2024/12/11 by Georges Gras, Gras, Georges
Mathematics · #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2412.08214
openalex publication_date 2024/12/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
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 ℤp-extension k\rm ac of k; for this, using a new approach based on the Logp-function (Theorems 2.3, 3.4), we determine the first layer k1\rm ac of k\rm ac over k, and we show that some partial capitulation may exist in k1\rm ac, even when k\rm ac/k is totally ramified. We have conjectured that this phenomenon of capitulation is specific of the ℤp-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\rm 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(√(3m)), 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 k1\rm ac, and the main invariants attached to the fields k, k^*, k1\rm ac; some relations with Iwasawa's invariants are discussed (Theorem 9.6).