On Greenberg's generalized conjecture for families of number fields

2025/05/12 by Do, Thong Nguyen Quang
#11R23 11R34 11R37 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2505.07529

Abstract

For a number field k and an odd prime p, let k be the compositum of all the \mathbb Z_p-extensions of k, Λ the associated Iwasawa algebra, and X(k) the Galois group over k of the maximal abelian unramified pro-p-extension of k. Greenberg's generalized conjecture (GGC for short) asserts that the Λ-module X(k) is pseudo-null. Very few theoritical results toward GGC are known. We show here that for an imaginary k, GGC is implied by certain pseudo-nullity conditions imposed on a special \mathbb Z²_p-extension of k, and these conditions are partially or entirely fullfilled by certain families of number fields.

Citations

• Characteristic ideals and Iwasawa theory
• Nombres de Weil, sommes de Gauss et annulateurs Galoisiens

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