# The lifted root number conjecture for fields of prime degree over the rationals: an approach via trees and Euler systems

Cornelius Greither^{[1]}; Radiu Kučera^{[2]}

- [1] Universität des Bundeswehr München, Fakultät fur Informatik, Institut für theoretische Informatik und Mathematik, 85577 Neubiberg (Allemagne)
- [2] Masarykova Univerzita, P{ř}íprodov{ě}decká Fakulta, Janá{č}kovo nám 2a, 663 95 Brno (République Tchèque)

Annales de l’institut Fourier (2002)

- Volume: 52, Issue: 3, page 735-777
- ISSN: 0373-0956

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topGreither, Cornelius, and Kučera, Radiu. "The lifted root number conjecture for fields of prime degree over the rationals: an approach via trees and Euler systems." Annales de l’institut Fourier 52.3 (2002): 735-777. <http://eudml.org/doc/115993>.

@article{Greither2002,

abstract = {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 $\Omega (3)$-localization. Following the lead of Ritter and Weiss, we prove the Lifted Root Number
Conjecture for the case that $F=\{\mathbb \{Q\}\}$ 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 $\{\mathbb \{Z\}\}_p$.
At the end we prove a generalization of the well-known Rédei-Reichardt theorem and
explain the close link with our theory.},

affiliation = {Universität des Bundeswehr München, Fakultät fur Informatik, Institut für theoretische Informatik und Mathematik, 85577 Neubiberg (Allemagne); Masarykova Univerzita, P\{ř\}íprodov\{ě\}decká Fakulta, Janá\{č\}kovo nám 2a, 663 95 Brno (République Tchèque)},

author = {Greither, Cornelius, Kučera, Radiu},

journal = {Annales de l’institut Fourier},

keywords = {lifted Chinburg conjecture; Euler systems; combinatorics; trees},

language = {eng},

number = {3},

pages = {735-777},

publisher = {Association des Annales de l'Institut Fourier},

title = {The lifted root number conjecture for fields of prime degree over the rationals: an approach via trees and Euler systems},

url = {http://eudml.org/doc/115993},

volume = {52},

year = {2002},

}

TY - JOUR

AU - Greither, Cornelius

AU - Kučera, Radiu

TI - The lifted root number conjecture for fields of prime degree over the rationals: an approach via trees and Euler systems

JO - Annales de l’institut Fourier

PY - 2002

PB - Association des Annales de l'Institut Fourier

VL - 52

IS - 3

SP - 735

EP - 777

AB - 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 $\Omega (3)$-localization. Following the lead of Ritter and Weiss, we prove the Lifted Root Number
Conjecture for the case that $F={\mathbb {Q}}$ 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 ${\mathbb {Z}}_p$.
At the end we prove a generalization of the well-known Rédei-Reichardt theorem and
explain the close link with our theory.

LA - eng

KW - lifted Chinburg conjecture; Euler systems; combinatorics; trees

UR - http://eudml.org/doc/115993

ER -

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