On the maximal unramified pro-2-extension of ℤ₂-extension of certain real biquadratic fields

2024/09/20 by Mohamed Mahmoud Chems-Eddin, Chems-Eddin, Mohamed Mahmoud
Mathematics · #math.NT

paper · pdf · doi:10.48550/arxiv.2409.13574

Abstract

For any positive integer n, we show that there exists a real number field k (resp. k') of degree 2ⁿ⁺² whose 2-class group is isomorphic to ℤ/2ℤ× ℤ/2ℤ such that the Galois group of the maximal unramified extension of k (resp. k') over k (resp. k') is abelian (resp. non abelian, more precisely isomorphic to Q₈ or D₈, the quaternion and the dihedral group of order 8 respectively). In fact, we construct the first examples in the literature of families of real biquadratic fields for which the layers of the cyclotomic ℤ₂-extension satisfy the previous conditions and whose unramified abelian 2-Iwasawa modules are isomorphic to ℤ/2ℤ× ℤ/2ℤ; hence these fields satisfy Greenberg's conjecture.

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