On Greenberg's generalized conjecture for imaginary quartic fields

2020/01/31 by N. Takahashi, Takahashi, Naoya
Mathematics · #11R16 (Secondary) #11R23 (Primary) 11R29 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2001.11768

openalex publication_date 2020/01/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For an algebraic number field K and a prime number p, let \widetildeK/K be the maximal multiple ℤ_p-extension. Greenberg's generalized conjecture (GGC) predicts that the Galois group of the maximal unramified abelian pro-p extension of \widetildeK is pseudo-null over the completed group ring ℤ_p[ [\mathopGal\nolimits(\widetildeK/K)] ]. We show that GGC holds for some imaginary quartic fields containing imaginary quadratic fields and some prime numbers.

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