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Zbl 1041.11074
Greither, Cornelius; Kučera, Radan
The lifted root number conjecture for fields of prime degree over the rationals: an approach via trees and Euler systems. (English)
[J] Ann. Inst. Fourier 52, No. 3, 735-777 (2002). ISSN 0373-0956; ISSN 1777-5310/e

Summary: The so-called Lifted Root Number Conjecture is a strengthening of Chinburg's $\Omega(3)$-conjecture for Galois extensions $K/F$ of number fields. It is certainly more difficult than the $\Omega(3)$-advantage of behaving well under localization. Following the lead of {\it J. Ritter} and {\it A. R. Weiss} [Acta Arith. 90, 313--340 (1999; Zbl 0932.11071)], we prove the Lifted Root Number Conjecture for the case that $F=\Bbb Q$ and the degree of $K/F$ is an odd prime, with another small restriction on ramification. The very explicit calculations with cyclotomic units use trees and some classical combinatorics for bookkeeping. An important point is the following: While dealing with our Euler systems, we have to keep track of the action of the Galois group, whose order is not invertible in the coefficient ring $\underline{\Bbb Z_p}$. At the end we prove a generalization of the well-known Rédei-Reichardt theorem [{\it L. Rédei} and {\it H. Reichardt}, J. Reine Angew. Math. 170, 69--74 (1933; Zbl 0007.39602)] and explain the close link with our theory.

MSC 2000:: *11R33 Integral representations related to algebraic numbers
11R37 Class field theory for global fields
11R18 Cyclotomic extensions

Keywords: lifted Chinburg conjecture; Euler systems

Citations: Zbl 1002.11082; Zbl 1014.11066; Zbl 1142.11076; Zbl 0932.11071; Zbl 0007.39602

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