$\mathbb{Z}_2$-extension of real quadratic fields with $\mathbb{Z}/2\mathbb{Z}$ as $2$-class group at each layer

H Laxmi; Anupam Saikia

$\mathbb{Z}_2$-extension of real quadratic fields with $\mathbb{Z}/2\mathbb{Z}$ as $2$-class group at each layer

Number Theory 2024-04-09 v2

Authors: H Laxmi , Anupam Saikia

Abstract

Let $K= \mathbb{Q}(\sqrt{d})$ be a real quadratic field with $d$ having three distinct prime factors. We show that the $2$ -class group of each layer in the $\mathbb{Z}_2$ -extension of $K$ is $\mathbb{Z}/2\mathbb{Z}$ under certain elementary assumptions on the prime factors of $d$ . In particular, it validates Greenberg's conjecture on the vanishing of the Iwasawa $\lambda$ -invariant for a new family of infinitely many real quadratic fields.

Keywords

real quadratic fields z2-extension 2-class group iwasawa lambda-invariant greenberg's conjecture

Cite

@article{arxiv.2310.03543,
  title  = {$\mathbb{Z}_2$-extension of real quadratic fields with $\mathbb{Z}/2\mathbb{Z}$ as $2$-class group at each layer},
  author = {H Laxmi and Anupam Saikia},
  journal= {arXiv preprint arXiv:2310.03543},
  year   = {2024}
}

Comments

14 pages, 3 pages

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