EXTENSIONS OF NUMBER FIELDS WITH WILD RAMIFICATION OF BOUNDED DEPTH

FARSHID HAJIR AND CHRISTIAN MAIRE

Abstract. We consider p-extensions of number fields such that the filtration of the Galois group by higher ramification groups is of prescribed finite length. We extend the basic properties of such towers (similar to the tamely ramified case but with more complications) to this more general setting; for instance, we show that these towers are “asymptotically good,” (we give an explicit bound for the root discriminant). We study the difficult problem of bounding the relation-rank of the Galois groups in question; by using results of Gordeev and Wingberg, we show that they can tend to infinity when the set of ramified primes is fixed but the length of the ramification filtration becomes large. We discuss the p-adic representations of these Galois groups in the context of the Fontaine-Mazur and provide some examples

Introduction

Fix a prime number p, a number field K, and a finite set S of primes of K. Let S_{p} be the set of all primes of K of residue characteristic p. Inside a fixed algebraic closure \overline{K} of K, let K_{S} be the maximal p-extension of K unramified outside S, and put G_{S} = Gal(K_{S}/K). The study of G_{S} is governed by a dichotomy between the tame (S ∩ S_{p} = ∅) and wild (S ∩ S_{p} ≠ ∅) cases.

One distinguishing feature of this dichotomoy is the following: in the tame case, every open subgroup of G_{S} has finite abelianization; on the other hand, if S_{p} ⊆ S, then G_{S} has a surjection onto ℤ_{p}^{r_{2}+1} (induced by the ℤ_{p}-extensions of K), where r_{2} is the number of imaginary places of K; (for surjections of G_{S} to ℤ_{p} when S ⊂ S_{p}, see [M]). In fact, a conjecture of Fontaine and Mazur [FM], predicts that, in the tame case, G_{S} has no infinite p-adic analytic quotients.

A second, and subtly related, distinguishing feature is the following: for 𝔭 ∈ S − S_{p}, the filtration of G_{S} by higher ramification groups at 𝔭 has length at most 2, i.e. D^{(1)}(K_{S}/K, 𝔭) vanishes, whereas in the case of wild ramification in an infinite pextension, it is often the case that the higher ramification groups of all indices are non-trivial (for example in a ℤ_{p}-extension).

In this paper, we study a generalization of tame towers, namely towers with wild ramification of bounded “depth”. To be precise, let ν : S → [0, ∞] be an arbitrary map (which will serve to limit the depth of ramification). Now define the group G_{S,ν} as the quotient of G_{S} by the closed normal subgroup generated by higher ramification groups (conjugacy classes), in the upper numbering, D^{(ν_{𝔭})}(K_{S}/K, 𝔭) as 𝔭 runs over S. The fixed field K_{S,ν} of this subgroup, with Galois group Gal(K_{S,ν}/K) = G_{S,ν}, is the compositum of all finite p-extensions of K having vanishing ν_{𝔭}^{th} ramification group at 𝔭 for all 𝔭 ∈ S. If the image of ν is simply ν(S) = {∞}, then G_{S,ν} is nothing but G_{S}, and much – most notably an estimate for its relation-rank – is known about this group [Sh], [K2], [NSW].

2000 Mathematics Subject Classification. Primary: 11R29, 11R37, 11S37. Key words and phrases.

Root discriminant, class field towers, wild ramification.

The case of greatest interest for us is when ν is “finite”, i.e. ν(S) ⊆ [0, ∞), because this ensures that the number fields in the tower K_{S,ν}/K form an “asymptotically good” family (in the sense of Tsfasman and Vladut [TV]), i.e. the root discriminant of these number fields remains bounded. The basic reason is that the exponent of a prime 𝔭 in the relative discriminant of L/K, where L is contained in K_{S}, is a sum of orders of ramification groups and therefore grows linearly in [L : K]. Thus, the root discriminant of the fields in K_{S}/K remains bounded. This is discussed in detail in section 5. For applications of asymptotically good families in the number field and function field settings, see [AM], [HM], and especially [TV].

We investigate the group G_{S,ν} via generators and relations. While calculating its generator-rank d(G_{S,ν}) = dim_{𝔽_{p}} H^{1}(G_{S,ν}, 𝔽_{p}) reduces to a standard calculation in class field theory, estimating r(G_{S,ν}) = dim_{𝔽_{p}} H^{2}(G_{S,ν}, 𝔽_{p}) presents an essential technical difficulty we have not been able to overcome. This is explained in section 6.

We are, however, able to make certain interesting observations regarding the relationrank of G_{S,ν}. First, there are non-trivial cases (i.e. where ν_{𝔭} ≥ 2 for some 𝔭 ∈ S ∩ S_{p}) where G_{S,ν} is finite, and, of course, G_{S,ν} can be infinite since already G_{∅} can be infinite. However, it is not at all clear that, in general, G_{S,ν} is even finitely presentable (has finite relation-rank), even if ν is finite. In fact, using results of Wingberg and Gordeev, we construct a family of examples with infinite ν achieving r(G_{S,ν}) = ∞. These examples allow us to show that for fixed S and growing, but finite, ν, the relation-rank can become arbitrarily large.

In the final section, further questions about the structure of G_{S,ν} are discussed, especially as regards p-adic representations of these groups. Fontaine and Mazur conjecture that when S ∩ S_{p} = ∅, G_{S} has no infinite p-adic analytic quotients. When ν is finite, we suspect G_{S,ν} is subject to the same conclusion, and provide some supporting evidence. A related question is whether every p-adic representation of G_{S,ν} is potentially semistable at p (in the sense of Fontaine) when ν is finite. We may summarize our questions as follows.

Question. Suppose K is a number field and L/K is a Galois extension such that Gal(L/K) is an infinite p-adic analytic group. Does it follow that

a) the root discriminant of the fields in this tower tend to infinity?

b) for some prime 𝔓 of L, all the higher ramification groups D^{y}(L/K, 𝔓) are non-trivial (for all 0 < y < ∞)?

c) there exists a p-adic representation ρ : Gal(L/K) → GL_{n}(E) (for some E/ℚ_{p} finite and some n ≥ 1) which is not potentially semistable?

The organization of the paper is as follows. The first two sections comprise a preliminary chapter of definitions and properties of ramification groups. We define the extensions of interest to us in section 3, and calculate the behavior of root discriminants in them in section 4. In sections 5 and 6, we use theorems of Sen, Gordeev, and Wingberg to explore the structure of the groups G_{S,ν}. In particular, we pinpoint the difficulty in extending the method for calculating the relation rank of G_{S} to that of G_{S,ν}, and we construct a family of G_{S,ν} with infinite ν which are not finitely presentable. Section 7 investigates a possible extension of Conjecture 5a of [FM] with some supporting evidence: We give examples of G_{S,ν} which do not admit infinite analytic quotients (Corollaries 40 and 41).

The first author was partially supported by a CSUSM Faculty Development Grant. The second author was supported by a Postdoctoral fellowship position at MSRI ^{1} in Berkeley, by CNRS, and by Laboratoire A2X ( Bordeaux I).

Part 1. Preliminaries

1. The setup

Let p be a prime number. Consider a number field K, a finite set S of primes of K equipped with an indexing function ν : S → [0, ∞] sending ν to ν(𝔭) = ν_{𝔭}; we will call the pair (S, ν) an indexed set for K. If ν(S) ⊆ [0, ∞), we will call (S, ν) a finitely indexed set. Sometimes it will be convenient to extend ν to all places of K by setting ν_{𝔭} = 0 for 𝔭 ∉ S. If ν(𝔭) = 0 or ∞ for all 𝔭 ∈ S ∩S_{p}, then the pair (S, ν) is abbreviated by S.

We fix once and for all an algebraic closure \overline{K}; an “extension of K” will mean a subfield of this fixed algebraic closure. Let 𝔭 be a prime ideal of K. We will use the following notation.

- S_{p} is the set of primes of K of residue characteristic p; S −S_{p} means S −(S ∩S_{p}).

- K_{𝔭} is the completion of K at 𝔭;

- O_{𝔭} the ring of integers of K_{𝔭};

- U_{𝔭} the group of units of O_{𝔭};

- U_{𝔭}^{(i)} = {u ∈ U_{𝔭}|u − 1 ∈ 𝔭^{⌈i⌉} for i ≥ 0; U_{𝔭}^{(0)} = U_{𝔭}; U_{𝔭}^{∞} = {1};

- J_{K} is the group of idèles of K;

- U_{S} = ∏_{𝔭∉S} U_{𝔭}; U_{S,ν} = ∏_{𝔭} U_{𝔭}^{(ν_{𝔭})}; these products are extended over the infinite places 𝔭 as well, where, for such a place, U_{𝔭} = U_{𝔭}^{0} = K_{𝔭}^{×}.

- K̂ is the maximal p-extension of K, with Galois group G = Gal(K̂/K);

- K̂_{𝔭} is the maximal p-extension of K_{𝔭} in a fixed Galois closure \overline{K_{𝔭}} of K_{𝔭}, with Galois group D_{𝔭} = Gal(K̂_{𝔭}/K_{𝔭});

- N_{S,ν} is the closed normal subgroup of G generated by all higher ramification groups D^{ν_{𝔭}}(K̂/K, 𝔭) for all K-primes 𝔭;

- G_{S,ν} = G/N_{S,ν} = Gal(K_{S,ν}/K) is the Galois group of the maximal p-extension of K unramified outside S and with ramification of depth at most ν_{𝔭} for 𝔭 ∈ S;

- d(G) = p-rkG = dim_{𝔽_{p}} G/[G, G]G^{p} is the p-rank of a group G;

- for a pro-p group, r(G) = dim_{𝔽_{p}} H^{2}(G, 𝔽_{p}) is the minimal number of relations for defining G as a pro-p group;

- d_{S,ν} = d(G_{S,ν}), r_{S,ν} = r(G_{S,ν});

- μ_{p} is the group of pth roots of unity in \overline{K};

- δ_{𝔭} = 1 if K_{𝔭} contains μ_{p}, 0 otherwise;

- δ(𝔭, ν_{𝔭}) is 1 if μ_{𝔭} ⊆ U_{𝔭}^{(ν_{𝔭})}, and is 0 otherwise;

- δ_{K} = 1 if K contains μ_{p}, 0 otherwise;

- θ_{S} = 0 unless δ_{K} = 1 and S is empty, in which case, θ_{S} = 1.

Let S′ be another finite set of prime ideals of K. For later use, we introduce the groups - Δ = {x ∈ K^{×}| (x) is a pth power in the group of fractional ideals of K}

1Research at MSRI is supported in part by NSF grant DMS-9701755

- Δ_{S} = {x ∈ Δ| x ∈ K_{𝔭}^{×}^{p} ∀𝔭 ∈ S}/K^{×}^{p};

- Δ_{S,ν} = {x ∈ Δ| x ∈ K_{𝔭}^{×}^{p}U_{𝔭}^{(ν_{𝔭})} ∀𝔭 ∈ S}/K^{×}^{p};

- Δ_{S}^{S′} = {x ∈ K^{×}| x ∈ U_{𝔭}K_{𝔭}^{×}^{p} ∀𝔭 ∉ S′, x ∈ K_{𝔭}^{×}^{p} ∀𝔭 ∈ S}/K^{×}^{p}.

Using Kummer theory, one has

Proposition 1. Assume μ_{p} ⊆ K.

- If S_{p} ⊆ S, then p-rk(Cl_{K}^{S,+}) = p-rk(Δ_{S}), where Cl_{K}^{S,+} is the S-class group of K in the narrow sense.

- If S′ ∪ S = S_{p}, then p-rk(Δ_{S}^{S′}) = p-rk(G_{S′}^{S}), where G_{S′}^{S} is the Galois group over K of the maximal p-extension of K unramified outside S′ in which S splits completely.

2. Higher Ramification groups

For the convenience of the reader, we collect here some definitions and standard properties of higher ramification groups. The reader is invited to consult this section only as needed, and is referred to Serre [Se1] for more details.

2.1. Ramification groups with lower numbering. Consider a finite Galois extension L/K of number fields and a prime 𝔓 of L lying over 𝔭 of K.

Definition 2. The decomposition group D(L/K, 𝔓) ⊆ Gal(L/K) of L/K at 𝔓 is the stabilizer of 𝔓 and can be identified with the local Galois group D(𝔓/𝔭) = Gal(L_{𝔓}/K_{𝔭}). For x ≥ −1, we define the higher ramification group of index x in the lower numbering D_{x}(L/K, 𝔓) ⊆ D(𝔓/𝔭) by

D_{x}(L/K, 𝔓) = {σ ∈ D(𝔓/𝔭)| v_{𝔓}(σ(α) − α) ≥ x + 1, ∀α ∈ O_{L}}.

In the sequel, we will sometimes abbreviate the data (L/K, 𝔓) as (𝔓/𝔭), for instance D_{x}(𝔓/𝔭) in place of D_{x}(L/K, 𝔓), etc. Note that the ramification groups of index −1, 0 are the decomposition and inertia groups of L/K at 𝔓, respectively. The higher ramification groups give a finite decreasing filtration of the decomposition group: D_{x}(L/K, 𝔓) ⊆ D_{y}(L/K, 𝔓), for x ≥ y and D_{x}(L/K, 𝔓) = {1} for all large enough x.

2.2. Ramification groups with upper numbering. Consider the map φ = φ_{L/K,𝔓} = φ_{𝔓/𝔭} from [−1, ∞) to [−1, ∞) defined as follows: For −1 ≤ u ≤ 0, φ(u) = u; for u ≥ 0, let m = ⌊u⌋ and put

φ(u) = \frac{1}{g_{0}} (g_{1} + ⋯ + g_{m} + (u − m)g_{m+1}) ,

where g_{i} = |D_{i}(L/K, 𝔓)|; in particular,

φ(m) = −1 + \frac{1}{g_{0}} ∑_{i=0}^{m} g_{i}.

Proposition 3. The function φ is continuous, piecewise linear, strictly increasing, concave, and satisfies φ(0) = 0, φ(x) ≤ x.

Let ψ = ψ_{L/K,𝔓} = ψ_{𝔓/𝔭} : [−1, ∞) → [−1, ∞) be the inverse of φ.

Proposition 4. The map ψ is continuous, piecewise linear, strictly increasing, convex, and satisfies ψ(0) = 0, ψ(y) ≥ y. If n is an integer, then ψ(n) is an integer.

Definition 5. We define the ramification group of index y ≥ −1 in the upper numbering by D^{y}(L/K, 𝔓) = D_{ψ(y)}(L/K, 𝔓).

Remark 2.1. We note that D_{x}(L/K, 𝔓) = D^{φ(x)}(L/K, 𝔓), D^{y}(L/K, 𝔓) ⊆ D_{y}(L/K, 𝔓), and for y ≥ z, D^{y}(L/K, 𝔓) ⊆ D^{z}(L/K, 𝔓). The ramification groups of 𝔓^{σ}, σ ∈ Gal(L/K), are the conjugates by σ of the ramification groups for 𝔓. We let D_{x}(L/K, 𝔭) (D^{y}(L/K, 𝔭)) denote the conjugacy class in Gal(L/K) of D_{x}(L/K, 𝔓) (D^{y}(L/K, 𝔓)).

Now let us consider the behavior of these groups in a tower L′/L/K where L′/K and L/K are finite Galois extensions, 𝔓′ is a prime of L′ lying over 𝔓 of L and 𝔭 of K.

Proposition 6. We have

1. φ_{𝔓′/𝔭} = φ_{𝔓/𝔭} ∘ φ_{𝔓′/𝔓}, and ψ_{𝔓′/𝔭} = ψ_{𝔓′/𝔓} ∘ ψ_{𝔓/𝔭};

2. D_{x}(𝔓′/𝔓) = D_{x}(𝔓′/𝔭) ∩ Gal(L′/L);

3. D^{y}(𝔓/𝔭) = D^{y}(𝔓′/𝔭)D(𝔓′/𝔓)/D(𝔓′/𝔓); in other words, D^{y}(𝔓/𝔭) is the image of D^{y}(𝔓′/𝔭) in Gal(L/K).

The groups in the lower numbering behave well under taking subgroups, whereas the groups in the upper numbering behave well under taking quotients. As noted in [Se1], the restriction property allows one to define the ramification groups D^{y}(L/K, 𝔓) with upper numbering for all profinite extensions L/K. We now list some simple consequences of the above proposition, which we will use later.

Proposition 7. 1. If 𝔓/𝔭 is unramified, then D^{y}(𝔓/𝔭) = D^{y}(𝔓′/𝔓).

2. If D^{y}(𝔓′/𝔓) and D^{y}(𝔓/𝔭) are trivial, then D^{y}(𝔓′/𝔭) is also trivial.

3. If L′′ is a Galois extension of K such that LL′′ = L′, and D^{y}(𝔓/𝔭) = D^{y}(𝔓′′/𝔭) = {1}, where 𝔓′′ = 𝔓′ ∩ L′′, then D^{y}(𝔓′/𝔭) = {1}.

Proof. 1. As 𝔓/𝔭 is unramified, the lower numbering ramification groups of 𝔓′/𝔓 are the same as those of L′/K. The map ψ_{𝔓′/𝔭} is then the same as ψ_{𝔓′/𝔓}. Then one has:

\begin{aligned}D^{y}(𝔓′/𝔭) & = & D_{ψ_{𝔓′/𝔭}(y)}(𝔓′/𝔭) = D_{ψ_{𝔓′/𝔓}(y)}(𝔓′/𝔭) \\ & = & D_{ψ_{𝔓′/𝔓}(y)}(𝔓′/𝔓) = D^{y}(𝔓′/𝔓). \end{aligned}

2. The restriction property shows that D^{y}(𝔓′/𝔭) ⊆ Gal(L′′/L). Moreover D^{y}(𝔓′/𝔭) = D_{ψ_{𝔓′/𝔭}(y)}(𝔓′/𝔭). Now using the transitivity of ψ and other elementary properties, we have D_{ψ_{𝔓′/𝔭}(y)}(𝔓′/𝔭) ⊆ D_{ψ_{𝔓′/𝔓}(y)}(𝔓′/𝔭). Thus,

D_{ψ_{𝔓′/𝔭}(y)}(𝔓′/𝔭) ⊆ Gal(L′′/L) ∩ D_{ψ_{𝔓′/𝔓}(y)}(𝔓′/𝔭) = D_{ψ_{𝔓′/𝔓}(y)}(𝔓′/𝔓) = D^{y}(𝔓′/𝔓).

The triviality of D^{y}(𝔓′/𝔓) allows us to conclude.

3. Follows easily from the restriction property.

Proposition 8. Suppose L/K is a finite p-extension.

1) If 𝔭 has residue characteristic ℓ ≠ p, D^{y}(𝔓/𝔭) vanishes for all y > 0.

2) If 𝔭 has residue characteristic p, and D^{y}(𝔓/𝔭) vanishes for some y ≤ 1, then 𝔓/𝔭 is unramified.

Proof. 1) Since x = ψ_{𝔓/𝔭}(y) > 0, we have D^{y}(𝔓/𝔭) = D_{x}(𝔓/𝔭) = D_{1}(𝔓/𝔭). Since L/K is a p-extension, 𝔓/𝔭 is at most tamely ramified and the wild inertia group D_{1}(𝔓/𝔭) is trivial.

2) We claim that x := ψ_{𝔓/𝔭}(1) = 1. We already know that x is a positive integer. Let us show that x ≤ 1. We have

1 = φ_{𝔓/𝔭}(x) = \frac{g_{1} + ⋯ + g_{x}}{g_{0}} ,

where g_{i} = |D_{i}(𝔓/𝔭)|. As D_{0}(𝔓/𝔭)/D_{1}(𝔓/𝔭) is trivial, g_{0} = g_{1}. One obtains:

1 + \frac{x − 1}{g_{0}} ≤ 1,

and so x = 1. Now we see that if y ≤ 1, then

D_{1}(𝔓/𝔭) = D_{ψ_{𝔓/𝔭}(1)}(𝔓/𝔭) ⊆ D_{ψ_{𝔓/𝔭}(y)}(𝔓/𝔭) = D^{y}(𝔓/𝔭) = {1},

and this implies the triviality of D_{0}(𝔓/𝔭) because this is a p-group and D_{0}(𝔓/𝔭)/D_{1}(𝔓/𝔭) has order prime to p.

Remark 2.2. If 𝔓 has residue characteristic p, ψ_{𝔓/𝔭}(y) = y for y ∈ [0, 1].

Proposition 9. Suppose L/K is an abelian extension. Then D_{1}(𝔓/𝔭) vanishes if and only if D^{1}(𝔓/𝔭) does.

Proof. One direction is trivial because D^{1}(𝔓/𝔭) ⊆ D_{1}(𝔓/𝔭). We know that the image of principal units under the local reciprocity map generates D^{1}(𝔓/𝔭). Thus, if this group is trivial, by using the structure of the group of local units we deduce the fact that the inertia group has order prime to the residue characteristic of 𝔭, and then D_{1}(𝔓/𝔭) is trivial.

Proposition 10. Suppose D^{y}(𝔓′/𝔭) = {1} for some y. Then D^{z}(𝔓′/𝔓) = {1} for z = ψ_{𝔓/𝔭}(y).

Proof. One has:

D^{z}(𝔓′/𝔓) = D_{ψ_{𝔓′/𝔓}(z)}(𝔓′/𝔓) = D_{ψ_{𝔓′/𝔭}(y)}(𝔓′/𝔓)
= D_{ψ_{𝔓′/𝔭}(y)}(𝔓′/𝔭) ∩ Gal(L′/L)
= D^{y}(𝔓′/𝔭) ∩ Gal(L′/L)
= {1}.

Part 2. Wild ramification of bounded depth

3. Definition and simple properties of towers with bounded ramification

Let S be a finite set of primes of K, equipped with a map ν : S → [0, ∞] sending 𝔭 to ν_{𝔭}.

Definition 11. Suppose L/K is a finite Galois extension and 𝔭 is a prime of K. We say that the ramification of L/K at 𝔭 is of depth at most y if D^{y}(𝔓/𝔭) vanishes for all primes 𝔓 of L lying over 𝔭.

The ramification of L/K at 𝔭 is of depth at most 0 (at most 1) means that 𝔭 is unramified (at most tamely ramified) in L/K.

Definition 12. Let K_{S,ν} be the compositum, inside a fixed algebraic closure of K, of all finite p-extensions L of K unramified outside S and with the property that, for every 𝔭 ∈ S, the ramification of L/K at 𝔭 is of depth at most ν_{𝔭}. Let G_{S,ν} = Gal(K_{S,ν}/K).

By Proposition 7, K_{S,ν} is the maximal p-extension of K unramified outside S and ramified to depth at most ν_{𝔭} for every 𝔭 ∈ S. If K_{S} is the maximal p-extension of K unramified outside S, then K_{S,ν} is the fixed field corresponding to the closed normal subgroup of G_{S} = Gal(K_{S}/K) generated by all D^{(ν_{𝔭})}(K_{S}/K, 𝔭). If ν vanishes identically, then K_{S,ν} is simply the Hilbert p-class field tower of K. By Proposition 8, we can and do assume without loss of generality that

𝔭 ∈ S − S_{p} ⇒ ν_{𝔭} = 1, 𝔭 ∈ S ∩ S_{p} ⇒ ν_{𝔭} > 1.

Definition 13. Suppose L/K is a finite extension contained in K_{S,ν}. We lift the indexed set (S, ν) of K to an indexed set (S, ν)_{L} = (S_{L}, ν_{L}) of L by

S_{L} = {𝔓 ⊆ O_{L}|𝔓 divides 𝔭O_{L} for some 𝔭 ∈ S}, ν_{L}(𝔓) = ψ_{𝔓/𝔭}(ν_{𝔭}).

Let us note that for a tower F/L/K, the indexed sets (S, ν)_{F} and (S_{L}, ν_{L})_{F} coincide, thanks to the transitivity of ψ.

Suppose ν is finite, and K_{1} is the maximal abelian extension of K contained in K_{S,ν}; this is the field associated by class field theory to the idèle subgroup:

U_{S,ν} = ∏_{𝔭∉S} U_{𝔭} ∏_{𝔭∈S} U_{𝔭}^{(ν_{𝔭})}.

Thus, K_{1} is the ray class field of K modulo 𝔪_{S,ν} = ∏_{𝔭∈S} 𝔭^{⌈ν_{𝔭}⌉}; in particular, K_{1}/K is finite.

To simplify notation, let us write (S_{1}, ν_{1}) for (S_{K_{1}}, ν_{K_{1}}). Define inductively a tower of finite abelian extensions as follows: K_{n+1} is the maximal abelian extension of K_{n} contained in K_{S_{n},ν_{n}} and (S_{n+1}, ν_{n+1}) is the lift of (S_{n}, ν_{n}) to K_{n+1}. Let K_{∞} = ∪_{n}K_{n}. By maximality for each step, K_{∞}/K is a Galois pro-p-extension.

Theorem 14. Suppose (S, ν) is a finitely indexed set for K. With notations as above, we have

1. K_{∞} = K_{S,ν};

2. If K/K_{0} is a Galois extension such that (S, ν) is Gal(K/K_{0})-stable, then K_{S,ν}/K_{0} is a Galois extension;

3. If L/K is a finite Galois extension contained in K_{S,ν}, then K_{S,ν} = L_{S_{L},ν_{L}};

4. The open subgroups of G_{S,ν} have finite abelianization.

Proof. 1) For 𝔭 ∈ S, choose a compatible system 𝔓_{n} ∈ S_{n}, lying over 𝔭. Recall that ν_{n}(𝔓_{n}) = ψ_{𝔓_{n}/𝔭}(ν_{𝔭}). To increase the readability of what follows, let us use the temporary notation D[j](𝔓/𝔭) = D^{j}(𝔓/𝔭) and D{j}(𝔓/𝔭) = D_{j}(𝔓/𝔭) for the upper and lower ramification groups respectively. Consider a positive integer n such that K_{n} ⊆ K_{S,ν} (this is the case for n = 1). We calculate:

D[ν_{𝔭}](𝔓_{n+1}/𝔭) ∩ Gal(K_{n+1}/K_{n}) = D{ψ_{𝔓_{n+1}/𝔭}(ν_{𝔭})}(𝔓_{n+1}/𝔭) ∩ Gal(K_{n+1}/K_{n})
= D{ψ_{𝔓_{n+1}/𝔭}(ν_{𝔭})}(𝔓_{n+1}/𝔓_{n})
= D{ψ_{𝔓_{n+1}/𝔓_{n}}(ψ_{𝔓_{n}/𝔭}(ν_{𝔭}))}(𝔓_{n+1}/𝔓_{n})
= D[ψ_{𝔓_{n}/𝔭}(ν_{𝔭})](𝔓_{n+1}/𝔓_{n})
= {1}.

But, by the restriction property of higher ramification groups, and the fact that D[ν_{𝔭}](𝔓_{n}/𝔭) vanishes, D[ν_{𝔭}](𝔓_{n+1}/𝔭) ⊆ Gal(K_{n+1}/K_{n}), hence is trivial. By induction, K_{∞} ⊆ K_{S,ν}.

To show the reverse inclusion, it suffices to show that if L is a finite abelian p-extension of K_{n} (for an arbitrary n) which is contained in K_{S,ν}, then L ⊆ K_{n+1}. If 𝔓 is a prime of L dividing 𝔓_{n}, we have:

D[ν_{n}(ν_{𝔭})](𝔓/𝔓_{n}) = D[ψ_{𝔓_{n}/𝔭}](𝔓/𝔓_{n})
= D{ψ_{𝔓/𝔓_{n}} ∘ ψ_{𝔓_{n}/𝔭}(ν_{𝔭})}(𝔓/𝔓_{n})
= D{ψ_{𝔓/𝔭}(ν_{𝔭})}(𝔓/𝔭) ∩ Gal(L/K_{n})
= D[ν_{𝔭}](𝔓/𝔭)
= {1}.

By the maximality of K_{n+1}/K_{n}, we then have L ⊆ K_{n+1}, completing the proof of 1.

2) is clear.

3) Proposition 10 gives K_{S,ν} ⊆ L_{S_{L},ν_{L}}. We show the reverse inclusion. Let (L_{n})_{n} be the sequence of maximal abelian extensions giving ∪_{n}L_{n} = L_{S_{L},ν_{L}}. Put L′ = L_{n} for an arbitrary fixed n. By 2), L′/K is Galois. One gets:

{1} = D[ν_{L}(𝔓)](𝔓′/𝔓) = D{ψ_{𝔓′/𝔓}(ν_{L}(𝔓))}(𝔓′/𝔓)
= D{ψ_{𝔓′/𝔭}(ν_{𝔭})}(𝔓′/𝔓)
= D{ψ_{𝔓′/𝔭}(ν_{𝔭})}(𝔓′/𝔭) ∩ Gal(L′/L)
= D[ν_{𝔭}](𝔓′/𝔭) ∩ Gal(L′/L).

Moreover, D[ν_{𝔭}](𝔓/𝔭) = {1}, and so D[ν_{𝔭}](𝔓′/𝔭) ⊆ Gal(L′/L). In conclusion D[ν_{𝔭}](𝔓′/𝔭) = {1}, and L_{n} = L′ ⊆ K_{S,ν}.

4) If H is an open subgroup of G_{S,ν}, its fixed field L is a finite extension of K and the abelianization of H = G_{S_{L},ν_{L}} is isomorphic to a ray class group, hence is finite.

Remark 3.1. 1. A compact topological group whose open subgroups have finite

abelianization is sometimes called “FAB.”

2. The naive lift of ν from K to L namely 𝔓 ↦ ν_{𝔭} (for a prime 𝔓 of L lying over 𝔭 of K) would lead to examples where L_{S_{L},ν_{L}} ≠ K_{S,ν}. Here is a simple example for p = 2: let K = ℚ, L = ℚ(\sqrt{3}) and F = K(\sqrt{2}). We let 𝔓, 𝔭 be the unique primes of F, L (respectively) dividing 2. One checks that D^{(3)}(𝔓/2) vanishes but that D^{(3)}(𝔓/𝔭) does not.

3. We should remark that the towers (of function fields) constructed in Perret [P] are based on this naive lift of (S, ν). They are contained in K_{S,ν} by Proposition 7), and thanks to Hasse-Arf, have a slightly lower root discriminant than the tower K_{S,ν}/K, cf. Remark 4.1. The Galois group of Perret’s tower, however, being less natural, is probably very difficult to study. For example, Neiderreiter and Xing [NX] have shown that the relation-rank estimate conjectured by Perret is not correct, at least for function fields over 𝔽_{2}.

4. By the above theorem, the fields K_{n} are simply the fields fixed by the “commutator series” of G_{S,ν}. Also, K_{∅} = the maximal unramified p-extension of K, is contained in K_{S,ν}, for an arbitrary (S, ν).

We conclude this section with a calculation of the generator rank of G_{S,ν}. By the Burnside Basis theorem, this reduces to calculating the p-rank of its maximal abelian quotient, which can be expressed, by class field theory, to the p-rank of an appropriate ray class group. The reduction to abelian extensions also allows us to assume, without loss of generality, that ν takes integer values (thanks to Hasse-Arf). We also note that if min_{𝔭∈S} ν_{𝔭} is large enough, then p-rk(G_{S,ν}) = p-rk(G_{S}).

Theorem 15. The generator-rank d_{S,ν} of G_{S,ν} is the p-rank of the ray class group Cl_{K,𝔪} where 𝔪 = ∏_{𝔭∈S} 𝔭^{⌈ν_{𝔭}⌉}. It satisfies

d_{S,ν} = p-rkΔ_{S,ν} + p-rk\frac{U_{∅}}{U_{S,ν}U_{∅}^{p}} − p-rkE_{K}

= p-rkΔ_{S,ν} + ∑_{𝔭∈S−S_{p}} δ_{𝔭} + ∑_{𝔭∈S∩S_{p}} p-rk\frac{U_{𝔭}^{(1)}}{U_{𝔭}^{(ν_{𝔭})}} − p-rkE_{K}

Proof. The main observation, as mentioned above, is that the fixed field K_{1} of (the closure of) the commutator subgroup [G_{S,ν}, G_{S,ν}] is the field associated by class field theory to the idèle subgroup:

U_{S,ν} = ∏_{𝔭∉S} U_{𝔭} ∏_{𝔭∈S} U_{𝔭}^{(ν_{𝔭})}.

We recall that we defined in section 1, certain number groups Δ_{S,ν}. We leave to the reader the verification of the exactness of the sequence:

1 ⟶ Δ_{S,ν} ⟶ Δ_{∅} ⟶ \frac{U_{∅}}{U_{∅}^{p}U_{S,ν}} ⟶ \frac{J_{K}}{U_{S,ν}J_{K}^{p}} ⟶ \frac{J_{K}}{K^{×}U_{∅}J_{K}^{p}} ≃ Cl_{K}/Cl_{K}^{p} ⟶ 1.

Moreover, it is easy to establish that p-rkΔ_{K} = p-rkCl_{K} + p-rkE_{K}, using the obvious map from Δ to the ideal classes killed by p. Putting all of this together, we obtain the desired formula.

Corollary 16. The genertor rank d_{S} of G_{S} satisfies:

d_{S} = p-rkΔ_{S} + ∑_{𝔭∈S−S_{p}} δ_{𝔭} + ∑_{𝔭∈S} p-rkU_{𝔭}^{(1)} − p-rkE_{K}.

Proof. For 𝔭 ∈ S, take ν_{𝔭} = ∞. This is the well-known formula of Shafarevich [Sh].

4. Behavior of root discriminant

We now suppose we have a finitely indexed set (S, ν) of K. We will give an upper bound for the root discriminant of the number fields in the tower K_{S,ν}/K.

Definition 17. Let K be a number field of degree n, and discriminant disc_{K}. The root discriminant rd_{K} of K is defined by

rd_{K} = |disc_{K}|^{1/n}.

Theorem 18. In the p-extension K_{S,ν}/K, the root discriminant is bounded. More precisely, if L is a finite extension of K contained in K_{S,ν}, then

rd_{L} ≤ rd_{K} ∏_{𝔭∈S−S_{𝔭}}(N_{K/ℚ}𝔭)^{1/[K:ℚ]} ∏_{𝔭∈S∩S_{𝔭}}(N_{K/ℚ}𝔭)^{(ν_{𝔭}+1)/[K:ℚ]}.

Proof. It suffices to consider K ⊆ L ⊆ K_{S,ν} such that L/K is Galois. Fix one of the r_{𝔭} primes 𝔓 of L dividing 𝔭 ∈ S; since L/K is Galois, the ramification index e_{𝔓/𝔭} = e_{𝔭} and residue degree f_{𝔓/𝔭} = f_{𝔭} depend only on 𝔭. Let φ = φ_{𝔓/𝔭}, ψ = ψ_{𝔓/𝔭}.

By the restriction property, D^{ν_{𝔭}}(𝔓/𝔭) is trivial for all 𝔭 in S; moreover, L/K is unramified outside S. By definition,

D^{ν_{𝔭}}(𝔓/𝔭) = D_{ψ(ν_{𝔭})}(𝔓/𝔭) = {1}.

Let n = ψ(ν_{𝔭}) and m = ⌊n⌋. The valuation v_{𝔓}(𝔡_{L/K}) at 𝔓 of the different of L/K is given by Hilbert’s formula [Se1]:

v_{𝔓}(𝔡_{L/K}) = g_{0} + g_{1} + ⋯ + g_{m−1} − m,

where g_{j} = |D_{j}(𝔓/𝔭)|; note that g_{j} = 1 for j ≥ m. In particular, for 𝔭 ∈ S − S_{p} the ramification is tame, so we have

v_{𝔓}(𝔡_{L/K}) = e_{𝔭} − 1.

From the definitions of ψ, φ, one has:

ν_{𝔭} = φ(n) = \frac{g_{1} + ⋯ + g_{m} + (n − m)}{g_{0}}.

Moreover,

v_{𝔓}(𝔡_{L/K}) = g_{0} + g_{1} + ⋯ + g_{m} − (m + 1)
= g_{0}(ν_{𝔭} + 1) − (m + 1) − (n − m)
= g_{0}(ν_{𝔭} + 1) − (n + 1)
≤ (e_{𝔭} − 1)(ν_{𝔭} + 1),

since g_{0} = e_{𝔓/𝔭} and ν_{𝔭} = φ(n) ≤ n.

Putting the local data together and recalling that e_{𝔭}f_{𝔭}r_{𝔭} = [L : K], we have

N_{L/K}𝔡_{L/K}^{1/[L:K]} ≤ ∏_{𝔭∈S−S_{p}} 𝔭^{(e_{𝔭}−1)f_{𝔭}r_{𝔭}/[L:K]} ∏_{𝔭∈S∩S_{p}} 𝔭^{(ν_{𝔭}+1)(e_{𝔭}−1)f_{𝔭}r_{𝔭}/[L:K]}
≤ ∏_{𝔭∈S−S_{p}} 𝔭^{(1−1/e_{𝔭})} ∏_{𝔭∈S∩S_{p}} 𝔭^{(ν_{𝔭}+1)(1−1/e_{𝔭})}.

We are now done, since

rd_{L} = rd_{K}N_{L/ℚ}(𝔡_{L/K})^{1/[L:ℚ]} = rd_{K}N_{K/ℚ}(N_{L/K}(𝔡_{L/K})^{1/[L:K]})^{1/[K:ℚ]}.

Remark 4.1. For the analogue of Perret’s tower in the number field case, one has the following root discriminant bound:

rd_{L} ≤ rd_{K} ∏_{𝔭∈S}(N_{K/ℚ}𝔭)^{⌈ν_{𝔭}⌉/[K:ℚ]}.

The details, which are essentially in [P], are left to the reader.

5. The relation-rank of G_{S} and G_{S,ν}

We put

r_{S} = dim_{𝔽_{p}} H^{2}(G_{S}, 𝔽_{p}), r_{S,ν} = dim_{𝔽_{p}} H^{2}(G_{S,ν}, 𝔽_{p}),

for the relation-rank of G_{S} and G_{S,ν}, respectively. An estimate for r_{S} was given by Shafarevich [Sh] in 1963. It turns out that estimating r_{S,ν} is a more delicate problem than estimating r_{S}. In fact, using results of Gordeev and Wingberg, we can find examples, with ν_{𝔭} finite for some 𝔭 and infinite for others, where G_{S,ν} is not finitely presentable! The main problem, then, is

Question 1. a) Do we have r_{S,ν} < ∞ for finitely indexed sets (S, ν)?

b) If so, give an explicit upper bound r_{S,ν} ≤ f(S, ν).

Below, we will first indicate briefly why the proof of Shafarevich does not easily generalize to our case, and then we will give examples with r_{S,ν} = ∞ (where (S, ν) is not finitely indexed), which leads to a family of examples showing that r_{S,ν} can tend to infinity for finitely indexed sets S, ν, where S is fixed, but max_{𝔭∈S} ν_{𝔭} tends to infinity.

5.1. Shafarevich’s relation-rank bound. The fundamental artithmetic result about G_{S} is an estimate for its relation rank due to Shafarevich [Sh].

Theorem 19. We have

r_{S} − d_{S} ≤ p-rkE_{K} − δ_{K} + θ_{S} − ∑_{𝔭∈S_{p}}[K_{𝔭} : ℚ_{p}].

We note in passing that when S_{p} ⊆ S, we in fact know the relation rank exactly: r_{S} − d_{S} = −(r_{2} + 1) where r_{2} is the number of imaginary places of K (cf. [H] and, for p = 2, [Sch]).

We will sketch the proof of Theorem 19 given by Koch (cf. [K1], [K2]), indicating a difficulty one encounters in attempting to generalize it by replacing S with (S, ν).

Recall the group of (S, ν)-unit idèles is

U_{S,ν} = ∏_{𝔭} U_{𝔭}^{(ν_{𝔭})},

where we have extended ν to all places by setting ν_{𝔭} = 0 for 𝔭 ∉ S.

The local reciprocity map ρ_{𝔭} for a local Galois extension L_{𝔓}/K_{𝔭} is defined on L_{𝔓}∩K_{𝔭}^{ab}, i.e in D(𝔓/𝔭)/[D(𝔓/𝔭), D(𝔓/𝔭)]. Moreover, by the restriction property of higher ramification groups, we have

D^{ν_{𝔭}}(L_{𝔓} ∩ K_{𝔭}^{ab}/K_{𝔭}) = \frac{D^{ν_{𝔭}}(𝔓/𝔭)}{D^{ν_{𝔭}}(𝔓/𝔭) ∩ [D(𝔓/𝔭), D(𝔓/𝔭)]}.

Also, ρ_{𝔭}(U_{𝔭}^{(ν_{𝔭})}) = D^{ν_{𝔭}}(L_{𝔓} ∩ K_{𝔭}^{ab}/K_{𝔭}).

Since D(𝔓/𝔭)/D^{0}(𝔓/𝔭) is cyclic, we have

[D(𝔓/𝔭), D(𝔓/𝔭)] = [D(𝔓/𝔭), D^{0}(𝔓/𝔭)].

This seemingly minor simplification will turn out to be quite important, as we will see in a moment. Recall that K_{S} is the maximal p-extension of K unramified outside S, with Galois group G_{S} = Gal(K_{S}/K); it will be convenient here to think of K_{S} as K_{S,ν} with ν_{𝔭} = ∞ for 𝔭 ∈ S.

Let G be the Galois group over K of its maximal p-extension K̂. We let D_{𝔭} be the Galois group of the maximal p-extension K̂_{𝔭} of K_{𝔭}. Let N_{S,ν} be the closed normal subgroup of G generated by all higher ramification groups D_{𝔭}^{ν_{𝔭}}, for 𝔭 ∈ S.

Definition 20. Let B_{S,ν} = ker (H^{2}(G_{S,ν}, 𝔽_{p}) → H^{2}(G, 𝔽_{p})) .

The exact sequence

1 ⟶ N_{S,ν} ⟶ G ⟶ G_{S,ν} ⟶ 1

gives, via Hochschild-Serre, the exact sequence

\{1\} ⟶ B_{S,ν}^{∗} ⟶ \frac{N_{S,ν}}{N_{S,ν}^{p}[G, N_{S,ν}]}⟶^{α} \frac{G}{G^{p}[G, G]},

allowing us to identify ker(α) with the dual of B_{S,ν}.

Proposition 21. 1) There is a natural injection

H^{2}(G, 𝔽_{p}) ↪ ⊕_{𝔭∉S}H^{2}(D_{𝔭}, 𝔽_{p}) ⊕_{𝔭∈S} H^{2}(D_{𝔭}, 𝔽_{p}).

Moreover if K contains μ_{p}, we can omit an arbitrary place in the sum.

2) If K_{𝔭} does not contain μ_{p}, H^{2}(D_{𝔭}, 𝔽_{p}) is trivial.

3) If K_{𝔭} contains μ_{p}, D_{𝔭} is a group with only one relation. So p-rkH^{2}(D_{𝔭}, 𝔽_{p}) = 1

4) In the composite map

H^{2}(G_{S,ν}, 𝔽_{p}) ⟶ H^{2}(G, 𝔽_{p}) ⟶ ⊕_{𝔭∉S}H^{2}(D_{𝔭}, 𝔽_{p}) ⊕_{𝔭∈S} H^{2}(D_{𝔭}, 𝔽_{p}),

the image of H^{2}(G_{S,ν}, 𝔽_{p}) in ⊕_{𝔭∉S}H^{2}(D_{𝔭}, 𝔽_{p}) is trivial.

Proof. This is proved in [K2]; for part 4), Koch treats only G_{S} but the same argument applies to G_{S,ν}. See also [Se2], [Sh], [K1])

We have therefore proved

Theorem 22. For a finite set S of places of K and an arbitrary indexing function ν of S, we have

r_{S,ν} = p-rkH^{2}(G_{S,ν}, 𝔽_{p}) ≤ ∑_{𝔭∈S} δ_{𝔭} − δ_{K} + θ_{S} + p-rkB_{S,ν}.

Thus, in order to complete the proof of theorem 19, (or to answer Question 1), it suffices to bound p-rkB_{S,ν} from above. Let us attempt to follow the proof given by Shafarevich and Koch for bounding B_{S} by relating it to the kernel of the natural map

η : U_{S,ν}/U_{S,ν}^{p} ⟶ J_{K}/K^{×}J_{K}^{p}.

Let us observe, first of all, that ker(η) is under control.

Lemma 23. We have

p-rk ker(η) ≤ p-rkE_{K} + d_{S,ν} + ∑_{ν_{𝔭}≠∞^{𝔭∈S∩Sp}}[K_{𝔭} : ℚ_{p}] + δ(𝔭, ν_{𝔭}) − ∑_{𝔭∈S−S_{p}} δ_{𝔭} − ∑_{𝔭∈S∩S_{p}} p-rk\frac{U_{𝔭}^{(1)}}{U_{𝔭}^{(ν_{𝔭})}} ,

where δ(𝔭, ν_{𝔭}) is 1 if μ_{p} ⊆ U_{𝔭}^{(ν_{𝔭})} and is 0 otherwise.

Proof. There is an exact sequence

\{1\} ⟶ \frac{U_{S,ν} ∩ U_{∅}^{p}}{U_{S,ν}^{p}} ⟶ ker(η) ⟶^{ϕ} Δ_{S,ν} ⟶ \{1\}.

where ϕ is defined as follows. We have

ker(η) = \frac{U_{S,ν} ∩ K^{×}J_{K}^{p}}{U_{S,ν} ∩ U_{∅}^{p}} .

Let u ∈ U_{S,ν} ∩ K^{×}J_{K}^{p}; write u = x.j^{p} with x ∈ K^{×}, j ∈ J_{K}. We let ϕ(u) be the image of x in Δ_{S,ν}. The kernel of ϕ is easily seen to be (U_{S,ν} ∩ U_{∅}^{p})/U_{S,ν}^{p}, which is trivial when ν(S) = {0, ∞}, and whose p-rank in general is bounded by

p-rk ker(ϕ) = ∑_{𝔭∈S∩S_{p}} p-rkU_{𝔭}^{(ν_{𝔭})} ∩ U_{𝔭}^{p}
≤ ∑_{𝔭∈S∩S_{p}} p-rkU_{𝔭}^{(ν_{𝔭})} = ∑_{ν_{𝔭}≠∞^{𝔭∈S∩Sp}}[K_{𝔭} : ℚ_{p}] + δ(𝔭, ν_{𝔭}).

We now only need apply the formula of Theorem 15.

Now, there are natural surjections

U_{S,ν} \underrightarrow{∏_{𝔭} ρ_{𝔭}} ∏_{𝔭} \frac{D_{𝔭}^{ν_{𝔭}}}{[D_{𝔭}, D_{𝔭}] ∩ D_{𝔭}^{ν_{𝔭}}} ⟵^{β} ∏_{𝔭} \frac{D_{𝔭}^{ν_{𝔭}}}{[D_{𝔭}, D_{𝔭}^{ν_{𝔭}}]} ⟶ \frac{N_{S,ν}}{[G, N_{S,ν}]}.  (1)

Note that β is “pointing the wrong way.” In the classical case (where ν_{𝔭} is either 0 or ∞), β is an isomorphism ([D_{𝔭}, D_{𝔭}] = [D_{𝔭}, D_{𝔭}^{0}]), so we can define a surjective global map

U_{S} ⟶^{ρ} N_{S}/N_{S}^{p}[G, N_{S}]).

We then have a commutative diagram:

B_{S}^{∗} ↪ \frac{N_{S}}{N_{S}^{p}[G, N_{S}] ∩ N_{S}} ⟶^{α} \frac{G}{G^{p}[G, G]}
∃ρ′′ ↑  ρ ↑  ρ′ ↑
ker(η) ↪  \frac{U_{S}}{U_{S}^{p}}  ⟶^{η} \frac{J_{K}}{K^{×}J_{K}^{p}}

where ρ is as defined above. As ρ′ is injective, there is a natural surjection ρ′′ : ker(η) ⟶ B_{S,ν}^{∗}. Putting this together with Theorem 22 and Lemma 23, we have proved theorem 19.

To summarize, in the general case, what is lacking is a bound for the dimension of B_{S,ν}, for which it would suffice to have a natural map ρ^{?} : U_{S,ν}/U_{S,ν}^{p} ⟶ N_{S,ν}/N_{S,ν}^{p}[G, N_{S,ν}] with finite cokernel. The classical approach sketched above does not work because the map β (an isomorphism in the classical case) has infinite cokernel in general, by the following remarkable theorem of Gordeev [G].

Theorem 24 ([G]). Let K_{𝔭} be a finite extension of ℚ_{p}. Let K̂_{𝔭} be the maximal pextension of K_{𝔭} and put D_{𝔭} = Gal(K̂_{𝔭}/K_{𝔭}). For ν_{𝔭} > 1,

r(D_{𝔭}/D_{𝔭}^{ν_{𝔭}}) = dim_{𝔽_{p}} H^{2}(D_{𝔭}/D_{𝔭}^{ν_{𝔭}}) = ∞.

Corollary 25. The map β of (1) has infinite cokernel. Indeed, with hypotheses as in the theorem,

dim_{𝔽_{𝔭}} \frac{D_{𝔭}^{ν_{𝔭}}}{[D_{𝔭}, D_{𝔭}^{ν_{𝔭}}](D_{𝔭}^{ν_{𝔭}})^{p}} = ∞, dim_{𝔽_{p}} \frac{D_{𝔭}^{ν_{𝔭}}}{[D_{𝔭}, D_{𝔭}] ∩ D_{𝔭}^{ν_{𝔭}}(D_{𝔭}^{ν_{𝔭}})^{p}} ≤ [K_{𝔭} : ℚ_{p}] + 1.

Proof. Let us take the exact sequence

1 ⟶ D_{𝔭}^{ν_{𝔭}} ⟶ D_{𝔭} ⟶ D_{𝔭}/D_{𝔭}^{ν_{𝔭}} ⟶ 1.

Applying the Hochschild-Serre sequence, we get

(ker (H^{2}(D_{𝔭}/D_{𝔭}^{ν_{𝔭}}, 𝔽_{p}) → H^{2}(D_{𝔭}, 𝔽_{p})))^{∗} ↪ \frac{D_{𝔭}^{ν_{𝔭}}}{[D_{𝔭}, D_{𝔭}^{ν_{𝔭}}](D_{𝔭}^{ν_{𝔭}})^{p}} ⟶ ⋯

But D_{𝔭} is a p-group with p-rkH^{2}(D_{𝔭}, 𝔽_{p}) = 0 or 1. Moreover by Theorem 24, D_{𝔭}/D_{𝔭}^{ν_{𝔭}}

has infinite relation-rank. Then the p-rank of \frac{D_{𝔭}^{ν_{𝔭}}}{[D_{𝔭}, D_{𝔭}^{ν_{𝔭}}](D_{𝔭}^{ν_{𝔭}})^{p}} is infinite. The quotient

\frac{D_{𝔭}^{ν_{𝔭}}}{[D_{𝔭}, D_{𝔭}] ∩ D_{𝔭}^{ν_{𝔭}}(D_{𝔭}^{ν_{𝔭}})^{p}} is the restriction of D_{𝔭}^{ν_{𝔭}} to the maximal abelian p-extension of

K_{𝔭}, and so has p-rank at most [K_{𝔭} : ℚ_{p}] + 1.

5.2. Not all G_{S,ν} are finitely presentable. The groups G_{S,ν} are finitely generated, so they are finitely presentable in the category of pro-p groups if and only if they have finite Euler characteristic χ_{2}(G_{S,ν}) = r_{S,ν} − d_{S,ν}. Note that, for finitely indexed (S, ν), χ_{2}(G_{S,ν}) is non-negative since G_{S,ν} has finite abelianization in that case.

The main result of this part gives infinitely many examples (with p = 2, ν_{𝔭} finite for some 𝔭 but infinite for others) for which r_{S,ν} = χ_{2}(G_{S,ν}) = ∞.

Theorem 26. Let p = 2, q is a prime with q ≡ 7 (mod 16). Put K = ℚ(\sqrt{−q}), in which 2 splits: 2O_{K} = 𝔭_{1}𝔭_{2}. Let i be a positive integer, and define ν : S = {𝔭_{1}, 𝔭_{2}} → [0, ∞] by ν(𝔭_{1}) = i, ν(𝔭_{2}) = ∞. Then r(G_{S,ν}) = ∞.

Remark 5.1. (i) If we take ν(𝔭_{1}) = ν(𝔭_{2}) = 2, then G_{S,ν} is finite for q = 7 by Theorem 18 and discriminant bounds. Assuming the Generalized Riemann Hypothesis, the same is true for ν(𝔭_{1}) = ν(𝔭_{2}) = 3.

(ii) Note that G_{S,ν} (with S, ν as in the theorem) has G_{{𝔭_{2}}} has as a quotient, which in turn has ℤ_{2} as a quotient.

To show Theorem 26 we have to recall a result of Wingberg [W].

Definition 27. Let K be a number field, and L/K be a pro-p-extension. We will say that L/K is maximal local at 𝔭 if the natural composite map:

D_{𝔭} →^{res} Gal(L_{𝔓}/K_{𝔭}) ↪ Gal(L/K),

is an isomorphism.

Theorem 28 ([W]). Let S be a finite set of places of K containing all the places above p and the archimedean places. Let K_{S} be the maximal p-extension of K unramified outside S. Then, K_{S}/K is maximal local at 𝔭 ∈ S_{p} if and only if the following conditions are satisfied:

1. for p = 2, K is totally imaginary,

2. ∑_{𝔮∈S\{𝔮}} δ_{𝔮} = δ_{K}, 3. Δ_{S−{𝔭}}^{{𝔭}} = {1},

4. r_{2} = [K_{𝔭} : ℚ_{p}], where r_{2} is the number of imaginary places of K.

Corollary 29. With notation as above, and p = 2, K_{S}/K is maximal local at 𝔭 ∈ S_{p} if and only if

1. K is totally imaginary;

2. S = {𝔭, 𝔮} where 2O_{K} = 𝔭𝔮;

3. There does not exist a non-trivial 2-extension of K unramified outside 𝔭 and in which 𝔮 splits,

4. r_{2} = [K_{𝔭} : ℚ_{p}].

Proof. There exists a non-trivial 2-extension of K, 𝔮-decomposed and unramified oustide 𝔭, if and only if Δ_{S−{𝔮}}^{{𝔮}} ≠ {1}, because K contains μ_{p} (by Proposition 1).

We need to verify the above properties for the fields appearing in Theorem 26. We do so in the following Proposition, which can be proved directly via a long calculation; we will instead derive it more easily by making use of a result in a book of Gras to appear [Gr2].

Proposition 30. Let K = ℚ(\sqrt{−q}) where q ≡ 7 mod 8 is a prime, 2O_{K} = 𝔭_{1}𝔭_{2}. There exists a non-trivial 2-extension of K unramified outside 𝔭_{1} and for which 𝔭_{2} splits completely if and only if p ≡ −1 mod 16.

Proof. Let n be the order of 𝔭_{1} in Cl_{K}: note that n is odd. There exists integers a and b, such that α = (a+b\sqrt{−q})/2 (a, b have the same parity, and are in fact odd) generates 𝔭_{1}^{n}: 𝔭_{1}^{n} = (α). Call the non-trivial element of Gal(K/ℚ) τ, so that α^{τ} = (a−b\sqrt{−q})/2 and 𝔭_{2}^{n} = (α^{τ}). We can handle the case q = 7 by hand. Now suppose q > 7. We find a sequence of equivalent conditions as follows.

Claim. For q > 7, the following are equivalent:

1. There exists non trivial 2-extension of K unramified outside 𝔭_{1} and for which 𝔭_{2} splits completely,

2. [K_{𝔭_{1}}(\sqrt{−1}, \sqrt{α^{τ}}) : K_{𝔭_{1}}] = 2,

3. (2, α^{τ})_{𝔭_{1}} = 1,

4. α^{τ} ≡ ±1 (mod𝔭_{1}^{3}),

5. a^{2} ≡ 1 mod 16,

6. q ≡ 15 mod 16.

Proof of Claim. The equivalence of 1 and 2 is from Gras’ book (Chapter V). We then have α^{τ} ∈ K_{𝔭_{1}}(\sqrt{−1}) ⟺ α^{τ} ≡ ±1 (mod 𝔭_{1}^{3}) ⟺ (2, α^{τ})_{𝔭_{1}} = 1, because α^{τ} is a local unit in 𝔭_{1} (see [Se1] for example). Now condition 4 is equivalent to α ≡ 0 (mod 𝔭_{1}^{3}) or α + α^{τ} ≡ ±1 (mod 𝔭_{1}^{3}), i.e. to a ≡ ±1 (mod 8) or a^{2} ≡ 1 mod 16. For the final step, we want to show that this is equivalent to q ≡ 15 mod 16. We take the norm of 2α: 4ℕα = a^{2} + b^{2}q = 2^{n+2}, to see that 2^{n+2} is a square modb, and as n ≥ 1 is odd, 2 is a square mod b. Now by the quadratic reciprocity law, 2 is a square (mod b) implies b ≡ ±1mod8. As q > 7, 2 could not be a norm in O_{K}/ℤ, so n ≥ 3. Now a^{2} + b^{2}q = 2^{n+2} gives a^{2} + b^{2}q ≡ 0 (mod 16) or a^{2} + q ≡ 0 (mod 16). This concludes the proof of the claim, of Proposition 30 and Theorem 26.

Proof. [Theorem 26] With S = {𝔭_{1}, 𝔭_{2}}, K_{S}/K is maximal local at 𝔭_{2} by Proposition 30 and Corollary 29. Thus, the global Galois group G_{S,ν} is isomorphic to the local group appearing in Gordeev’s Theorem 24, which has infinite relation rank.

5.3. Variation of r_{S,ν} for fixed S. Let us consider what happens as we vary ν for K = ℚ(\sqrt{−q}), S = \{𝔭_{1}, 𝔭_{2}\} as above. We introduce some notation. For integers i, j > 1, let ν_{i,j} : S = {𝔭_{1}, 𝔭_{2}} → [0, ∞) be defined by ν_{i,j}(𝔭_{1}) = i, ν_{i,j}(𝔭_{2}) = j. For simplification, we write

G_{i,j} = G_{S,ν_{i,j}}.

Definition 31. For a finitely generated pro-p-group G, the partial Euler-Poincare characteristic χ_{2}(G) is defined by :

χ_{2} = p-rkH^{2}(G, 𝔽_{p}) − p-rkH^{1}(G, 𝔽_{p}) = r(G) − p-rkG.

Corollary 32. For fixed i, we have sup_{j} r(G_{i,j}) = sup_{j} χ_{2}(G_{i,j}) = ∞.

Proof. We have G_{i,∞} = lim_{j^{←}} G_{i,j}. Consequently,

H^{2}(G_{i,∞}, 𝔽_{p}) = lim_{j^{→}} H^{2}(G_{i,j}, 𝔽_{p}).

If for some j, r(G_{i,j}) = ∞, then there is nothing to prove. Now suppose r(G_{i,j}) ≤ M < ∞ for all j. Then H^{2}(G_{i,∞}, 𝔽_{p}) is a direct limit of abelian groups of cardinality at most M, hence is finite. This contradicts Theorem 26 (r(G_{i,∞}) = ∞). The claim for sup_{j} χ_{2}(G_{i,j}) follows since d(G_{i,j}) is bounded by d(G_{i,∞}).

Note that sup_{j}(G_{i,j}) = ∞ implies that G is not analytic. This is a particular instance of the following theorem.

Theorem 33. Let Γ = lim_{i^{←}} Γ_{i} be a finitely generated projective limit of pro-p-groups

Γ_{i}, such that the natural map of restriction from Γ to Γ_{i} is surjective for sufficiently large i. Assume that sup_{i} χ_{2}(Γ_{i}) = ∞. Then Γ is not p-adic analytic.

Proof. The homomorphism of restriction from Γ to Γ_{i} is continuous and the kernel H_{i} is the inverse image of a closed subgroup of Γ_{i} and so is closed in Γ. Thus, for large i, we have:

1 ⟶ H_{i} ⟶ Γ ⟶ Γ_{i} ⟶ 1.

By using the Hochschild-Serre sequence plus duality we have:

(ker(H^{2}(Γ_{i}, 𝔽_{p}) → H^{2}(Γ, 𝔽_{p})))^{∗} ↪ \frac{H_{i}}{H_{i}^{p}[Γ, H_{i}]} ⟶ \frac{Γ}{Γ^{p}[Γ, Γ]} ⟶ \frac{Γ_{i}}{Γ_{i}^{p}[Γ_{i}, Γ_{i}]} ⟶ 1.

Passing to the p-rank, we obtain:

p-rkH_{i} ≥ p-rkH_{i}/[H_{i}, Γ]
≥ χ_{2}(Γ_{i}) − χ_{2}(Γ).  (2)

If sup_{i} χ_{2}(Γ_{i}) = ∞, then sup_{i} p-rkH_{i} = ∞, and as H_{i} is closed in Γ, we find that

sup{p-rkH| H ⊆ Γ, H closed} = ∞.

By a result of Lubotzky and Mann [LM], this implies that Γ is not analytic.

Remark 5.2. 1) Consider G_{S}, the Galois group over K = ℚ(\sqrt{−q}) (with prime q ≡ 7 mod 16)) of the maximal 2-extension unramified outside 2. Then G_{S} is not analytic, since it has non-analytic quotients G_{i,∞}. Note that we have G_{S} = lim_{j^{←}} G_{i,∞}, with

χ_{2}(G_{i,∞}) = ∞, but χ_{2}(G_{S}) = −(r_{2} + 1) = −2 is finite.

2) We give a simple example of the Theorem. Take K = ℚ, S = {p}. Then G_{S} is a pro-p group with one generator and without relation, i.e. G_{S} ≃ ℤ_{p}. For any finite indexing ν of S, we have G_{S,ν} is finite with vanishing Euler characteristic.

3) Theorem 33 suggests that the study of the growth of the Euler characteristic χ_{2}(Γ/H) for open subgroups H of Γ, where Γ = G_{S} or Γ = G_{S,ν} would yield interesting information about these groups. So far, one knows little about the growth of Euler characteristics for these groups. When S = S_{p}, Shafarevich’s relation rank estimate together with the Golod-Shafarevich theorem show that G_{S} is not analytic (unless it is isomorphic to ℤ_{p}), so the Euler characteristic of finite quotients tends to infinity in that case.

Part 3. The Fontaine-Mazur conjecture and p-adic representations of G_{S,ν}

6. A Theorem of Sen

Suppose F is a finite extension of ℚ_{p}, and E/F is a totally ramified normal extension such that G = Gal(E/F) is an infinite p-adic analytic group. The group G has two natural filtrations given by its central series (G_{1} = G, G_{j+1} = [G, G_{j}]), and the higher ramification groups D^{i}(E/F). It was conjectured by Serre and proven by Sen [S] that these filtrations are closely related as follows.

Theorem 34 (Sen). Let F be a finite extension of ℚ_{p}, with ramification index e = e_{F/ℚ_{p}}. Suppose E/F is a totally ramified p-extension with p-adic analytic Galois group G. Then there exists a constant c, such that for all n ≥ 0,

D^{ne+c}(E/F) ⊆ G_{n} ⊆ D^{ne−c}(E/F).

Corollary 35. Suppose F is a finite extension of ℚ_{p} and E/F is a p-extension with p-adic analytic Galois group. Let F^{un} be the maximal unramified p-extension of F, and assume that E∩F^{un} is of finite degree over F. Assume, moreover, that the ramification is bounded in E/F, i.e. there exists i > 0 such that D^{i}(E/F) = {1}. Then E/F is finite.

Proof. Let F_{0} = E ∩ F^{un}. Let G = Gal(E/F_{0}); G is a p-adic analytic group. As F_{0}/F is unramified, the groups D^{i}(E/F_{0}) and D^{i}(E/F) are the same (cf. Proposition 7), and by assumption, they vanish. By Theorem 34, there exists a constant c such that

D^{ne+c}(E/F_{0}) ⊆ Gal(E/F_{0})_{n} ⊆ D^{ne−c}(E/F_{0})).

Put m = ⌈(i + c)/e⌉ and j = me − c. Then j ≥ i, so D^{j}(E/F_{0}) is trivial, hence G_{m} (the mth term of the central series of G = Gal(E/F_{0})) is trival as well. A group with a finite central series filtration and whose open subgroups have finite abelianization must be finite, so we will know that Gal(E/F_{0}) (and hence Gal(E/F)) is finite, once we prove the following lemma.

Lemma 36. Let E be a Galois extension of local field F, totally ramified and with bounded ramification. Then for every open subgroup H of Gal(E/F), H^{ab} is finite.

Proof. By Proposition 10, E/E^{H} is totally ramified and has bounded ramification, so it suffices to prove that G = Gal(E/F) has finite abelianization. Let F_{1} be the fixed field of [G, G]. By assumption, there exists i such that D^{i}(E/F) = {1}. By the property of restriction, D^{i}(F_{1}/F) = {1}. As E/F is totally ramified, F_{1}/F is totally ramified (and abelian). By class field theory, G^{ab} is generated by the image of the units U of F under the local reciprocity map ρ, and the image of U^{i} is precisely D^{i}(F_{1}/F), which is trivial. We have a map, therefore,

U/U^{i} →^{ρ} G^{ab},

which implies the finiteness of G^{ab}.

7. The Conjecture of Fontaine-Mazur

Let us call a finitely generated pro-p group p-adically finite if it has no infinite p-adic analytic quotients. In other words, all p-adic representations of it into GL_{n}(ℚ_{p}) have open kernel. Recall the conjecture of Fontaine-Mazur.

Conjecture 1 ([FM], Conjecture 5a). Let K be a number field, S a finite set of places of K disjoint from S_{p}. Then G_{S} is p-adically finite.

For partial results on this topic, see Boston [B1], [B2], [B3] and Hajir [Ha]. The above conjecture of Fontaine and Mazur is only a special case of their vast program for characterizing the representations which “come from algebraic geometry,” meaning those isomorphic to a subquotient of the action of Gal(\overline{K}/K) on an étale cohomology group H_{ét}^{q}(X\overline{K}, ℚ_{p}(r)), where X is a smooth projective variety over K. They call a p-adic representation of Gal(\overline{K}/K) “geometric” if is satisfies two conditions: 1) it is unramified outside a finite set of places of K, and 2) its restriction to every decomposition group is potentially semistable. Their main conjecture, then, is that an irreducible p-adic representation is geometric if and only if it is potentially semistable. One direction of this conjecture (“algebro-geometric representations are potentially semistable”) has a much longer history, and is better understood (see, for example, Tsuji [T]).

By a theorem of Grothendieck, a tamely ramified p-adic representation is always potentially semistable; thus, when S ∩S_{p} = ∅, every p-adic representation ρ of G_{S} should come from algebraic geometry, and then the Hodge and Tate conjectures imply that ρ has finite image. This is equivalent to the assertion of Conjecture 1.

Given the group-theoretic (Theorem 14) and arithmetic (Theorem 18) analogies between G_{S} (when S ∩ S_{p} = ∅) and G_{S,ν} (for finitely indexed (S, ν)), as well as Sen’s Theorem and its Corollary 35, it is tempting to guess, that the property of being padically finite which the tame groups G_{S} are predicted to enjoy (by the Fontaine-Mazur conjecture), also applies to the groups G_{S,ν}. We now list this and some closely related ideas as questions. Let (S, ν) be a finitely indexed set of places of K.

Question 2. Is G_{S,ν} p-adically finite?

Question 3. Is every irreducible p-adic representation of G_{S,ν} everywhere potentially semistable?

Question 4. Does every irreducible p-adic representation of Gal(\overline{K}/K) coming from algebraic geometry and factoring through G_{S,ν} have finite image?

Question 5. Suppose L/K is an infinite p-extension in which the root discriminant remains bounded. Does every irreducible p-adic representation of Gal(L/K) have finite image?

Remark 7.1. 1. Positive answers to Questions 3 and 4 would show that the groups G_{S,ν} with bounded ramification, and not just the tame ones, are predicted by the main conjecture of Fontaine-Mazur ([FM], Conjecture 1) to be p-adically finite.

2. By theorem 18, Question 5 is more general than Question 4. When L/K is abelian, Question 5 has a positive answer (cf. Frey, Perret, Stichtenoth [FPS]).

3. The formulation of these questions in the introduction suggests that what is required is a better understanding the ramification in Galois extensions of global fields with p-adic analytic Galois group.

In the remainder of this section, we provide some partial evidence for a positive answer to Question 2.

7.1. Global extensions of local type. In section 6, we applied a fruitful notion introduced and studied by Wingberg [W]; here we will use further results from that paper.

Definition 37. A p-extension L of a number field K is said to be of local type if there exists a place 𝔭 of K such that the natural morphism:

D_{𝔭} →^{res} Gal(L_{𝔓}/K_{𝔭}) ↪ Gal(L/K),

is surjective (for some 𝔓 of L dividing 𝔭).

Let S be a finite set of primes of K and let S̃ be S if p ≠ 2 and the union of S with the real infinite places of K if p = 2. Let KS̃ the maximal p-extension of K, unramified for primes outside S (if p = 2, real places are allowed to become complex in KS̃). Wingberg [W] gives a necessary and sufficent condition for KS̃/K to be of local type.

Theorem 38. The extension KS̃/K is of local type at 𝔭_{0} if and only if

∑_{𝔭∈S̃\\{𝔭_{0}\}} δ_{𝔭} − δ_{K} + d(Δ_{S̃\\{𝔭_{0}\}}^{\{𝔭_{0}\}}) + r_{2} = \{ \begin{matrix}[K_{𝔭_{0}} : ℚ_{p}], 𝔭_{0}|p, \\ 0, otherwise. \end{matrix}

One corollary of this theorem is that if L/K is of local type, then for all intermediate fields K ⊆ F ⊆ L, F/K is of local type.

Remark 7.2. There is a slight misprint in the statement of this theorem in [W].

7.2. Structure of Gal(K_{S,ν}/K). We now give some evidence for Conjecture 2, namely examples of infinite wild extensions with bounded ramification admitting no infinite p-adic analytic quotients.

Theorem 39. Suppose (S, ν) is a finitely indexed set and let S ∩ S_{p} = {𝔭_{1}, ⋯, 𝔭_{s}} be the places in S above p. Define recursively: S_{0} = S_{S−S_{p}}, S_{i} = S_{i−1} ∪{𝔭_{i}}, S_{s} = S. We equip each S_{i} with natural indexing maps ν_{i}, for i = 1, ⋯, s: ν_{i} agrees with ν on S_{i} and equals 0 outside S_{i}. Assume that for i = 0, ⋯, s − 1, the extensions K_{S_{i+1},ν_{i+1}}/K_{S_{i},ν_{i}} are of local type at 𝔓_{i}, some prime of K_{S_{i},ν_{i}} dividing 𝔭_{i}. If K_{S−S_{p},ν_{0}}/K is finite (or if Conjecture 1 holds), then G_{S,ν} is p-adically finite.

Corollary 40. We consider the situation of theorem 26: p = 2, K = ℚ(\sqrt{−q}), q ≡ 7 mod 16 is prime, 2O_{K} = 𝔭_{1}𝔭_{2}. Take 0 < i, j < ∞, S = {𝔭_{1}, 𝔭_{2}}, and put ν_{i,j}(𝔭_{1}) = i, ν_{i,j}(𝔭_{2}) = j. Then G_{S,ν_{i,j}} is 2-adically finite.

Remark 7.3. For large i, j , G_{S,ν_{i,j}} is a group with three generators.

Corollary 41. Suppose p be an odd regular prime, and put K = ℚ(ζ_{p}). We have the factorization: pO_{K} = 𝔭^{p−1}. Let 1 < i < ∞, S = {𝔭} and ν_{i} defined by ν_{i}(𝔭) = i. The group G_{S,ν_{i}} has no infinite p-adic analytic quotients. (For large i, G_{S,ν_{i}} is a p-group with (p + 1)/2 generators).

These corollaries follow from Theorems 38 and 39. Note that we do not need to assume Conjecture 1 here, since the unramified part is trivial.

Proof. of Theorem 39. Let K ⊆ L ⊆ K_{S,ν} is an intermediate field such that Gal(L/K) is p-adic analytic. Put L_{i} = L ∩ K_{S_{i},ν_{i}}; in particular, L_{0} = L ∩ K_{S−S_{p}}, and L_{s} = L. The groups Gal(L_{i}/K) are all analytic. We proceed by induction on s.

• s = 0: By Conjecture 1, L_{0}/K is finite.

• Suppose that L_{i}/K is finite; we will show that L_{i+1}/K is finite. By hypothesis, the extension K_{S_{i+1},ν_{i+1}}/K_{S_{i}} is of local type at 𝔓_{i}, and this implies that L_{i+1}/L_{i} is of local type at 𝔓_{i} ∩ L_{i}.

Let F_{i} be the subfield of L_{i+1}/L_{i} fixed by an inertia group of 𝔓_{i} ∩ L_{i} in this extension. Then Gal(F_{i}/L_{i}) is cyclic or pro-p-cyclic and is unramified at 𝔭_{i}. We will show that Gal(F_{i}/L_{i}) is finite and then cyclic. In the extension F_{i}/K the ramification is bounded for all places 𝔭 of K. By Proposition 10 the ramification is bounded in F_{i}/L_{i} also, because [L_{i} : K] < ∞. And so Gal(F_{i}/L_{i}) is isomorphic to the ray group for a certain finite modulus and so is finite. We apply Corollary 35 to conlcude that [L_{i+1} : L_{i}] < ∞.

Remark 7.4. 1. Further evidence for Conjecture 2 can be obtained by adapting to our case the group-theoretic method of Boston [B1] and [B1].

2. Let L be an intermediate field of K_{S,ν}/K such that Gal(L/K) is p-adic analytic. Suppose that for all primes 𝔭 ∈ S_{p}, the inertia groups D^{0}(𝔓/𝔭) in L/K are all torsion. Since GL_{n}(ℚ_{p}) has no infinite torsion subgroups, these inertia groups are finite and so L/K is potentially tamely ramified: there exists a finite extension F/K, such that L/F is tamely ramified. It then would follow from the Fontaine-Mazur Conjecture that L/F is finite.

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Farshid Hajir Dept. of Mathematics California State University, San Marcos San Marcos CA 92096

fhajir@csusm.edu

Christian Maire Laboratoire A2X Université Bordeaux I Cours de la Libération 33405 Talence Cedex France

maire@math.u-bordeaux.fr

Current address:

Mathematical Sciences Research Institute 1000 Centennial Drive Berkeley, Califoria 94720 USA

maire@msri.org