Initial layer of the anti-cyclotomic [formula omitted]-extension of [formula omitted] and capitulation phenomenon.
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| Title: | Initial layer of the anti-cyclotomic [formula omitted]-extension of [formula omitted] and capitulation phenomenon. |
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| Authors: | Gras, Georges1 (AUTHOR) g.mn.gras@wanadoo.fr |
| Source: | Journal of Number Theory. Mar2026, Vol. 280, p634-701. 68p. |
| Subjects: | Quadratic fields, Class groups (Mathematics), Cyclotomic fields, Cubic equations, Number theory |
| Abstract: | Let k = Q (− m) be an imaginary quadratic field. We consider the properties of capitulation of the p -class group of k in the anti-cyclotomic Z p -extension k ac of k ; for this, using a new approach based on the Log p -function (Theorems 2.3, 3.4), we determine the first layer k 1 ac of k ac over k , and we show that some partial capitulation may exist in k 1 ac , even when k ac / k is totally ramified. We have conjectured that this phenomenon of capitulation is specific of the Z p -extensions of k , distinct from the cyclotomic one. For p = 3 , we characterize a sub-family of fields k (Normal Split cases) for which k ac 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 k ⁎ = Q (3 m) , 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 k 1 ac , and the main invariants attached to the fields k , k ⁎ , k 1 ac ; some relations with Iwasawa's invariants are discussed (Theorem 9.6). [ABSTRACT FROM AUTHOR] |
| Copyright of Journal of Number Theory is the property of Academic Press Inc. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
| FullText | Text: Availability: 0 |
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| Header | DbId: egs DbLabel: Engineering Source An: 188782714 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: Initial layer of the anti-cyclotomic [formula omitted]-extension of [formula omitted] and capitulation phenomenon. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Gras%2C+Georges%22">Gras, Georges</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> g.mn.gras@wanadoo.fr</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Journal+of+Number+Theory%22">Journal of Number Theory</searchLink>. Mar2026, Vol. 280, p634-701. 68p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Quadratic+fields%22">Quadratic fields</searchLink><br /><searchLink fieldCode="DE" term="%22Class+groups+%28Mathematics%29%22">Class groups (Mathematics)</searchLink><br /><searchLink fieldCode="DE" term="%22Cyclotomic+fields%22">Cyclotomic fields</searchLink><br /><searchLink fieldCode="DE" term="%22Cubic+equations%22">Cubic equations</searchLink><br /><searchLink fieldCode="DE" term="%22Number+theory%22">Number theory</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: Let k = Q (− m) be an imaginary quadratic field. We consider the properties of capitulation of the p -class group of k in the anti-cyclotomic Z p -extension k ac of k ; for this, using a new approach based on the Log p -function (Theorems 2.3, 3.4), we determine the first layer k 1 ac of k ac over k , and we show that some partial capitulation may exist in k 1 ac , even when k ac / k is totally ramified. We have conjectured that this phenomenon of capitulation is specific of the Z p -extensions of k , distinct from the cyclotomic one. For p = 3 , we characterize a sub-family of fields k (Normal Split cases) for which k ac 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 k ⁎ = Q (3 m) , 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 k 1 ac , and the main invariants attached to the fields k , k ⁎ , k 1 ac ; some relations with Iwasawa's invariants are discussed (Theorem 9.6). [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Journal of Number Theory is the property of Academic Press Inc. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1016/j.jnt.2025.09.004 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 68 StartPage: 634 Subjects: – SubjectFull: Quadratic fields Type: general – SubjectFull: Class groups (Mathematics) Type: general – SubjectFull: Cyclotomic fields Type: general – SubjectFull: Cubic equations Type: general – SubjectFull: Number theory Type: general Titles: – TitleFull: Initial layer of the anti-cyclotomic [formula omitted]-extension of [formula omitted] and capitulation phenomenon. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Gras, Georges IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 03 Text: Mar2026 Type: published Y: 2026 Identifiers: – Type: issn-print Value: 0022314X Numbering: – Type: volume Value: 280 Titles: – TitleFull: Journal of Number Theory Type: main |
| ResultId | 1 |
