Bounds for étale capitulation kernels.

*(English)*Zbl 1163.11347From 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 \(K_2\).

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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\textit{J. Assim} and \textit{A. Movahhedi}, \(K\)-Theory 33, No. 3, 199--213 (2004; Zbl 1163.11347)

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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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