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

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• 作者：Georges Gras ^{g.mn.gras@wanadoo.fr" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Ideal class groups ; Kida's formula ; Iwasawa theory ; Class field theory ; p-ramification ; Ambiguous classes
• 刊名：Journal of Number Theory
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：171
• 期：Complete
• 页码：213-226
• 全文大小：380 K

文摘

We disprove, by means of numerical examples and theoretical arguments, illustrated with class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301998&_mathId=si1.gif&_user=111111111&_pii=S0022314X16301998&_rdoc=1&_issn=0022314X&md5=3722af86593da037dd61cda4230271cb" title="Click to view the MathML source">p=3class="mathContainer hidden">class="mathCode">$p=3$, the existence of a Riemann–Hurwitz formula for the p-ranks of relative class groups in a p-ramified p  -extension class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301998&_mathId=si2.gif&_user=111111111&_pii=S0022314X16301998&_rdoc=1&_issn=0022314X&md5=78adbca67146b8b3320f88286ea0cfca" title="Click to view the MathML source">K/kclass="mathContainer hidden">class="mathCode">$K/k$ of number fields of CM-type containing class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301998&_mathId=si3.gif&_user=111111111&_pii=S0022314X16301998&_rdoc=1&_issn=0022314X&md5=3bc73f80f273a1fe2b76aad053fa0fc8" title="Click to view the MathML source">μ_pclass="mathContainer hidden">class="mathCode">${\mu }_p$ 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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301998&_mathId=si1.gif&_user=111111111&_pii=S0022314X16301998&_rdoc=1&_issn=0022314X&md5=3722af86593da037dd61cda4230271cb" title="Click to view the MathML source">p=3class="mathContainer hidden">class="mathCode">$p=3$.