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, Volume 52 (2002) no. 3, p. 735-777
The so-called Lifted Root Number Conjecture is a strengthening of Chinburg’s Ω(3)- conjecture for Galois extensions K/F of number fields. It is certainly more difficult than the Ω(3)-localization. Following the lead of Ritter and Weiss, we prove the Lifted Root Number Conjecture for the case that F= 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 p . At the end we prove a generalization of the well-known Rédei-Reichardt theorem and explain the close link with our theory.
La “conjecture de Chinburg relevée” (Lifted Root Number Conjecture, LRNC) est une version beaucoup plus forte de la conjecture Ω(3) de Chinburg concernant les extensions galoisiennes K/F de corps de nombres. Tout en étant plus difficile que la conjecture Ω(3), la conjecture LRNC a l’avantage de se comporter très bien sous localisation. Avec une démarche de Ritter et Weiss comme point de départ, nous démontrons LRNC dans le cas où F= et où le degré de K/F est premier impair (de plus il y a une petite restriction sur la ramification). Nos calculs très explicites avec des unités cyclotomiques font intervenir des arbres et de la combinatoire classique comme outils d’organisation. Soulignons encore que nous devons, en travaillant avec le système d’Euler, tenir compte de l’action du groupe de Galois, groupe dont l’ordre n’est pas inversible dans l’anneau de coefficients p . À la fin, nous donnons une généralisation du théorème classique de Rédei et Reichardt et explicitons le lien étroit avec notre théorie.
DOI : https://doi.org/10.5802/aif.1900
Classification:  11R18,  11R33,  11R37,  05C05
Keywords: lifted Chinburg conjecture, Euler systems, combinatorics, trees
@article{AIF_2002__52_3_735_0,
     author = {Greither, Cornelius and Ku\v cera, Radiu},
     title = {The lifted root number conjecture for fields of prime degree over the rationals: an approach via trees and Euler systems},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {52},
     number = {3},
     year = {2002},
     pages = {735-777},
     doi = {10.5802/aif.1900},
     zbl = {1041.11074},
     mrnumber = {1907386},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2002__52_3_735_0}
}
Greither, Cornelius; 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, Volume 52 (2002) no. 3, pp. 735-777. doi : 10.5802/aif.1900. http://www.numdam.org/item/AIF_2002__52_3_735_0/

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