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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.
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]
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Database: Engineering Source
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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]
ISSN:0022314X
DOI:10.1016/j.jnt.2025.09.004