Galois co-descent for étale wild kernels and capitulation

Abstract. Fix a Galois extension E/F of totally real number fields such that the Galois group G has exponent 2. Let S be a finite set of primes of F containing the infinite primes and all those which ramify in E, let S E denote the primes of E lying above those in S, and let O S E denote the ring of S E -integers of E. We then compare the Fitting ideal of especially for biquadratic extensions, where we compute the index of the higher Stickelberger ideal. We find a sufficient condition for the Fitting ideal to contain the higher Stickelberger ideal in the case where E is a biquadratic extension of F containing the first layer of the cyclotomic Z 2 -extension of F , and describe a class of biquadratic extensions of F = Q that satisfy this condition.

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“…Under our assumptions on S, it follows more generally from Kahn's theorem of [6, 5.1] as observed in [9] that the transfer map from …”

Section: Proposition 21 (Stickelberger Ideals Under Change Of Extensmentioning

confidence: 89%

Values at s = −1 of L -functions for multi-quadratic extensions of number fields and annihilation of the tame kernel

Sands

Simons²

2007

Journal of the London Mathematical Society

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“…In the case i D 2 the group 2 .F;Z p .2// is isomorphic to the p-part of the classical wild kernel WK 2 .F / in algebraic K-theory, and therefore the groups 2 .F;Z p .i // for i 2 are also denoted in the literature by WK ét 2i 2 .F / and called higher étale wild kernels ([14], [6], [10]). …”

Section: éTale Cohomologymentioning

confidence: 99%

On λ-invariants of number fields and étale cohomology

Kolster¹,

Movahhedi²

2013

J K-Theor

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For an odd prime p we prove a Riemann-Hurwitz type formula for odd eigenspaces of the standard Iwasawa modules over F . p 1 /, the field obtained from a totally real number field F by adjoining all p-power roots of unity. We use a new approach based on the relationship between eigenspaces and étale cohomology groups over the cyclotomic Z p -extension F 1 of F . The systematic use of étale cohomology greatly simplifies the proof and allows to generalize the classical result about the minus-eigenspace to all odd eigenspaces.

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“…tout du moins pour i = 1 (voir [7]). Ici, H 1 (G F , Z p (i − 1))[p] désigne le sous-groupe des élé-ments tués par p. D (i) (F ) contient visiblement F ×p et l'on considère habituellement le i ième noyau de Tate modulo p:…”

Section: Définitionunclassified

Noyaux de Tate et capitulation

Vauclair

2008

Journal of Number Theory

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Following Kahn, and Assim and Movahhedi, we look for bounds for the order of the capitulation kernels of higher K-groups of S-integers into abelian S-ramified p-extensions. The basic strategy is to change twists inside some Galois-cohomology groups, which is done via the comparison of Tate Kernels of higher order. We investigate two ways: a global one, valid for twists close to 0 (in a certain sense), and a local one, valid for twists close to 1 in cyclic extensions. The global method produces lower bounds for abelian p-extensions which are S-ramified, but not Z p -embeddable. The local method is close to that of [J. Assim, A. Movahhedi, Bounds for étale capitulation kernels, K-Theory 33 (2004) 199-213], but is improved to take into consideration what happens when S consists of only the p-places. In contrast to the first one, one can expect this second method to produce nontrivial lower bounds in certain Z p -extensions. For example, we construct Z p -extensions in which the capitulation kernel is as big as we want (when letting the twist vary). We also include a complete solution to the problem of comparing Tate Kernels.

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Galois co-descent for étale wild kernels and capitulation

Cited by 27 publications

References 14 publications

Values at s = −1 of L -functions for multi-quadratic extensions of number fields and annihilation of the tame kernel

Values at s = −1 of L -functions for multi-quadratic extensions of number fields and annihilation of the tame kernel

On λ-invariants of number fields and étale cohomology

Noyaux de Tate et capitulation

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