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Bounds for étale capitulation kernels. (English) Zbl 1163.11347
From the 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 $$p$$ and the infinite primes of $$F$$, let $$o_F^S$$ denote the ring of $$S$$-integers of $$F$$. It is well known that the étale $$K$$-theory groups of $$o_F^S$$ are closely related to the arithmetic of the field $$F$$, with respect to the prime $$p$$. In particular, the odd étale $$K$$-groups, $$K_{2i-1}^{\text{ét}}(o_F^S)$$, $$i\geq 1$$, play a role similar to $$U_F\otimes\mathbb Z_p$$ where $$U_F$$ is the group of units of $$F$$, and the even étale $$K$$-groups $$K_{2i-2}^{\text{ét}}(o_F^S)$$ contain remarkable subgroups (the étale wild kernels) which behave like the $$p$$-part of the class group of $$F$$. In particular, if $$E$$ is a Galois $$p$$-extension of $$F$$ with Galois group $$G$$ which is unramified outside $$S$$, then the kernel and the cokernel of the natural map $$f_i:K_{2i-2}^{\text{ét}}(o_F^S)\to (K_{2i-2}^{\text{ét}}(o_F^S))^G$$ are described by the cohomology of $$K_{2i-1}^{\text{ét}}(o_F^S)$$. Although the Galois module structure of $$K_{2i-1}^{\text{ét}}(o_F^S)$$ is not well understood, we know that when $$G$$ is cyclic, the Herbrand quotient $$h(G,K_{2i-1}^{\text{ét}}(o_F^S))$$ is trivial, i.e. $$\ker(f_i)$$ and $$\text{coker}(f_i)$$ have the same order. 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 $$\text{coker}(f_i)$$ as explained by B. Kahn [K-Theory 7, No. 1, 55–100 (1993; Zbl 0780.12007), Section 4], by means of the number of the real and complex embeddings of the number field $$F$$.
In contrast, as mentioned in the introduction of loc. cit., giving a lower bound for the rank of $$\ker(f_i)$$ and $$\text{coker}(f_i)$$ proves to be difficult. Our purpose in the present paper is to give partial answers to this problem in the case where $$F$$ contains a primitive pth root of unity $$\zeta_p$$ and $$G$$ is a cyclic group of order $$p$$. We prove, under some mild hypotheses, that the rank of $$\ker(f_i)$$ is bounded by means of the number of some specified tamely ramified primes, more precisely, by the maximal number $$t$$ of the tamely ramified primes in $$E/F$$ which belong to a ‘primitive’ set for $$(F,p)$$ (Theorem 3.8). In particular, if $$S$$ contains a ‘maximal’ primitive set, then the rank of $$\ker(f_i)$$ (as well as that of $$\text{coker}(f_i)$$ is exactly $$1+r_2$$, where $$r_2$$ is the number of non-conjugate complex embeddings of $$F$$.
To prove the result, we first relate the rank of $$\text{coker}(f_i)$$ to the norm index of the Tate kernel $$D_E^{(i)}$$ of $$E$$ in $$D_F^{(i)}$$. When $$i=2$$, the Tate kernel $$D_F^{(2)}$$ consists of the non-zero elements $$a$$ of $$F^\bullet$$, such that $$\{a,\zeta_p\}$$ is trivial in $$K_2(F)$$. It is known that $$D_F^{(2)}/F^{\bullet p}$$ is of order $$p^{1+r_2}$$ and Greenberg shows, under Leopoldt’s conjecture, that $$D_F^{(2)}$$ coincides with the Kummer radical $$A_F$$ of the compositum of the first layers of $$\mathbb Z_p$$-extensions of $$F$$, if $$F$$ contains the $$p^n$$th roots of unity for $$n$$ large enough or if we replace $$F$$ by a sufficiently high layer of the cyclotomic $$\mathbb Z_p$$-extension of $$F$$ [R. Greenberg, Am. J. Math. 100, 1235–1245 (1978; Zbl 0408.12012)]. We prove that, under the same hypotheses, this is also the case for all $$i\geq 2$$: $$D_F^{(i)}=A_F$$. Accordingly, we replace $$D_F^{(i)}$$ by $$A_F$$, and finally obtain our result by showing that $$p^t$$ is an upper bound for the norm index $$[A_F:A_F\cap E_{E/F}(E^\bullet)]$$. It is interesting to note that when $$F$$ contains only one prime above $$p$$, this norm index turns out to be exactly $$p^t$$.
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 $$p$$, such that the rank of the kernel (or the cokernel) takes any prescribed value between zero and the trivial upper bound $$1+r_2$$.
In this paper, we do not go into the case where the extension $$E/F$$ is $$p$$-ramified. Note that the case where $$E$$ is contained in the cyclotomic $$\mathbb Z_p$$-extension of $$F$$ is certainly further complicated by its link to Greenberg’s famous conjecture in Iwasawa theory [R. Greenberg, Am. J. Math. 98, 263–284 (1976; Zbl 0334.12013)].
Finally, the recent results announced by Voevodsky on the Bloch-Kato conjecture would imply the Quillen-Lichtenbaum conjecture for odd prime $$p$$, so that the results of the paper could be formulated in terms of the $$p$$-primary parts of Quillen-$$K$$-groups, which would be closer to Kahn’s original approach on $$K_2$$.

##### MSC:
 11R70 $$K$$-theory of global fields 19F27 Étale cohomology, higher regulators, zeta and $$L$$-functions ($$K$$-theoretic aspects)
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##### References:
 [8] Iwasawa, K.: On Zl-extensions of algebraic number fields, Ann. Math. (1973). · Zbl 0285.12008 [10] Kahn, B.: On the Lichtenbaum-Quillen Conjecture, in J.F. Jardine (ed.), Algebraic K-theory and Algebraic Topology, Nato Proc. Lake Louise 407, Kluwer (1993), pp. 147-166. · Zbl 0885.19004 [13] Movahhedi, A.: Sur les p-extensions des corps p-rationnels, Thèse Paris 7 (1988). · Zbl 0723.11054 [15] Movahhedi, A. and Nguyen Quang Do, T.: Sur l?arithmétique des corps de nombres p-rationnels in Séminaire de Théorie des nombres, Paris 1987-88, 155-200, Progr. Math., 81, Birkhäuser, (1990).
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