No general Riemann-Hurwitz formula for relative p-class groups

JOURNAL OF NUMBER THEORY

卷:171

Article

Gras, Georges¹ 

[1] Chemin Chateau Gagniere, F-38520 Le Bourg Doisans, France

关键词: Ideal class groups;    Kida's formula;    Iwasawa theory;    Class field theory;    p-ramification;    Ambiguous classes;   

DOI  :  10.1016/j.jnt.2016.07.016

来源: Elsevier

【 摘 要 】

We disprove, by means of numerical examples and theoretical arguments, illustrated with p = 3, the existence of a Riemann-Hurwitz formula for the p-ranks of relative class groups in a p-ramified p-extension K/k of number fields of CM-type containing mu(p) in contradiction with a result published in 1996. In the cyclic case of degree p, under some assumptions on the p-class group of k and the decomposition of the p-places, we prove some results on the structure of the p-class group of K and justify that some theoretical structures do not exist in this particular situation. In this context, an analogue of Kida's formula is valid for the p-ranks if and only if the p-class group of K is reduced to the group of ambiguous classes, which is not always the case, as shown by a numerical table for p = 3. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

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