Bounds For Étale Capitulation Kernels II

Let p be an odd prime and E/F a cyclic p-extension of number fields. We give a lower bound for the order of the kernel and cokernel of the natural extension map between the even étale K-groups of the ring of S-integers of E/F , where S is a finite set of primes containing those which are p-adic. Bornes pour les noyaux de capitulations II Résumé Soit p un nombre premier impair et E/F une p-extension cyclique de corps de nombres. Nous donnons une minoration pour l’ordre du noyau et conoyau de l’application naturelle d’extension entre les K-groupes étales des anneaux de Sentiers de E/F où S est un ensemble fini de places contenant les places p-adiques.


Introduction
Let F be an algebraic number field and let p be an odd prime number.For a finite set S of primes of F containing the primes above p, let o S F denote the ring of S-integers of F .For a Galois p-extension E of F with Galois group G which is unramified outside S, the kernel and the cokernel of the natural functorial map between the even étale K-groups )) G are described by the cohomology of odd étale K-groups K ét 2i−1 (o S E ).So using Borel's results on the abelian group structure of odd K-groups, one can give an upper bound for the rank of the finite p-groups ker(f i ) and coker(f i ), as explained by B. KAHN [8, section 4], by means of the number of real and complex embeddings of the number field F .In [1], partially answering a question asked by B. KAHN loc.cit., we gave a lower bound for the order of ker(f i ) and coker(f i ), in the case where the extension E/F is cyclic of degree p in terms of tamely ramified primes.Our purpose in the present paper is to similarly treat the case where E/F is cyclic of degree p n , n ≥ 1.
When the number field F contains a primitive p-th root of unity ζ p , the classical Tate kernel D F consists of the non-zero elements a of F , such that the symbol {a, ζ p } is trivial in K 2 F .Obviously, D F lies between F • , the multiplicative group of non zero elements of F and F •p .It is known that the factor group D F /F •p is of rank 1 + r 2 , where r 2 is the number of complex embeddings of F [14].When F satisfies Leopoldt's conjecture at the prime p, the Kummer radical F of the compositum of the first layers of Z p -extensions of F has the same size : Answering a question raised by J. COATES [2], R. GREENBERG showed that even though in general A F = D F , they coincide when the base field F contains enough p-primary roots of unity [4].
More generally, when F contains the p n -th roots of unity, for each integer i ≥ 2, there exists a subgroup F of the compositum of the n-th layers of Z p -extensions of F , provided F contains enough p-primary roots of unity.We then obtain our lower bound by minorizing the norm index [A (n) in terms of the ramification indices in E/F of non-p-adic primes belonging to the same "primitive" set for (F, p) (Proposition 4.3).
At the end of the paper, we treat the case where the base field F is "p-regular" and all the tamely ramified primes in E/F belong to the same primitive set.In particular, we show that there are infinitely many cyclic extensions E/F of degree p n , such that the order of the kernel (or the cokernel) takes any prescribed value between 1 and the trivial upper bound p n(1+r 2 ) .

A lower bound via the Tate kernel
Suppose that E/F is a cyclic extension of degree p n with Galois group G, and that F contains the p n -th roots of unity µ p n .Denote by S the set of p-adic primes, as well as those which ramify in E/F .Throughout this paper i is an integer ≥ 2. The exact sequence where H 1 (F, ) denotes the first continuous cochain cohomology group of the absolute Galois group G F of F and, for any G F -module M , the notation M (i) is the i-fold Tate twisted module M [14].Thus there exists a subgroup the analogue of the Tate-kernel in the case of i = 2 and n = 1 -, such that Since the odd étale K-groups satisfy Galois descent, we have [1, Section 1]: ∩ N E/F (E • ), we have the following lower bound for the order of the kernel or the cokernel of the natural natural functorial map between the even étale K-groups ) is trivial, so that ker(f i ) and coker(f i ) have the same order): Proposition 2.1.Let E/F be a cyclic extension of degree p n of algebraic number fields containing µ p n .Then A detailed account of these generalized Tate kernels D (i,n) F can be found in [6,15], see also [9] for the case n = 1.

Tate kernel and Kummer radical
In this section, we fix a positive integer n and assume that our base number field F contains the p n -th roots of unity µ p n .Let µ p ∞ := ∪ m≥1 µ p m be the group of all p-primary roots of unity and F ∞ := F (µ p ∞ ) be the cyclotomic Z p -extension of F .Denote by F n the n-th layer in F ∞ and by Γ the Galois group Gal(F ∞ /F ).Fix a topological generator γ of Γ in order to identify the Iwasawa algebra Z p [[Γ]] with the power series algebra considered as a discrete group on which Γ acts through the first factor.Let F be the compositum of all Z p -extensions of F and A (n) Following Greenberg [4], Γ )} and one can establish as in [1, page 204] that for all i ≥ 2 Here Div stands for the maximal divisible subgroup.Let K ∞ be the maximal abelian pro-p-extension of F ∞ .Kummer theory yields a perfect pairing [7, Section 7] Now let M ∞ be, as usual, the maximal abelian pro-p-extension of F ∞ unramified outside p and Let N ∞ be the subfield of M ∞ fixed by the torsion submodule Tor Λ (X ∞ ).Denote by N the subgroup of K corresponding to the field N ∞ by the above pairing.For every integer i, we then have a perfect pairing It is well known that X is a submodule of Λ r 2 of finite index.The quotient module H F := Λ r 2 /X is isomorphic as an abelian group to the kernel of the natural map K 2 F n −→ K 2 F ∞ , for n large [2].The exponent of the finite group H F will play an important role in what follows and will be henceforth denoted by p e .
From the above pairing we see that for all i ∈ Z, The following lemma generalizes [1, Lemma 2.1] to the case of cyclic extensions of degree p n with which we are dealing: Proof.As in the proof of [1, Lemma 2.1], we have, for each integer i, Let Y i := X(i) ∩ T (Λ r 2 (i)) + p n X(i).We have to show that the two submodules Y i and Y j are the same for any two integers i and j such that j ≡ i (mod p r ).
Let κ be the cyclotomic character and recall that γ, which we have already fixed, is a topological generator of Γ. Denote the action of T on Λ r 2 (i) by T (i) := κ(γ) i γ − 1.Each element y ∈ Y i can be written as y = T (i) λ + p n x, with T (i) λ ∈ X, for a λ ∈ Λ r 2 and an x ∈ X. Write y = (T (i) − T (j) )λ + T (j) λ + p n x.Since, by hypothesis µ p n+e−r ⊂ F , we have κ(γ) ≡ 1(mod p n+e−r ).
Moreover p r dividing i − j, we obtain from the preceding congruence κ(γ) i−j ≡ 1(mod p n+e ).
Thus (T (i) − T (j) )Λ r 2 is contained in p n+e Λ r 2 .On the other hand, as an abelian group X/Y j (Z/p n Z) r 2 is of exponent p n , so the exponent of Λ r 2 /Y j is at most p n+e .Thus (T (i) − T (j) )Λ r 2 ⊂ Y j .The element T (j) λ of T (Λ r 2 (j)) is also in X because y, (T (i) − T (j) )λ and p n x are in X.We conclude that y is in Y j .The lemma follows.
By duality, the previous lemma then shows that under the same conditions In particular, putting j = 0: Recall now that for any rational integer i ≥ 2 [13] Div(N (i − 1) Γ ) = Div(K(i − 1) Γ ) and for any i = 1 the above equality is conjectured to be true (Greenberg, Schneider).The case i = 0 corresponds to the Leopoldt conjecture for the base number field F at the prime p.Thus we have the following corollaries: Corollary 3.2.For two integers i ≥ 2 and j ≥ 2, if j ≡ i (mod p r ) for an integer r ≤ n + e, then provided µ p n+e−r ⊂ F .Recall our assumption that F always contains at least µ p .
In the following corollaries, we put j = 0 and i ≥ 2.

Corollary 3.3. Assume the number field F contains µ p and satisfies
Leopoldt's conjecture at the prime p. Then Since µ p ⊂ F , for m large, the m-th layer F m of the cyclotomic Z pextension of F contains enough p-primary roots of unity and the condition µ p n+e−r ⊂ F m is automatically satisfied: Corollary 3.4.Assume that the layers F m of the cyclotomic Z p -extension of F satisfy Leopoldt's conjecture at the prime p.Then, we have The preceding corollaries generalize those of [1, Section 2] where the case of cyclic extensions of degree p is treated.

Bounds For The Higher étale capitulation Kernels
Let E/F be a cyclic extension of algebraic number fields of degree p n , containing µ p n , with Galois group G.The set S consists of a finite set of primes containing S p and those primes which ramify in E/F .Since the étale K-groups K ét 2i−1 F are finitely generated Z p -modules of rank r 2 and have cyclic torsion subgroup, we have the following upper bound for the kernel or the cokernel of the natural extension map where i ≥ 2 and r 2 is the number of complex places of F .We also recall that the maps f i are not injective once a non-p-adic prime ramifies in Assume that the number field F contains µ p n .Let Fn be the compositum of the n-th layers of the Z p -extensions of F .By the definition of the Kummer radical A (n) F , we have a perfect pairing Definition 4.1.([3, 10, 11, 12]) A set S of finite primes of F containing S p is called primitive for (F, p) if the Frobenius "attached" to the primes v in S − S p generate a direct summand in Gal( F /F ) of Z p -rank the cardinality of S − S p , where F is the compositum of all the Z p -extensions of F .
Let S −S p = {v 1 , v 2 , • • • , v s } be the set of non-p-adic primes which ramify in E/F .We extract from this a set S p ∪ {v 1 , v 2 , • • • , v t } primitive for (F, p).Denote by σ j := σ j ( Fn /F ) the Frobenius "attached" to the prime v j in the extension Fn /F .We consider Gal( Fn /F ) as a naturally free Z/p n Zmodule.By the definition of primitivity, the set Here δ F dentoes the default of Leopoldt's conjecture for (F, p).Introduce the dual basis {a 1 , • • • , a 1+r 2 +δ F } with respect to the above pairing: Here ζ p n is a fixed primitive p n -th root of unity.In particular, for each j, the prime v j remains inert in F ( p n √ a j ) and splits in Let v be any of the primes in Denote by w a prime of E above v.Let F v , E w be the completion of F and E at v and w respectively.The natural composite map showing that The following lemma gives the order of this cyclic group: Proof.By construction, all the a k for k = j belong to N Ew/Fv (E • w ) (since . Let ( , ) v be the Hilbert symbol in the local field F v with values in µ p n .For any integer α, we have the following equivalences: Since the extension F v ( p n √ a)/F v is unramified of degree p n , this last norm group consists of all elements whose valuation is exactly p n .Accordingly, a p α ∈ N Ew/Fv (E • w ) precisely when p n−α divides the valuation of b in F v .Finally, we have: w ) is exactly p e , as was to be shown.

Now consider the canonical map
where the set T := S p ∪ {v 1 , v 2 , • • • , v t } consists of a primitive set for (F, p) inside S. The map ϕ is obviously injective.On the other hand, by the construction of the dual basis a j , we have Therefore, the map ϕ is in fact an isomorphism.Now by the previous lemma, the target group is of order p e 1 +•••+et where p e j ≥ p is the ramification index of the non-p-adic prime v j in the cyclic p-extension E/F .Accordingly Proposition 4.3.Let E/F be a cyclic extension of degree p n containing µ p n .Let {v 1 , • • • , v t } consist of a set of tamely ramified primes in E/F belonging to a primitive set for (F, p).We then have the following lower bound for the norm index in the Kummer radical A (n) F of the n-th layers of the Z p -extensions of F : where p e j is the ramification index of v j in E/F .Combining this proposition with the results of the previous sections we get the following lower bound for the kernel or the cokernel of the natural map f Proof.We successively have In the classical case of i = 2, we necessarily have r = 0 and obtain: Corollary 4.5.Let F be a number field satisfying Leopoldt's conjecture at the prime p and let µ p n ⊂ F .Let E/F be a cyclic extension of degree p n .Let {v 1 , • • • , v t } consist of a maximal set of tamely ramified primes in E/F belonging to a primitive set for (F, p).Denote by p e j ≥ p the ramification index of v j in E/F .If µ p n+e ⊂ F , then we have the following lower bound for the kernel and the cokernel of the natural extension map of the tame kernels f : K 2 (o S F ) −→ K 2 (o S E ) G .A set T primitive for (F, p) is said to be maximal when T − S p is as large as possible.When F satisfies Leopoldt's conjecture, this is the case where T − S p contains exactly 1 + r 2 primes, r 2 being the number of nonconjugate complex embeddings of F .When amongst totally and tamely ramified primes in E/F one can extract a set {v 1 , • • • , v 1+r 2 } sitting in a primitive set, then the method developed here gives the exact size of | ker(f i )| = |coker(f i )|: Corollary 4.6.Let F be a number field satisfying Leopoldt's conjecture at the prime p and let µ p n ⊂ F .Let E/F be a cyclic extension of degree p n .Assume there exists a primitive set T for (F, p) which is maximal, and such that each v ∈ T − S p is totally ramified in E/F .Then | ker(f i )| = |coker(f i )| = p n(1+r 2 ) , provided µ p n+e−r ⊂ F for an integer r ≤ n + e such that p r | i.
To finish, we establish that for each non-negative integer t ≤ 1 + r 2 , there exist cyclic extensions E/F of degree p n where the order of ker(f i ) is exactly p nt .Start with the following short exact sequence prescribed ramified index p ev in E/F .Thus, according to the preceding proposition, for each p-regular number field F with r 2 non-conjugate complex embeddings, and for each p-power (given in advance) p m ≤ p n(1+r 2 ) , we can find infinitely many cyclic extensions E of F of degree p n , such that | ker(f i ) |=| coker(f i ) |= p m .
and the order of coker(f i ) is minorized by the norm index in the generalized Tate kernel D (i,n) F (Proposition 2.1).Following Greenberg's method, one can show that, once again under Leopoldt's conjecture, D (i,n) F turns out to be the Kummer radical A (n) by definition of e, F v ( p n−e √ b) being the maximal unramified extension of F v contained in E w = F v ( p n √ b), we conclude that the order of the class of a in A (n) which we are interested in.Theorem 4.4.Let F be a number field satisfying Leopoldt's conjecture at the prime p.Let E/F be a cyclic extension of degree p n .Let {v 1 , • • • , v t } consist of a set of tamely ramified primes in E/F belonging to a primitive set for (F, p).Denote by p e j ≥ p the ramification index of v j in E/F and by p e the exponent of H