Journal of Number Theory最新文献_第2页

Initial layer of the anti-cyclotomic Zp-extension of Q(−m) and capitulation phenomenon 初始层的抗切圆zp扩展Q（−m）和投降现象

IF 0.7 3区数学

Journal of Number Theory Pub Date : 2025-10-01 DOI: 10.1016/j.jnt.2025.09.004

Georges Gras

{"title":"Initial layer of the anti-cyclotomic Zp-extension of Q(−m) and capitulation phenomenon","authors":"Georges Gras","doi":"10.1016/j.jnt.2025.09.004","DOIUrl":"10.1016/j.jnt.2025.09.004","url":null,"abstract":"<div><div>Let <math><mi>k</mi><mo>=</mo><mi>Q</mi><mo>(</mo><msqrt><mrow><mo>−</mo><mi>m</mi></mrow></msqrt><mo>)</mo></math> be an imaginary quadratic field. We consider the properties of capitulation of the p-class group of k in the anti-cyclotomic <math><msub><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow></msub></math>-extension <math><msup><mrow><mi>k</mi></mrow><mrow><mi>ac</mi></mrow></msup></math> of k; for this, using a new approach based on the <math><msub><mrow><mi>Log</mi></mrow><mrow><mi>p</mi></mrow></msub></math>-function (Theorem 2.3, Theorem 3.4), we determine the first layer <math><msubsup><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>ac</mi></mrow></msubsup></math> of <math><msup><mrow><mi>k</mi></mrow><mrow><mi>ac</mi></mrow></msup></math> over k, and we show that some partial capitulation may exist in <math><msubsup><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>ac</mi></mrow></msubsup></math>, even when <math><msup><mrow><mi>k</mi></mrow><mrow><mi>ac</mi></mrow></msup><mo>/</mo><mi>k</mi></math> is totally ramified. We have conjectured that this phenomenon of capitulation is specific of the <math><msub><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow></msub></math>-extensions of k, distinct from the cyclotomic one. For <math><mi>p</mi><mo>=</mo><mn>3</mn></math>, we characterize a sub-family of fields k (Normal Split cases) for which <math><msup><mrow><mi>k</mi></mrow><mrow><mi>ac</mi></mrow></msup></math> is not linearly disjoint from the Hilbert class field (Theorem 5.1). No assumptions are made on the splitting of 3 in k and in <math><msup><mrow><mi>k</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>=</mo><mi>Q</mi><mo>(</mo><msqrt><mrow><mn>3</mn><mi>m</mi></mrow></msqrt><mo>)</mo></math>, nor on the structures of their 3-class groups. Four pari/gp programs (7.1, 7.2, 7.3, 7.4 depending on the classification of Definition 2.10) are given, computing a defining cubic polynomial of <math><msubsup><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>ac</mi></mrow></msubsup></math>, and the main invariants attached to the fields k, <math><msup><mrow><mi>k</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math>, <math><msubsup><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>ac</mi></mrow></msubsup></math>; some relations with Iwasawa's invariants are discussed (Theorem 9.6).</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 634-701"},"PeriodicalIF":0.7,"publicationDate":"2025-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145267235","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}