arrow] N abelian over M, and [g.sub.h] := deg([N.sup.g] [right arrow] N) [member of] N [union] {[infinity]} is called the relative genus number.

In addition, for a finite branched Galois cover h : N [right arrow] M, we define the same notions by considering the maximal unbranched cover [N.sup.g] [right arrow] N obtained as a composite (in the sense of Galois theory) of h and a branched abelian cover of M instead.

Now the first theorem is presented as follows:

Theorem 3.2. Let h : N [right arrow] M be a finite branched Galois cover of 3-manifolds branched over L = [??][K.sub.i], and let [h.sub.0] : [N.sub.0] [right arrow] M denote the maximal abelian subcover of h. Then the branch indices [e.sub.i] of [K.sub.i] in h satisfy

[g.sub.h] = #[H.sub.1](M) [[product].sub.i] [e.sub.i]/deg([h.sub.0])

Proof. Let [([N.sup.g]).sub.0] [right arrow] [N.sub.0] [right arrow] M denote the maximal subcovers of [N.sup.g] [right arrow] N [right arrow] M abelian over M. Then [g.sub.h] = deg([([N.sup.g]).sub.0] [right arrow] [N.sub.0]). Indeed, let [Y.sup.g] [right arrow] Y [right arrow] X and [([Y.sup.g]).sub.0] [right arrow] [Y.sub.0] [right arrow] X denote their restrictions to the exteriors of the branch links, let D([[pi].sub.1](X)) denote the commutator group of [[pi].sub.1](X), and put A := Ker([[pi].sub.1](Y) [right arrow] [[pi].sub.1](N)). Then by definition, [[pi].sub.1]([Y.sup.g]) is the smallest subgroup of [[pi].sub.1](X) satisfying [[pi].sub.1]([Y.sup.g]) > A and [[pi].sub.1]([Y.sup.g]) = [[pi].sub.1](Y) [intersection] P for some P > D([[pi].sub.1](X)). Thus [[pi].sub.1]([Y.sup.g]) = [[pi].sub.1](Y) [intersection] (D([[pi].sub.1] (X)) x A), [[pi].sub.1]([Y.sub.0]) = [[pi].sub.1](Y) x D([[pi].sub.1](X)), and [[pi].sub.1]([([Y.sup.g]).sub.0]) = [[pi].sub.1]([Y.sup.g]) x D([[pi].sub.1](X)). Hence [[pi].sub.1] ([Y.sup.g]) = [[pi].sub.1]([([Y.sup.g]).sub.0]) [intersection] [[pi].sub.1](Y), [[pi].sub.1]([Y.sub.0]) = [[pi].sub.1](Y) x [[pi].sub.1]([([Y.sup.g]).sub.0]) and Gal([([N.sup.g]).sub.0] [right arrow] [N.sub.0]) = [[pi].sub.1]([Y.sub.0])/[[pi].sub.1]([([Y.sup.g]).sub.o]) [congruent to] [[pi].sub.1](Y)/ [[pi].sub.1]([Y.sup.g]) = [g.sub.h].

The set of meridians of [h.sup.-1](L) generates B := Ker([H.sub.1](Y) [??] [H.sub.1](N)). By the definition of the relative genus cover, the covers [([N.sup.g]).sub.0] [right arrow] [N.sub.0] [right arrow] M correspond to the subgroups [h.sub.*](B) < [h.sub.*]([H.sub.1](Y)) < [H.sub.1](X). Since Gal([h.sub.0]) [congruent to] [H.sub.1](X)/[h.sub.*]([H.sub.1](Y)), we have Gal([N.sup.g]/N) [congruent to] [h.sub.*]([H.sub.1](Y))/[h.sub.*](B) and [g.sub.h] = #([h.sub.*]([H.sub.1](Y))/[h.sub.*](B)) = #([H.sub.1](X)/[h.sub.*](B))/deg([h.sub.0]).

Now suppose that L is a t-component link, and let ([[mu].sub.L]) < [H.sub.1](X) denote the meridian group. If M is not a [QHS.sup.3], then the formula is clear by a surjection [H.sub.1](X)/[h.sub.*](B) [??] [H.sub.1](M). If M is a [QHS.sup.3], then the Mayer Vietoris long exact sequence yields the exact sequence 0 [right arrow] ([[mu].sub.L]) [right arrow] [H.sub.1](X) [right arrow] [H.sub.1](M) [right arrow] 0. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] denote the tubular neighborhood of [K.sub.i]. Then [[pi].sub.1]([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) [congruent to] [Z.sup.2] is abelian, and so is the decomposition group. Since [[product].sub.i] [e.sub.i] Z [congruent to] [h.sub.*](B) < <[[mu].sub.L]> [congruent to] [Z.sup.t], we have an exact sequence 0 [right arrow] <[[mu].sub.L]>/[h.sub.*](B)[right arrow] [H.sub.1](X)/[h.sub.*](B) [right arrow] [H.sub.1](M) [right arrow] 0 with <[[mu].sub.L]>/[h.sub.*](B) [congruent to] [[product].sub.i], Z/[e.sub.i]Z. Hence [g.sub.h] deg([h.sub.0])= #([H.sub.1](X)/[h.sub.*](B)) = #[H.sub.1](M) [[product].sub.i][e.sub.i], and the assertion holds.

Corollary 3.3. Let h : N [right arrow] M be a finite branched Galois cover of 3-manifolds. Then M is a [QHS.sup.3] if and only if [g.sub.h] is finite.

Proof. If M is not a [QHS.sup.3], then #[H.sub.1](M) = [infinity] and so is [g.sub.h].If M is a [QHS.sup.3], then by Theorem 3.2, [g.sub.h] < [infinity] (while N is not necessarily a [QHS.sup.3]).

Corollary-Definition 3.4. Let h : N [right arrow] M be a finite branched Galois cover of 3-manifolds. If the branch link L consists of null-homologous components, then there are a natural splitting [H.sub.1](X) [congruent to] [H.sub.1](M) [direct sum] <[[mu].sub.L]) ([Uek, Lemma 4.4]) and a well-defined homomorphism [PHI] : [H.sub.1](N) [right arrow] [H.sub.1](X)/[h.sub.*](B) [??] [H.sub.1](M) [direct sum] [[product].sub.i], [e.sub.i]Z/[e.sub.i]Z with [PHI]([c]) = ([h(c)], (lk(c, [K.sub.i]) mod [[e.sub.i]).sub.i]) for any c [member of] Hom([S.sup.1],N). We say that a, b [member of] [H.sub.1](N) belong to the same genus over M if [PHI](a) = [PHI](b). This generalizes the notion of genus over [S.sup.3] ([Mor12, Chapter 6.2]).

Remark. Let h : N [right arrow] M be a finite branched Galois cover of [QHS.sup.3]. Then, since [N.sup.g] [right arrow] N is unbranched and abelian, gh|#[H.sub.1](N) holds. This fact will be used in the study of Iwasawa invariants in [section]5.

Remark. Let h : N [right arrow] M be a finite cyclic branched cover of [QHS.sup.3] with G = Gal(h), and fix finite CW-structures on them compatible with h. Then [Uek14, Proposition 16] states that gh = #[H.sub.1][(N).sup.G] = [gamma] [[product].sub.i][Q.sub.i] [e.sub.i])#[H.sub.1](M), [gamma] = #[[??].sup.0](G, [Z.sub.2](N))/ #[[??].sup.1](G, [Z.sub.2](N)). By the theorem above, we obtain [gamma] = 1/deg(h).

Next, we study the relation between the coinvariant group and the relative genus number.

For the trivial action of a group G on Z, we write [H.sub.i](G): = [H.sub.i](G, Z). We have [H.sub.1](G) = [G.sup.ab]. If G is finite, then [H.sub.2](G) is finite, and if G is cyclic in addition, then [H.sub.2](G) = 0. Further, for a path-connected space X, we have [H.sub.2]([[pi]1 (X)) = Coker[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (Hopf's theorem, [Bro94]). Now we have

Theorem 3.5. Let h : N [right arrow] M be a finite branched Galois cover over a [QHS.sup.3] with G = Gal(h), and put b := ([h.sub.0*] (B0) : [h.sub.*](B)) with the notation of Theorem 3.2. Then [g.sub.h] = #([H.sub.1][(N).sub.G]/[bar.[delta]]([H.sub.2](G)))/[sup.b] for some map [bar.[delta]] with 1 [less than or equal to] b [less than or equal to] deg(N [right arrow] [N.sub.0]).

If h is abelian, then b = 1. If G = <[sigma]>, then [g.sub.h] = #[H.sub.1][(N).sub.G] = #[H.sub.1](N)/(1 - [sigma])[H.sub.1](N) = #[H.sub.1][(N).sup.G].

Proof. By the Hochschild-Serre spectral sequence ([Bro94, VII-6]), the short exact sequence [H.sub.1] [right arrow] [[pi].sub.1] (Y) [right arrow] [[pi].sub.1] (X) [right arrow] G [right arrow] 1 yields an exact sequence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. By the Hurewicz isomorphism [[pi].sub.1][(X).sup.ab] [congruent to] [H.sub.1](X), we have an exact sequence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Since [h.sub.*](Iq[H.sub.1](Y)) = 0, we have an exact sequence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since [h.sub.*](IGB) = 0, there is an induced surjection [h.sub.*]: [B.sub.G] [??] [h.sub.*](B). Since [( ).sub.G] = [H.sub.0]( ), an exact sequence 0 [right arrow] B [right arrow] H1(Y) [right arrow] H1(N) [right arrow] 0 yields an exact sequence ... [right arrow] [B.sub.G] [right arrow] [H.sub.1] [(Y).sub.G] [right arrow] [H.sub.1][(N).sub.G] [right arrow] 0. Thus we have a commutative diagram

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

consisting of exact sequences. Let [bar.[delta]]: [H.sub.2](G) [right arrow] [H.sub.1][(N).sub.G] denote the natural map. Then by the snake lemma, we have [bar.[delta]]([H.sub.2](G)) = Ker([H.sub.1][(N).sub.G] [right arrow] [h.sub.*]([H.sub.1](Y))/[h.sub.*](B)). Hence by Theorem 3.2, we have #[H.sub.1][(N).sub.G]/ [bar.[delta]]([H.sub.2](G)) = #[h.sub.*]([H.sub.1](Y))/[h.sub.*](B) = [g.sub.h]b.

Now h': N [right arrow] [N.sub.0] satisfies [h.sup'.sub.*](B) < Ker([B.sub.0] [??] [H.sub.1]([Y.sub.0]) [??] [H.sub.1] ([Y.sub.0])/[h.sup.'.sub.*] ([H.sup.1](Y))), and hence b [less than or equal to] #[B.sub.0]/ [h.sup.'.sub.*](B) [less than or equal to] #[H.sub.1]([Y.sub.0])/[h.sup.'.sub.*]([H.sub.1](Y)) [less than or equal to] deg(h').

Finally, we give an alternative proof of Theorem 3.2, which is rather parallel to the one in [Fur67], by employing Niibo's idele. This idele theory was initially introduced by Niibo in [Nii14], and was modified and generalized in [NU]. We first recall definitions and results in [NU]. Let M be a 3-manifold, and let K [subset] M be an infinite link equipped with a tubular neighborhood [V.sub.K] = [[??].sub.K[subset]K][V.sub.K]. Let [A.sub.M,K] denote the set of all the abelian covers of M branched over a finite sublink of K. We call K a very admissible link of M if for any h : N [right arrow] M [member of] [A.sub.M,K], [H.sub.1](N) is generated by components of the preimage [h.sup.-1](K). For any link L consisting of countably many tame components in a 3-manifold M, there exists such a K including L ([NU, Theorem 2.3]). Let (M, K) be such a pair. Then the idele group [I.sub.M] = [[product].sub.K[subset]K] [H.sub.1]([partial derivative][V.sub.K]) is defined as the restricted product with respect to the meridian subgroups [??][mu]K[??] < [H.sub.1]([partial derivative][V.sub.K]). In other words, we put [I.sub.M]: = {([a.sub.K]) [member of] [[product].sub.K[subset]K] [H.sub.1]([partial derivative][Y.sub.K]) | [u.sub.K] ([a.sub.K]) = 0 for almost all K}, where [u.sub.K]: [H.sub.1]([partial derivative][V.sub.K]) [right arrow] [H.sub.1]([V.sub.K]) is a natural map. The principle idele group is defined by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where L runs through all the finite sublinks of K. The unit idele group [U.sub.M] < [I.sub.M] is the subgroup consisting of formal infinite sums of meridians [[summation].sub.K[subset]K]: [a.sub.K][[mu].sub.K]([a.sub.K] [member of] Z). By [NU, Lemma 5.7], we have [I.sub.M]/([P.sub.M] + [U.sub.M]) [congruent to] [H.sub.1](M). By [NU, Theorem 7.4] (the existence theorem), there is a natural bijective correspondence between certain subgroups [P.sub.M] < H < [I.sub.M] and h's. For each h : N [right arrow] M [member of] [A.sub.M,K] and a very admissible link [h.sup.-1] (K) of N, we define [I.sub.N], [P.sub.N], [U.sub.N]. By [NU, Theorem 5.5] (the global reciprocity law), there is a natural isomorphism [I.sub.M]/([P.sub.M] + [h.sub.*]([I.sub.N])) [congruent to] Gal(h).

An alternative proof of Theorem 3.2. We fix a very admissible link K of M containing L = [[??].sub.i][K.sub.i]. Since [H.sup.*]:= [P.sub.M] + [h.sub.*] ([P.sub.N] + [U.sub.N]) < [I.sub.M] corresponds to [([N.sup.g]).sub.0] [right arrow] M, we have deg([h.sub.0])[g.sub.h] = [[I.sub.M] : [H.sup.*]] = [[I.sub.M] : [P.sub.M] + [U.sub.M]] x [[P.sub.M] + [U.sub.M] : [P.sub.M] + [h.sub.*] ([P.sub.N] + [U.sub.N])] = #[H.sub.1] (M) x [[P.sub.M] + [U.sub.M] : [P.sub.M] + [h.sub.*] ([U.sub.N])] = #[H.sub.1](M)[[U.sub.M] : [h.sub.*]([U.sub.N])]/[[P.sub.M] [intersection] [U.sub.M] : [P.sub.M] [intersection] [h.sub.*] ([U.sub.N])]. If M is not a [QHS.sup.3], then the formula is clear. If M is a [QHS.sup.3], then exact sequences 0 [right arrow] ([[mu].sub.L']) [right arrow] [H.sub.1](M - L') [right arrow] [H.sub.1(M)] [right arrow] 0 for L' C K yield a natural injection [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Gal(h). Therefore [P.sub.M] [intersection] [U.sub.M] = 0 and the denominator is 1. Since [U.sub.M]/h*([U.sub.N]) = [congruent to] [PI]/[e.sub.i]Z, we have deg([h.sub.0])gh = #[H.sub.1](M) [[PI].sub.i][e.sub.i]

4. Iwasawa [mu]-invariants of [Z.sub.p]-fields. Let p be a prime number. In this section, we recall the theorems on the [mu]-invariants in p-extensions of [Z.sub.p]-fields given by Iwasawa in [Iwa73] with use of relative genus theory. We also refer to [Och14] for the details.

Let [v.sub.p] denote the p-adic valuation. Let [k.sub.[infinity]/k be a [Z.sub.p]-extension of a number field, and let [k.sub.n]/k denote the subextension of degree [p.sup.n] for each n. Then we have the Iwasawa class number formula ([Iwa59]):

[v.sub.p] (#Cl([k.sub.n])) = [lambda]n + [mu][p.sup.n] + v for n >> 0, for some constants [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] called the Iwasawa invariants. The value of [lambda] and whether [mu] = 0 or not depend only on [k.sub.[infinity]], and are independent of the choice of k.

There is a unique [Z.sub.p]-extension of [Q.sub.[infinity]] of Q. For a number field k, the composite [[k.sup.c.sub.[infinity]] = [kQ.sub.[infinity]] is called a cyclotomic [Z.sub.p]-field. In a cyclotomic [Z.sub.p]-extension [[k.sup.c.sub.[infinity]]/k, every non-p prime decomposes finitely. On the other hand, a number field k is called a CM-field if it is a totally imaginary quadratic extension of a totally real field [k.sup.+]. Such k has a [Z.sub.p]-extension K/k which is a limit of dihedral extensions of [k.sup.+], and every prime is inert in k/[k.sup.+] decomposes completely in K/k ([Iwa73]). (If p > 2, then a CM-field k has the anti-cyclotomic [Z.sup.[k:Q]/2].sub.p]-extension [[k.sup.ac.sub.[infinity]]/k, whose any sub-[Z.sub.p]-extension K/k is dihedral over [k.sup.+].)

Iwasawa conjectured that [mu] = 0 holds for every cyclotomic [Z.sub.p]-extension [[k.sup.ac.sub.[infinity]]/k, and it is true by Ferrero-Washington [FW79] if k is abelian over Q. If [mu] = 0, then the nature of a [Z.sub.p]-field "resembles" that of a function field. For general cases, however, there exist [Z.sub.p]-extensions with arbitrary large [mu]:

Theorem 4.1 ([Iwa73, [section]1]). Let k/Q be an extension of degree d containing primitive p-th roots of unity, and let [k[infinity]]/k be a [Z.sub.p]-extension. Suppose that there exist primes [p.sub.1],... ,[p.sub.t] in k which are completely decomposed in kM/k, and let k' = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [k'.sub.[infinity]] = k'k.sub.[infinity]. Then, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] holds.

Let k be the 4-th or p > 2-th cyclotomic field. Then k is a CM-field, and there is a [Z.sub.p]-extension K/k dihedral over [k.sup.+]. Since there are infinitely many primes inert in k/[k.sup.+], there are infinitely many primes completely decomposed in K/k. Therefore by the previous theorem, we obtain the following

Theorem 4.2 ([Iwa73, Theorem 1]). Let k be the cyclotomic field of p-th or 4-th roots of unity according as p > 2 or p = 2. Then, for any N [member of] N, there exist an extension k'/k of degree p and a [Z.sub.p]-extension [k'.sub.[infinity]/k'] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

On the other hand, the following tells that there are many [Z.sub.p]-fields with [mu] = 0.

Theorem 4.3 ([Iwa73, Theorem 2]). Let k be a number field (totally imaginary if p = 2), [k[infinity]/k a [Z.sub.p]-extension, k'/k a finite Galois p-extension, and put [k'.sub.[infinity]] = [k.sub.[infinity]k'. Suppose that every prime of k which is ramified in k'/k is finitely decomposed in [k.sub.[infinity]]/k. Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] if and only if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

5. Iwasawa [mu]-invariants of branched [Z.sub.p]-covers. In this section, we formulate analogues of Iwasawa's results recalled in the previous section.

Let L [subset] M be a link in [QHS.sup.3]. We call an inverse system of L-branched Z/[p.sup.n]Z-covers [??] = [{[h.sub.n] : [M.sub.n] [right arrow] M}.sub.n] a branched [Z.sub.p]-cover, and regard it as an analogue of a [Z.sub.p]-field. Put X := M - L. Then a surjective homomorphism from the pro-p completion of the fundamental group [tau] : [??] (X) [??] [Z.sub.p] corresponds to such [??]. Assume that [M.sub.n] is a [QHS.sup.3] for any n. Then we have an Iwasawa type formula ([HMM06], [KM08], [Uek]):

[v.sub.p] ([H.sub.1] ([M.sub.n])) = [lambda]n + [mu][p.sup.n] + v for n [??] 0,

for some [lambda] = [[lambda].sub.[??]], [mu] = [[lambda].sub.[??]] [member of] N, v = [v.sub.[??] [member of] Z. These constants are called the Iwasawa invariants.

Next, we review an analogous object of an extension of [Z.sub.p]-fields introduced in [Uek]. Let [??] = [{[h.sub.n] : [M.sub.n] [right arrow] M}.sub.n] be an L'-branched [Z.sub.p]-cover and M' = [{[h'.sub.n] : [M'.sub.n] M'}.sub.n] an L'-branched [Z.sub.p]-cover. Then a branched Galois cover f : [??] [right arrow] M of degree r is a compatible system of branched Galois covers [{[f.sub.n] : [M'.sub.n] [right arrow] [M.sub.n]}.sub.n] of degree r such that each induced map Gal([f.sub.n+1]) [right arrow] Gal([f.sub.n]) is an isomorphism. If L and L' are properly branched in [??] and [??], then L' = [f.sup.-1.sub.0](L). We can easily see that the branch links [S.sub.n] of [f.sub.n] satisify [S.sub.n] [subset] [h.sup.-1.sub.n] ([S.sub.0]). We put S' := [f.sup.-1.sub.0](S), Y := M - L [union] S, and Y' := M' - L' [union] S'. Then, there is a commutative diagram

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for the defining homomorphisms [tau], [tau]' of [??], [??]'. Conversely, if [f.sub.0] : M' [right arrow] M and such a diagram are given, then [??]' [right arrow] [??] is defined.

Let M be a [QHS.sub.3], let L = [union][K.sub.i] be a link which consists of null-homologous components in M, and let [[mu].sub.i] denote the meridian of [K.sub.i] for each i. Then, the branched [Z.sub.p]-cover [??] defined by [tau] : [[pi].sub.1] (M - L) [right arrow] Z; [for all][[mu].sub.i] [right arrow] 1 is called the total linking number (or TLN for short) [Z.sub.p]-cover over (M, L).

Let [summation] be a Seifert surface of L, that is, a compact orientable surface [summation] satisfying [partial derivative][summation] = L. Then M - [summation] gives a fundamental domain of each Z/[p.sup.n] Z-cover. Let K [subset] M - L be a knot, and assume that K and [summation] intersect transversally (perturb [summation] if necessary). Then the intersection number [iota] satisfies lk(K, L) = [iota](K, [summation]). By a standard argument similar to [Mor12, Chapter 4.1], the natural map [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] sends [K] to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Therefore, if lk(K, L) [not equal to] 0 and n [greater than or equal to] [[upsilon].sub.p](lk(K, L)), then each component of [h.sup.-1.sub.n] (K) consists of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] copies of K, and [h.sup.-1.sub.n] (K) is a [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]-component link. Otherwise, [h.sup.-1.sub.n](K) is a [p.sup.n]-component link. In particular, we have the following

Proposition 5.1. Let [??] = [{[h.sub.n] : [M.sub.n] [right arrow] M}.sub.n] be the TLN-[Z.sub.p]-cover over (M, L), and K [subset] M - L a knot. Then, lk(K, L) [not equal to] 0 holds if and only if K is finitely decomposed into [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] components in [h.sub.n] for all n [much greater than] 0, and lk(K, L) = 0 holds if and only if K is completely decomposed in all [h.sub.n].

If S is finitely decomposed in [??], then f : [??]' [right arrow] [??] resembles a p-extension of cyclotomic [Z.sub.p]-field. If S is completely decomposed in [??], then f : [??]' [right arrow] [??] resembles the case of anti-cyclotomic.

Now we present our main theorems.

Theorem 5.2 (arbitrary large [mu]). Let f : [??]' [right arrow] [??] be a branched Galois cover of degree p of [Z.sub.p]-covers of [QHS.sub.3]. Suppose that the branch link S of [f.sub.0] : M' [right arrow] M is a t-component link, and that S is completely decomposed in [??]. Then, [[mu].sub.[??]'] [greater than or equal to] t holds.

Proof. Since [f.sub.n] : [M'.sub.n] [right arrow] [M.sub.n] is of degree p and the branch link [S.sub.n] of [f.sub.n] is a [tp.sup.n]-component link [h.sup.-1.sub.n] (S), by Theorem 3.2, we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then, #[H.sub.1][([M'.sub.n]).sub.G]|#[H.sub.1]([M'.sub.n]) implies [[upsilon].sub.p] (#[H.sub.1]([M'.sub.n])) [greater than or equal to] [[upsilon].sub.p](#[H.sub.1]([M.sub.n])) + [tp.sup.n] - 1, and hence [[mu].sub.[??]'] [greater than or equal to] t.

Theorem 5.3 (many [mu] = 0). Let f : [??]' [right arrow] [??] be a branched Galois p-cover of branched [Z.sub.p]-covers of [QHS.sub.3], and suppose that any knot branched in [f.sub.0] : M' [right arrow] M is finitely decomposed in [??]. Then [[mu].sub.[??]] = 0 if and only if [[mu].sub.[??]'] = 0.

Proof. Since any finite p-group has a nontrivial center, we can reduce the argument to the case of degree p.

For a finite abelian p-group A, we put rank A := dim A [cross product] [F.sub.p]. If B < A, then rank B, rank A/B [less than or equal to] rank A [less than or equal to] rank B + rank A/B. If a group G = <[sigma]> [congruent to] Z/pZ acts on A, then [(1 - [sigma]).sup.p] acts on A [cross product] [F.sub.p] as zero, and rank A [less than or equal to] [[summation].sup.p-1.sub.i=0] rank [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [less than or equal to] p rank A/[A.sup.1-[sigma]] holds.

Now let [f.sub.0] : M' [right arrow] M be a branched cover of degree p with Gal([f.sub.0]) = <[sigma]>. Let A and A' denote the p-parts of [H.sub.1](M) and [H.sub.1](M') respectively, and put r = [r.sub.0] := rank A and r' = [r'.sub.0] := rank A'. Let s = [s.sub.0] denote the number of components of the branch link S of [f.sub.0]. Then genus theory yields r - 1 [less than or equal to] r' [less than or equal to] p(r + s). Indeed, let [M.sub.ab] [right arrow] M and [M'.sub.ab] [right arrow] M' denote the maximal unbranched abelian p-covers. Then, the relative genus cover [M'.sup.g] [right arrow] M of [f.sub.0] : M' [right arrow] M factors through [M.sub.ab] [right arrow] M by definition. Put [r.sub.g] := rank Gal([M'.sup.g]/M'). Then by Theorem 3.5, Gal([M'.sup.g]/M') [congruent to] A'/[A'.sup.1-[sigma]] holds, and hence r' [less than or equal to] [pr.sub.g]. For each i, let [T.sub.i] < Gal([M'.sub.ab]/M) denote the inertia group of [K.sub.i] in [M'.sup.g] [right arrow] M. Then, since [M.sup.'g] [right arrow] M' is unbranched, we have [T.sub.i] [congruent to] Z/pZ, and Gal([M.sup.'g]/ [M.sub.ab]) = [T.sub.1]...[T.sub.s]. Therefore [r.sub.g] [less than or equal to] rank Gal([M.sup.'g]/ M) (less than or equal to] r + rank Gal([M.sup.'g]/[M.sub.ab]) [less than or equal to] r + s. On the other hand, we have r [less than or equal to] rank Gal([M.sup.'g]/M) [less than or equal to] 1 + rank Gal([M.sup.'g]/M') [less than or equal to] 1 + r'.

Similarly, for each [f.sub.n] : [M'.sub.n] [right arrow] [M.sub.n], let [r.sub.n] and [r'.sub.n] denote the p-ranks of [H.sub.1]([M.sub.n]) and [H.sub.1]([M'.sub.n]) respectively, and let [s.sub.n] denote the number of component of the branch link [S.sub.n] of [f.sub.n]. By a similar argument, [r.sub.n] - 1 [less than or equal to] [r'.sub.n] [less than or equal to] p([r.sub.n] + [s.sub.n]) holds. Since [S.sub.n] [subset] [h.sup.-1.sub.n] (S) and S is finitely decomposed in [??], [{[s.sub.n]}.sub.n] is bounded. By Sakuma's exact sequence ([Uek, Proposition 4.11]), there is a finitely generated torsion [LAMBDA] = [Z.sub.p][[T]]-module [H.sub.[??]] with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for any n, where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. By the structure theorem of finitely generated [LAMBDA]-modules ([Uek, Lemma 3.1 (4)]), [[mu].sub.[??]] = 0 (resp. [[mu].sub.[??]'] = 0) is equivalent to that [{[r.sub.n]}.sub.n] (resp. {[r'.sub.n]}n) is bounded. Thus the assertion holds.

Example 5.4. Let L and S be distinct unknots in M = [S.sup.3] and let [f.sub.0] : M' [right arrow] M be the S-branched cover of degree p. Let [??] and M' denote the TLN-[Z.sub.p]-covers over (M, L) and (M', L') for L' = [f.sup.-1.sub.0] (L) respectively. Then a branched Galois cover f : M' [right arrow] [??] is defined and [[mu].sub.[??]] = 0 holds. If [??]' consists of [QHS.sup.3]'s and if lk(L, S)= p, then [[mu].sub.[??]'] = 0 by Theorem 5.3. If p = 2, then L' is the Hopf link. If p = 3, then L' is the Borromean ring.

If lk(L, S)= 0, then L' is a split link, its Alexander polynomial is zero, [??]' does not consist of [QHS.sup.3]'s, and [[mu].sub.[??]'] is not defined. Instead, let L = [K.sub.1] [union] [K.sub.2] be a Hopf link, let L [union] S be [6.sup.3.sub.3] in Rolfsen's table ([Rol76]) and fix orientations so that lk([K.sub.1], S) = l and lk([K.sub.2], S) = -1. Then [[mu].sub.[??]'] = 0 and lk(L, S) = 0. If p = 2, then [[mu].sub.[??]'] is defined, and [[mu].sub.[??]'] [greater than or equal to] 1 by Theorem 5.2. Indeed, L' = [K'.sub.1] [union] [K'.sub.2] is [4.sup.2.sub.1] in Rolfsen's table and [[mu].sub.[??]'] = 1 holds.

Theorems 5.2 and 5.3 give branched [Z.sub.p]-covers [??]' which are candidates for [mu] = 0 and [mu] [greater than or equal to] t. We can check whether [??]' consists of [QHS.sup.3]'s or not by using the Alexander polynomials. We note that various constructions of [??] with given [lambda], [mu], v are studied in [KM08] and [KM13].

Remark. A [Z.sub.p]-field with [mu] = 0 resembles a function field. Especially, as an analogue of the Riemann Hurwitz formula for a cover of Riemann surfaces, Kida's formula for a p-extension of [Z.sub.p]-fields with [mu] = 0 is known. In [Uek], following Iwasawa's second proof in [Iwa81], their analogue in the topological context was formulated. It describes the balance of Iwasawa [lambda]-invariants, covering degree, and branching indices. We employed representation theory of finite groups, and Tate cohomology of 2-cycles [[??].sup.i] (G, [Z.sub.2]([??])). Meanwhile, Kida's formula for [Z.sub.p]-fields extension was first proved with use of genus theory ([Kid80]). We expect an alternative proof for our formula by following Kida's proof.

Acknowledgments. The author is very grateful to Masanori Morishita for encouragement and advice, Tsuyoshi Ito for suggesting that he consider an analogue of [Iwa73], Takahiro Kitajima, Tomoki Mihara, Yasushi Mizusawa, Hirofumi Niibo, and the anonymous referee for helpful comments and enlightening discussions, and Yuki Imoto for checking his English. The author is partially supported by Grant-in-Aid for JSPS Fellows (25-2241).

doi: 10.3792/pjaa.92.67

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Jun UEKI

Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan

(Communicated by Shigefumi MORI, M.J.A., May 12, 2016)
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