Stability of 2-class groups in the ℤ₂-extension of certain real biquadratic fields

2025/01/22 by Laxmi, H, Saikia, Anupam
#11R11 #11R23 #11R29 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2501.12782

Abstract

Greenberg's conjecture on the stability of ℓ-class groups in the cyclotomic ℤ_ℓ-extension of a real field has been proven for various infinite families of real quadratic fields for the prime ℓ=2. In this work, we consider an infinite family of real biquadratic fields K. With some extensive use of elementary group theoretic and class field theoretic arguments, we investigate the 2-class groups of the n-th layers K_n of the cyclotomic ℤ₂-extension of K and verify Greenberg's conjecture. We also relate capitulation of ideal classes of certain sub-extensions of K_n to the relative sizes of the 2-class groups.

Related

• Greenberg's conjecture for real quadratic fields and the cyclotomic ℤ₂-extensions
• On the structure and stability of ranks of 2-class groups in cyclotomic ℤ₂-extensions of certain real quadratic fields
• Greenberg's conjecture and Iwasawa module of Real biquadratic fields I
• ℤ₂-extension of real quadratic fields with ℤ/2ℤ as 2-class group at each layer
• On the maximal unramified pro-2-extension of ℤ₂-extension of certain real biquadratic fields