On the Fontaine-Mazur Conjecture for CM-Fields

by Kay Wingberg at Heidelberg

In [3] Fontaine and Mazur conjecture (as a consequence of a general principle) that a number field k has no infinite unramified Galois extension such that its Galois group is a p-adic analytic pro-p-group. A counter-example to this conjecture would produce an unramified Galois representation with infinite image, that could not “come from geometry”. Some evidence for this conjecture is shown in [1] and [4].

Since every p-adic analytic pro-p-group contains an open powerful resp. uni form subgroup one is led to the question whether a given number field possesses an infinite unramified Galois p-extension with powerful resp. uniform Galois group. With regard to this problem, we would like to mention a result of Boston [1]:

Let p be a prime number and let k|k_{0} be a finite cyclic Galois extension of degree prime to p such that p does not divide the class number of k_{0}. Then, if the Galois group G(M|k) of an unramified Galois p-extension M of k, Galois over k_{0}, is powerful, it is finite.

In this paper we will prove a statement which is in some sense weaker as the above and in another sense stronger (and in view of the general conjecture very weak):

Let p be odd and let k be a CM-field with maximal totally real subfield k^{+} containing the group μ_{p} of p-th roots of unity. Let M = L(p) be the maximal unramified p-extension of k. Assume that the p-rank of the ideal class group Cl(k^{+}) of k^{+} is not equal to 1. Then, if the Galois group G(L(p)|k) is powerful, it is finite.

If the p-rank of Cl(k^{+}) is equal to 1, we have two weaker results. First, replacing the word powerful by uniform and assuming that the first step in the pcyclotomic tower of k is not unramified, then the statement above holds without any condition on Cl(k^{+}). Secondly, we consider the conjecture in the p-cyclotomic tower of the number field k. Denote the n-th layer of the cyclotomic ℤ_{p}-extension k_{∞} of k by k_{n} and let G(L_{n}(p)|k_{n}) be the Galois group of the maximal unramified p-extension L_{n}(p) of k_{n}. Then the following statement holds.

Let p ≠ 2 and let k be a CM-field containing μ_{p}. Assume that the Iwasawa μ-invariant of k_{∞}|k is zero. Then there exists a number n_{0} such that for all n ≥ n_{0} the following holds: If the Galois group G(L_{n}(p)|k_{n}) is powerful, then it is finite.

Similar results hold for the maximal unramified p-extension L_{S}(p) which is completely decomposed at all primes in S and for the maximal p-extension k_{S}(p) of k which is unramified outside S, if S contains no prime above p.

Of course, our main interest is the conjecture for general p-adic analytic groups. We will prove the following result.

Let p ≠ 2 and let k be a CM-field containing μ_{p} with maximal totally real subfield k^{+} and assume that μ_{p} ⊆ / k_{𝔭}^{+} for all primes 𝔭 of k^{+} above p. Then, if G(L_{k}(p)|k) is p-adic analytic, G(L_{k^{+}}(p)|k^{+}) is finite.

Unfortunately, we do not have Boston’s result for general analytic pro-p-groups. Otherwise, in the situation above it would follow that G(L_{k}(p)|k) is not an infinite p-adic analytic group.

1 A duality theorem

We use the following notation:

p  is a prime number,
k  is a number field,
S_{∞}  is the set of archimedean primes of k,
S  is a set of primes of k containing S_{∞},
E_{S}(k) is the group of S-units of k,
Cl_{S}(k) is the S-ideal class group of k,
L_{S}  is the maximal unramified extension of k

which is completely decomposed at S,

L_{S}(p) is the maximal p-extension of k inside L_{S},
L  is the maximal unramified extension of k,
L(p) is the maximal p-extension of k inside L.

We write E(k) for the group E_{S_{∞}}(k) of units of k and Cl(k) for the ideal class group Cl_{S_{∞}}(k) of k. Obviously,

L = L_{S_{∞}},  if k is totally imaginary,
L(p) = L_{S_{∞}}(p), if p ≠ 2 or k totally imaginary.

If K is an infinite algebraic extension of ℚ, then E_{S}(K) = \underrightarrow{lim}_{k} E_{S}(k) where k runs through the finite subextensions of K.

For a profinite group G, a discrete G-module M and any integer i the i-th Tate cohomology is defined by

Ĥ^{i}(G, M) = H^{i}(G, M) for i ≥ 1 and Ĥ^{i}(G, M) = lim_{⟵}_{U,def} Ĥ^{i}(G/U, M^{U}) for i ≤ 0,

where U runs through all open normal subgroups of G and the transition maps are given by the deflation, see [7].

Theorem 1.1 Let S be a set of primes of k containing S_{∞}. Then the following holds:

(i) There are canonical isomorphisms

Ĥ^{i}(G(L_{S}|k), E_{S}(L_{S})) ≅ Ĥ^{2−i}(G(L_{S}|k), ℚ/ℤ)^{∨}

for all i ∈ ℤ. Here ^{∨} denotes the Pontryagin dual. (ii) There are canonical isomorphisms

Ĥ^{i}(G(L_{S}(p)|k), E_{S}(L_{S}(p))) ≅ Ĥ^{2−i}(G(L_{S}(p)|k), ℚ_{p}/ℤ_{p})^{∨}

for all i ∈ ℤ.

Proof: Let C_{S}(L_{S}) be the S-idele class group of L_{S}. The subgroup C_{S}^{0}(L_{S}) of C_{S}(L_{S}) given by the ideles of norm 1 is a level-compact class formation for G(L_{S}|k) with divisible group of universal norms. From the duality theorem of Nakayama-Tate we obtain the isomorphisms

Ĥ^{i}(G(L_{S}|k), C_{S}(L_{S})) ≅ Ĥ^{2−i}(G(L_{S}|k), ℤ)^{∨}, i ∈ ℤ,

since Ĥ^{i}(G(L_{S}|k), C_{S}(L_{S})) ≅ Ĥ^{i}(G(L_{S}|k), C_{S}^{0}(L_{S})), see [7] proposition 4. Let K|k be a finite Galois extension inside L_{S}. From the exact sequence

0 ⟶ E_{S}(K) ⟶ J_{S}(K) ⟶ C_{S}(K) ⟶ Cl_{S}(K) ⟶ 0,

where J_{S}(K) denotes the group of S-ideles of K, which is a cohomological trivial G(K|k)-module (K|k is completely decomposed at S), we obtain isomorphisms

Ĥ^{i+1}(G(K|k), E_{S}(K)) ≅ Ĥ^{i}(G(K|k), D(K)),

where D(K) denotes the kernel of the surjection C_{S}(K) ↠ Cl_{S}(K), and a long exact sequences

⟶ Ĥ^{i}(G(K|k), D(K)) ⟶ Ĥ^{i}(G(K|k), C_{S}(K)) ⟶ Ĥ^{i}(G(K|k), Cl_{S}(K)) ⟶ .

If K′ is the maximal abelian extension of K in L_{S}, then G(L_{S}|K′) is an open subgroup of G(L_{S}|K) by the finiteness of the class number of K. The commutative diagram

\xymatrix{Cl_{S}(K′) \ar[r]^{norm} \ar[d]^{∼}_{rec} & Cl_{S}(K) \ar[d]^{∼}_{rec} \\ G(L_{S}|K′)^{ab} \ar[r]^{can} & G(L_{S}|K)^{ab}}

shows, since can is the zero map, that

Cl_{S}(K′) ⟶^{norm} Cl_{S}(K)

is trivial. It follows that

lim_{⟵}_{K} Ĥ^{i}(G(K|k), Cl_{S}(K)) = 0 for i ≤ 0.

Since all groups in the exact sequence above are finite, we can pass to the projective limit and we obtain isomorphisms

lim_{⟵}_{K} Ĥ^{i}(G(K|k), D(K)) ≅ Ĥ^{i}(G(L_{S}|k), C_{S}(L_{S})) for i ≤ 0,

and therefore isomorphisms

Ĥ^{i+1}(G(L_{S}|k), E_{S}(L_{S})) ≅ Ĥ^{i}(G(L_{S}|k), C_{S}(L_{S})) for i ≤ −1.

The last assertion also holds for i = 0: from the commutative diagram

Ĥ^{0}(G(K′|k), D(K′)) _{∼} ⟶^{δ} H^{1}(G(K′|k), E_{S}(K′))
↓^{def}  ↓
Ĥ^{0}(G(K|k), D(K)) _{∼} ⟶^{δ} H^{1}(G(K|k), E_{S}(K)),

where k ⊆ K ⊆ K′ are finite Galois extensions inside L_{S}, it follows that the limit lim_{⟵}_{K}H^{1}(G(K|k), E_{S}(K)) exists. Since

H^{1}(G(K|k), E_{S}(K)) ⊆ H^{1}(G(L_{S}|k), E_{S}(L_{S})) ≅ Cl_{S}(k)

and

\begin{aligned}lim_{⟵}_{K} Ĥ^{0}(G(K|k), D(K)) & ≅ & Ĥ^{0}(G(L_{S}|k), C_{S}(L_{S})) ≅ H^{2}(G(L_{S}|k), ℤ)^{∨} \\ & ≅ & H^{1}(G(L_{S}|k), ℚ/ℤ)^{∨} = G(L_{S}|k)^{ab} ≅ Cl_{S}(k), \end{aligned}

the projective limit lim_{⟵}_{K}H^{1}(G(K|k), E_{S}(K)) becomes stationary and is equal to

H^{1}(G(L_{S}|k), E_{S}(L_{S})).

For i ≥ 1 the exact sequence

0 ⟶ E_{S}(L_{S}) ⟶ J_{S}(L_{S}) ⟶ C_{S}(L_{S}) ⟶ 0

induces isomorphisms

H^{i}(G(L_{S}|k), C_{S}(L_{S})) ≅ H^{i+1}(G(L_{S}|k), E_{S}(L_{S})).

Putting all together, we obtain canonical isomorphisms

Ĥ^{i+1}(G(L_{S}|k), E_{S}(L_{S})) ≅ Ĥ^{2−i}(G(L_{S}|k), ℤ)^{∨} ≅ Ĥ^{1−i}(G(L_{S}|k), ℚ/ℤ)^{∨}

for all i ∈ ℤ. The proof for the field L_{S}(p) is analogous. □

Let k be a number field of CM-type with maximal totally real subfield k^{+} and let Δ = G(k|k^{+}) = ⟨σ⟩ ≅ ℤ/2ℤ. If p ≠ 2, we put as usual

M^{±} = (1 ± σ)M

for a ℤ_{p}[Δ]-module M. For a ℤ_{p}-module N let _{p}N = {x ∈ N | px = 0}.

Corollary 1.2 Let p be an odd prime number and let k be a CM-field. Let S be a set of primes of k^{+} containing S_{∞} and assume that no prime of S splits in the extension k|k^{+}. Then

dim_{𝔽_{p}} _{p}H^{2}(G(L_{S}(p)|k), ℚ_{p}/ℤ_{p})^{−} ≤ δ,

where δ is equal to 1 if k contains the group μ_{p} of p-th roots of unity and otherwise equal to 0.

Proof: By proposition 1.1, there is a Δ-invariant surjection

E_{S}(k) ↠ Ĥ^{0}(G(L_{S}(p)|k), E_{S}(L_{S}(p))) ≅ H^{2}(G(L_{S}(p)|k), ℚ_{p}/ℤ_{p})^{∨}

and so a surjection

(E_{S}(k)/p)^{−} ↠ (_{p}H^{2}(G(L_{S}(p)|k), ℚ_{p}/ℤ_{p})^{−})^{∨}.

Since no prime of S splits in the extension k|k^{+}, we have (E_{S}(k)/p)^{−} ≅ μ_{p}(k) which gives us the desired result. □

2 Powerful pro-p-groups with involution

Let p be a prime number. For a pro-p-group G the descending p-central series is defined by

G_{1} = G, G_{i+1} = (G_{i})^{p}[G_{i}, G] for i ≥ 1.

If a group Δ ≅ ℤ/2ℤ acts on G and p is odd, then we define

d(G)^{±} = dim_{𝔽_{p}}(G/G_{2})^{±} = dim_{𝔽_{p}} H^{1}(G, ℤ/pℤ)^{±}.

The following proposition also follows from Boston result (resp. its proof), but in our situation, where only an involution acts on G, we will give a simple proof.

Proposition 2.1 Let p ≠ 2 and let G be a finitely generated powerful pro-p-group with an action by the group Δ ≅ ℤ/2ℤ. Then the following holds:

If d(G)^{+} = 0, then G is abelian.

In particular, if d(G)^{+} = 0 and G^{ab} is finite, then G is finite.

Proof: Since G is powerful, we have

[G, G]/H ⊆ G^{p}H/H where H = ([G, G])^{p}[G, G, G].

From G/G_{2} = (G/G_{2})^{−} it follows that

[G, G]/H = ([G, G]/H)^{+} and G^{p}H/H = (G^{p}H/H)^{−},

since G/[G, G] = (G/[G, G])^{−} and G^{p} = {x^{p} | x ∈ G}, [2] theorem 3.6(iii), and so

(x^{p})^{σ} ≡ x^{−p} mod H for 1 ≠ σ ∈ Δ and x ∈ G.

We obtain

[G, G] ⊆ ([G, G])^{p}[G, G, G].

This implies [G, G] = 1. □

Lemma 2.2 Let p ≠ 2 and let G be a finitely generated pro-p-group with an action by the group Δ ≅ ℤ/2ℤ. Then the following inequalities hold:

d(G)^{+}· d(G)^{−} ≤ dim_{𝔽_{p}}(G_{2}/G_{3})^{−}−rank_{ℤ_{p}}(G^{ab})^{−}+dim_{𝔽_{p}} _{p}H^{2}(G, ℚ_{p}/ℤ_{p})^{−},
\binom{d(G)^{+}}{2} + \binom{d(G)^{−}}{2} ≤ dim_{𝔽_{p}}(G_{2}/G_{3})^{+}−rank_{ℤ_{p}}(G^{ab})^{+}+dim_{𝔽_{p}} _{p}H^{2}(G, ℚ_{p}/ℤ_{p})^{+}.

Proof: Let d^{±} = d(G)^{±}. From the exact sequences

0 ⟶ H^{1}(G/G_{2}, ℤ/pℤ) ⟶^{∼} H^{1}(G, ℤ/pℤ) ⟶ H^{1}(G_{2}, ℤ/pℤ)^{G}

⟶ H^{2}(G/G_{2}, ℤ/pℤ) ⟶ H^{2}(G, ℤ/pℤ)

and

0 ⟶(_{p}G^{ab})^{∨} ⟶ H^{2}(G, ℤ/pℤ) ⟶_{p}H^{2}(G, ℚ_{p}/ℤ_{p}) ⟶ 0

we obtain the inequalities

dim_{𝔽_{p}} H^{2}(G/G_{2}, ℤ/pℤ)^{±} ≤ dim_{𝔽_{p}}(G_{2}/G_{3})^{±} + dim_{𝔽_{p}}(_{p}G^{ab})^{±}
+ dim_{𝔽_{p}} _{p}H^{2}(G, ℚ_{p}/ℤ_{p})^{±}.

Let

G/G_{2} ≅ A_{1} ⊕ ⋯ ⊕ A_{d^{+}} ⊕ B_{1} ⊕ ⋯ ⊕ B_{d^{−}}

be a Δ-invariant decomposition into cyclic groups of order p such that A_{i} = A_{i}^{+} and B_{j} = B_{j}^{−}. For H^{2}(G/G_{2}, ℤ/pℤ) we obtain the Δ-invariant Künneth decomposition:

H^{2}(G/G_{2}, ℤ/pℤ) ≅ ⨁_{i=1}^{d^{+}} H^{2}(A_{i}, ℤ/pℤ)

⊕ ⨁_{i<j} H^{1}(A_{i}, ℤ/pℤ) ⊗ H^{1}(A_{j}, ℤ/pℤ)

⊕ ⨁_{i<j} H^{1}(B_{i}, ℤ/pℤ) ⊗ H^{1}(B_{j}, ℤ/pℤ)

⊕ ⨁_{i=1}^{d^{−}} H^{2}(B_{i}, ℤ/pℤ)

⊕ ⨁_{i,j} H^{1}(A_{i}, ℤ/pℤ) ⊗ H^{1}(B_{j}, ℤ/pℤ).

Counting dimensions yields

\begin{aligned}dim_{𝔽_{p}} H^{2}(G/G_{2}, ℤ/pℤ)^{+} & = & d^{+} + \binom{d^{+}}{2} + \binom{d^{−}}{2}, \\ dim_{𝔽_{p}} H^{2}(G/G_{2}, ℤ/pℤ)^{−} & = & d^{−} + d^{+}d^{−}. \end{aligned}

Since

d^{±} = rank_{ℤ_{p}}(G^{ab})^{±} + dim_{𝔽_{p}}(_{p}G^{ab})^{±},

we obtain the desired result. □

Proposition 2.3 Let p ≠ 2 and let G be a finitely generated powerful pro-p-group with an action by the group Δ ≅ ℤ/2ℤ. Then the following inequalities hold:

(i) d(G)^{+}· d(G)^{−} ≤ d(G)^{−} + dim_{𝔽_{p}} _{p}H^{2}(G, ℚ_{p}/ℤ_{p})^{−},

(ii) \binom{d(G)^{+}}{2} + \binom{d(G)^{−}}{2} ≤ d(G)^{+} + dim_{𝔽_{p}} _{p}H^{2}(G, ℚ_{p}/ℤ_{p})^{+}.

Proof: Since G is powerful, the Δ-invariant homomorphism

G/G_{2} −↠^{p} G_{2}/G_{3}

is surjective, see [2] theorem 3.6, and we obtain

dim_{𝔽_{p}}(G_{2}/G_{3})^{±} ≤ d(G)^{±}.

Using lemma 2.2, this proves the proposition. □

Now we analyze the case where G is a powerful pro-p-group which is a Poincaré group of dimension 3.

Proposition 2.4 Let p be odd and let P be a finitely generated powerful pro-p-group with an action of Δ ≅ ℤ/2ℤ.

(i) If P is uniform, then

\begin{aligned}dim_{𝔽_{p}} H^{2}(P, ℤ/pℤ)^{+} & = & \binom{d(P)^{+}}{2} + \binom{d(P)^{−}}{2}, \\ dim_{𝔽_{p}} H^{2}(P, ℤ/pℤ)^{−} & = & d(P)^{+}· d(P)^{−}. \end{aligned}

(ii) If P is uniform such that P^{ab} is finite and d(P)^{+} = 1, then

dim_{𝔽_{p}} _{p}H^{2}(P, ℚ_{p}/ℤ_{p})^{−} = 0.

(iii) If P is a Poincaré group of dimension 3 such that P^{ab} is finite, then

d(P)^{+} = 1 and d(P)^{−} = 2  or
d(P)^{+} = 3 and d(P)^{−} = 0.

Proof: Let P be uniform. By [2] definition 4.1 and theorem 4.26, we have

dim_{𝔽_{p}}(H^{1}(P_{2}, ℤ/pℤ)^{P})^{±} = d(P)^{±} and dim_{𝔽_{p}} H^{2}(P, ℤ/pℤ) = \binom{d(P)}{2}.

Counting dimensions shows that

dim_{𝔽_{p}} H^{2}(P/P_{2}, ℤ/pℤ) = dim_{𝔽_{p}} H^{1}(P_{2}, ℤ/pℤ)^{P} + dim_{𝔽_{p}} H^{2}(P, ℤ/pℤ),

and so the sequence

0 ⟶ H^{1}(P_{2}, ℤ/pℤ)^{P} ⟶ H^{2}(P/P_{2}, ℤ/pℤ) ⟶ H^{2}(P, ℤ/pℤ) ⟶ 0

is exact. Therefore

dim_{𝔽_{p}} H^{2}(P, ℤ/pℤ)^{±} = dim_{𝔽_{p}} H^{2}(P/P_{2}, ℤ/pℤ)^{±} − dim_{𝔽_{p}}(H^{1}(P_{2}, ℤ/pℤ)^{P})^{±},

which proves (i).

If P^{ab} is finite, then dim_{𝔽_{p}}(_{p}P^{ab})^{±} = d(P)^{±}, and so by (i)

dim_{𝔽_{p}} _{p}H^{2}(P, ℚ_{p}/ℤ_{p})^{−} = dim_{𝔽_{p}} H^{2}(P, ℤ/pℤ)^{−} − dim_{𝔽_{p}}(_{p}P^{ab})^{−}
= d(P)^{+}· d(P)^{−} − d(P)^{−}.

This gives us the desired result (ii).

Now let P be a powerful Poincaré group of dimension 3; in particular, P is torsionfree and therefore P is uniform, see [2] theorem 4.8. Since

dim_{𝔽_{p}} H^{1}(P, ℤ/pℤ) = dim_{𝔽_{p}} H^{2}(P, ℤ/pℤ)

and since P^{ab} is finite, the exact sequence

0 ⟶(_{p}P^{ab})^{∨} ⟶ H^{2}(P, ℤ/pℤ) ⟶_{p}H^{2}(P, ℚ_{p}/ℤ_{p}) ⟶ 0

shows that

(_{p}P^{ab})^{∨} ⟶^{∼} H^{2}(P, ℤ/pℤ).

It follows that

dim_{𝔽_{p}} H^{2}(P, ℤ/pℤ)^{±} = d(P)^{±},

and so by (i)

d(P)^{+}· d(P)^{−} = d(P)^{−}.

This proves (iii). □

3 On the Fontaine-Mazur Conjecture

We keep the notation of sections 1 and 2. Let

d_{k}^{±} = dim_{𝔽_{p}}(Cl(k)/p)^{±} = d(G(L(p)|k))^{±}.

Theorem 3.1 Let p be an odd prime number and let k be a CM-field such that (i) d_{k}^{−} ≠ 0, if μ_{p} ⊆ / k, (ii) d_{k}^{+} ≠ 1. Then, if the Galois group G(L(p)|k) of the maximal unramified p-extension L(p) of k is powerful, it is finite.

Proof: If d_{k}^{+} = 0, then the theorem follows from proposition 2.1. Therefore we assume that d_{k}^{+} ≥ 2 (assumption (ii)). From assumption (i) and Leopoldt’s Spiegelungssatz, see [8] theorem 10.11, it follows that d_{k}^{−} ≥ 1. From proposition 2.3 and corollary 1.2 we obtain the inequality

d_{k}^{+}d_{k}^{−} ≤ d_{k}^{−} + δ.

It follows that d_{k}^{+} = 2, d_{k}^{−} = 1 (and δ = 1), and so d(G(L(p)|k)) = 3.

If P = G(L(p)|k)_{i}, i large enough, then P is uniform, [2] theorem 4.2, and d(P) ≤ 3, [2] theorem 3.8. Suppose that P is non-trivial. Then P is a Poincaré group of dimension dim(P) = d(P) ≤ 3, see [5] chap.V theorem (2.2.8) and (2.5.8). But Poincaré groups of dimension dim(P) ≤ 2 have the group ℤ_{p} as homomorphic image, and so we can assume that dim(P) = d(P) = 3. Since G(L(p)|k) is powerful, we have a surjection

G(L(p)|k)/G(L(p)|k)_{2} ↠ G(L(p)|k)_{i}/G(L(p)|k)_{i+1}.

Furthermore, by [2] theorem 3.6(ii), G(L(p)|k)_{i+1} = (G(L(p)|k)_{i})_{2} = P_{2}, and so G(L(p)|k)_{i}/G(L(p)|k)_{i+1} = P/P_{2}. Therefore d(P)^{+} = 2 and d(P)^{−} = 1. By proposition 2.4(iii) we get a contradiction. □

If μ_{p} ⊆ k, then d_{k}^{+} = 1 is the only remaining case. Here we only get a weaker result. Let k_{∞} be the cyclotomic ℤ_{p}-extension of k and denote by k_{n} the n-th layer of k_{∞}|k.

Theorem 3.2 Let p ≠ 2 and let k be a CM-field containing μ_{p}. Assume that k_{1}|k is not unramified if d_{k}^{+} = 1. Then the Galois group G(L(p)|k) of the maximal unramified p-extension L(p) of k is not uniform.

Proof: Suppose that G = G(L(p)|k) is uniform. Using theorem 3.1, we may assume that d(G)^{+} = 1, and so, by proposition 2.4(ii),

dim_{𝔽_{p}} _{p}H^{2}(G, ℚ_{p}/ℤ_{p})^{−} = 0.

On the other hand, by theorem 1.1, we have a surjection

H^{2}(G, ℚ_{p}/ℤ_{p})^{∨} ≅ Ĥ^{0}(G, E(L(p))) ↠ Ĥ^{0}(G(K|k), E(K))

where K|k is a finite unramified Galois p-extension of CM-fields (recall that d(G)^{+} ≠ 0), and so a surjection

(H^{2}(G, ℚ_{p}/ℤ_{p})^{−})^{∨} ↠ Ĥ^{0}(G(K|k), E(K))^{−}.

Since K is of CM-type, it follows that

Ĥ^{0}(G(K|k), E(K))^{−} ≅ Ĥ^{0}(G(K|k), μ(K)(p)).

By our assumption, K is disjoint to k_{∞}, i.e. μ(K)(p) = μ(k)(p), and so

dim_{𝔽_{p}} Ĥ^{0}(G(K|k), μ(K)(p))/p = 1.

It follows that

dim_{𝔽_{p}} _{p}H^{2}(G, ℚ_{p}/ℤ_{p})^{−} = 1.

This contradiction proves the theorem. □

Remarks:

(1) The theorems 3.1 and 3.2 also hold in the following situation: Replace L(p) by L_{S}(p) and Cl by Cl_{S} where S ⊇ S_{∞} is a set of primes which do not split in the extension k|k^{+}. Use corollary 1.2 for S instead of S_{∞}.

(2) Theorem 3.1 is also true, if we replace L(p) by the maximal p-extension k_{S}(p) of k which is unramified outside a finite set S which contains S_{∞} but no prime above p. Instead of Cl(k) one has to take the ray class group C(k)/C^{𝔪}(k) mod 𝔪 = ∏_{𝔭∈S} 𝔭 (which is finite). In order to prove an analog of corollary 1.2, use the exact sequence

0 ⟶ E^{S}(K) ⟶ J_{S_{∞}}(K) × U_{S′}^{1}(K) ⟶ C_{S}(K) ⟶ C(K)/C^{𝔪}(K) ⟶ 0

where S′ = S\S_{∞} and U_{S′}^{1}(K) is the product over the principal units at the places of S′ and E^{S}(K) = ker(E(K) → U_{S′}(K)/U_{S′}^{1}(K)).

Now we consider the Galois groups G(L_{n}(p)|k_{n}) of the maximal unramified p-extension L_{n}(p) of k_{n} in the p-cyclotomic tower of k.

Theorem 3.3 Let p ≠ 2 and let k be a CM-field containing μ_{p}. Assume that the Iwasawa μ-invariant of the cyclotomic ℤ_{p}-extension k_{∞}|k is zero.

Then there exists a number n_{0} such that for all n ≥ n_{0} the following holds: If the Galois group G(L_{n}(p)|k_{n}) is powerful, then it is finite.

Proof: Let

1 ⟶ G_{∞} ⟶ G(L_{∞}(p)|k) ⟶ Γ ⟶ 1

where G_{∞} = G(L_{∞}(p)|k_{∞}) is the Galois group of the maximal unramified pextension L_{∞}(p) of k_{∞} and Γ = G(k_{∞}|k) = ⟨γ⟩. Let Γ_{n} = ⟨γ^{p^{n}}⟩, n ≥ 0, be the open subgroups of Γ of index p^{n}. By our assumption on the Iwasawa μ-invariant G_{∞} is a finitely generated pro-p-group.

Let n_{1} be large enough such that all primes of k_{n_{1}} above p are totally ramified in k_{∞}|k_{n_{1}} and let ⟨γ_{j}⟩ ⊆ G(L_{∞}(p)|k_{n_{1}}), j = 1, …, s, be the inertia groups of some extensions of the finitely many primes 𝔭_{1}, … 𝔭_{s} of k_{n_{1}} above p.

For n ≥ n_{1} let

M_{n} = (γ_{j}^{p^{n−n_{1}}}, j = 1, …, s) ⊆ G(L_{∞}(p)|k_{n})

be the normal subgroup generated by all conjugates of the elements γ_{j}^{p^{n−n_{1}}} and

N_{n} := M_{n} ∩ G_{∞} = (γ_{i}^{p^{n−n_{1}}}γ_{j}^{−p^{n−n_{1}}}, [γ_{j}^{p^{n−n_{1}}}, g], i, j = 1, …, s, g ∈ G_{∞}).

Then the commutative exact diagram

[figure: 1; N_{n}; M_{n}; Γ_{n}; 1; 1; G_{∞}; G(L_{∞}(p)|k_{n}); Γ_{n}; 1]

shows that

G_{∞}/N_{n} ≅ G(L_{n}(p)|k_{n})

and we have canonical surjections

G_{∞} ↠ G(L_{m}(p)|k_{m}) ↠ G(L_{n}(p)|k_{n})

for m ≥ n ≥ n_{1}.

Let n_{0} ≥ n_{1} be large enough such that

G_{∞}/(G_{∞})_{3} ⟶^{∼} G(L_{n}(p)|k_{n})/(G(L_{n}(p)|k_{n}))_{3}

for all n ≥ n_{0}, i.e.

G(L_{∞}(p)|k_{n})/(G_{∞})_{3} = G_{∞}/(G_{∞})_{3} × Γ_{n} ≅ G(L_{n}(p)|k_{n})/(G(L_{n}(p)|k_{n}))_{3} × Γ_{n}.

Then ⟨γ_{j}^{p^{n−n_{1}}}⟩ acts trivially on G_{∞}/(G_{∞})_{3} for all j ≤ s and N_{n} is contained in (G_{∞})_{3}.

Suppose that G(L_{n}(p)|k_{n}), n ≥ n_{0}, is powerful. Then

[G_{∞}, G_{∞}] ⊆ (G_{∞})^{p}N_{n}.

By assumption on n_{0} the group N_{n} is contained in (G_{∞})_{3}, and so

[G_{∞}, G_{∞}] ⊆ (G_{∞})^{p}[G_{∞}, [G_{∞}, G_{∞}]].

From this inclusion it follows that

[G_{∞}, G_{∞}] ⊆ (G_{∞})^{p},

thus G_{∞} is powerful.

Using proposition 2.1, we can assume that

d_{k_{n}}^{+} = dim_{𝔽_{p}}(Cl(k_{n})/p)^{+} ≥ 1.

Let K|k_{n} be an unramified Galois extension of degree p such that G(K|k_{n}) = G(K|k_{n})^{+} and let K_{∞} = k_{∞}K. Because of our definition of n_{1} the field K is not contained in k_{∞} and G(L_{∞}(p)|K_{∞}) is a normal subgroup of G(L_{∞}(p)|k_{∞}) of index p.

Using results of Iwasawa theory, [6] (11.4.13) and (11.4.8), we obtain

d(G(L_{∞}(p)|K_{∞}))^{−} = p(d(G(L_{∞}(p)|k_{∞}))^{−} − 1) + 1.

From [2] theorem 3.8 and the equality above it follows that

d(G(L_{∞}(p)|k_{∞}))^{+} + d(G(L_{∞}(p)|k_{∞}))^{−}
= d(G(L_{∞}(p)|k_{∞}))
≥ d(G(L_{∞}(p)|K_{∞}))
= d(G(L_{∞}(p)|K_{∞}))^{+} + d(G(L_{∞}(p)|K_{∞}))^{−}
= d(G(L_{∞}(p)|K_{∞}))^{+} + p(d(G(L_{∞}(p)|k_{∞}))^{−} − 1) + 1.

The maximal quotient G(L_{∞}(p)|k_{∞})_{Δ} of G(L_{∞}(p)|k_{∞}) with trivial action of Δ is also powerful and we have d(G(L_{∞}(p)|k_{∞})_{Δ}) = d(G(L_{∞}(p)|k_{∞}))^{+}. Using again [2] theorem 3.8, we get

d(G(L_{∞}(p)|k_{∞}))^{+} ≥ d(G(L_{∞}(p)|K_{∞}))^{+}.

Both inequalities together imply

d(G(L_{∞}(p)|k_{∞}))^{−} ≤ 1.

Using [6] (11.4.4), we finally obtain

d(G(L_{∞}(p)|k_{∞}))^{+}, d(G(L_{∞}(p)|k_{∞}))^{−} ≤ 1.

It follows that G(L_{n}(p)|k_{n}) is a powerful pro-p-group with d(G(L_{n}(p)|k_{n})) ≤ 2. If G(L_{n}(p)|k_{n}) is not finite, then it contains an open subgroup P which is a Poincaré group (see [5] chap.V theorem (2.2.8) and (2.5.8)) of dimension dim P = d(P) ≤ 2 (use again [2] theorem 3.8). But these groups have the group ℤ_{p} as homomorphic image. By the finiteness of the class number it follows that G(L_{n}(p)|k_{n}) is finite.

□

Remark: Theorem 3.3 also holds if we replace L(p) by L_{Σ}(p) and Cl by Cl_{Σ}, where Σ = S_{∞} ∪ S_{p} is the set of archimedean primes and primes above p, and if we assume that no prime of S_{p} splits in the extension k|k^{+}.

Now we consider the conjecture for general p-adic analytic groups. Let

1 ⟶ D ⟶ G ⟶ G ⟶ 1

be an exact sequence of pro-p-groups. For an open normal subgroup H of G we denote the pre-image of H in G by H. Thus we get a commutative exact diagram

[figure: 1; D; G; G; 1; 1; D; H; H; 1.]

Proposition 3.4 With the notation as above assume that

(i) G is finitely generated and cd_{p} G ≤ 2, (ii) cd_{p} G < ∞, (iii) the Euler-Poincaré characteristic of G is zero, i.e.

χ(G) = ∑_{i=0}^{2}(−1)^{i} dim_{𝔽_{p}} H^{i}(G, ℤ/pℤ) = 0 .

Then d(H) is unbounded for varying open normal subgroups H of G or cd_{p} G ≤ 2.

Proof: Suppose that dim_{𝔽_{p}} H^{1}(H, ℤ/pℤ) is bounded for varying H. Since χ(G) = 0, the same is true for dim_{𝔽_{p}} H^{2}(H, ℤ/pℤ). It follows that H^{i}(D, ℤ/pℤ) is finite for i = 1, 2. By [6] proposition (3.3.7), we obtain

cd_{p} G = cd_{p} G + cd_{p} D ≥ cd_{p} G.

This proves the proposition. □

As an application to our problem we get the following result for the maximal unramified p-extension L_{k}(p) of a number field k.

Theorem 3.5 Let p ≠ 2 and let k be a CM-field containing μ_{p} with maximal totally real subfield k^{+}. Assume that μ_{p} ⊆ / k_{𝔭}^{+} for all primes 𝔭 of k^{+} above p. Then the following holds:

either  (i) G(L_{k^{+}}(p)|k^{+}) is finite,
or  (ii) G(L_{k}(p)|k) is not p-adic analytic,

with other words, if G(L_{k}(p)|k) is p-adic analytic, then G(L_{k^{+}}(p)|k^{+}) is finite.

Proof: Suppose that (i) and (ii) do not hold. Then the maximal quotient G(L_{k^{+}}(p)|k^{+}) of the p-adic analytic group G(L_{k}(p)|k) with trivial action by Δ = G(k|k^{+}) is an infinite analytic group. Passing to a finite extension of k^{+}, we may assume that G(L_{k^{+}}(p)|k^{+}) is uniform (our assumptions on k are still valid). The dimension of G(L_{k^{+}}(p)|k^{+}) is greater or equal to 3, since otherwise it would have the group ℤ_{p} as quotient which is impossible by the finiteness of the class number.

If k_{S_{p}}^{+}(p) is the maximal p-extension of k^{+} which is unramified outside p, then cd_{p} G(k_{S_{p}}^{+}(p)|k^{+}) ≤ 2 and χ(G(k_{S_{p}}^{+}(p)|k^{+})) = 0, see [6] (8.3.17), (8.6.16) and (10.4.8). Applying proposition 3.4, we obtain that

dim_{𝔽_{p}} H^{1}(G(k_{S_{p}}^{+}(p)|K^{+}), ℤ/pℤ) = dim_{𝔽_{p}} H^{1}(G(k_{S_{p}}(p)|K^{+}(μ_{p})), ℤ/pℤ)^{+}

is unbounded, if K^{+} varies over the finite Galois extension of k^{+} inside L_{k^{+}}(p). By [6] theorem (8.7.3) and the assumption that μ_{p} ⊆ / k_{𝔭}^{+} for all primes 𝔭|p, it follows that

d(G(L_{k}(p)|K^{+}(μ_{p})) = dim_{𝔽_{p}} Cl(K^{+}(μ_{p}))/p
≥ dim_{𝔽_{p}}(Cl_{S_{p}}(K^{+}(μ_{p}))/p)^{−}
= dim_{𝔽_{p}} H^{1}(G(k_{S_{p}}(p)|K^{+}(μ_{p})), ℤ/pℤ)^{+} − 1

is unbounded for varying K^{+} inside L_{k^{+}}(p) and therefore G(L_{k}(p)|k) is not p-adic analytic. This contradiction proves the theorem. □

References

[1] Boston, N. Some cases of the Fontaine-Mazur conjecture II. J. Number Theory 75 (1999), 161-169

[2] Dixon, J.D., du Sautoy, M.P.F., Mann, A., Segal, D. Analytic pro-p Groups LMS Lectures Notes 157, Cambridge University Press 1991

[3] Fontaine, J-M., Mazur B. Geometric Galois representations. In: Elliptic Curves, Modular Forms, & Fermat’s Last Theorem, edited by J. Coates, and S.T. Yau. International Press, Boston 1995

[4] Hajir F. On the groth of p-class groups in p-class towers J. Algebra 188 (1997), 256-271

[5] Lazard, M. Groupes analytiques p-adiques. Inst. Hautes Études Scientifiques, Publ. Math. 26 1965

[6] Neukirch, J., Schmidt, A., Wingberg, K. Cohomology of Number Fields. Springer 2000

[7] Schmidt, A. On Poitou’s duality theorem. J. reine angew. Math. 517 (1999), 145-160

[8] Washington, L.C. Introduction to Cyclotomic Fields. Springer 1982 (Second ed. 1997)

Mathematisches Institut der Universität Heidelberg Im Neuenheimer Feld 288 69120 Heidelberg Germany e-mail: wingberg@mathi.uni-heidelberg.de

16