On the Iwasawa $\lambda$-invariant of the cyclotomic $\mathbb{Z}_2$-extension of a family of real quadratic fields in which $2$ splits

TL;DR

This paper proves Greenberg's conjecture for certain real quadratic fields with specific splitting conditions at 2, showing the Iwasawa lambda-invariant is zero using a combination of criteria and capitulation arguments.

Contribution

It introduces a new approach combining Greenberg's criterion with capitulation and a square-class computation to establish the lambda-invariant result.

Findings

Proves λ₂(K)=0 for specified real quadratic fields.

Uses a novel square-class computation of the Hasse unit index.

Combines Greenberg's criterion with capitulation arguments.

Abstract

We study Greenberg's conjecture for cyclotomic $\mathbb{Z}_2$-extensions of real quadratic fields. Let $K=\mathbb{Q}(\sqrt{pq})$, where $$ p\equiv 1 \mod 8,\qquad q\equiv 9 \mod {16},\qquad \left(\frac{p}{q}\right)=-1. $$ Under the additional assumptions $$ \left(\frac{2}{p}\right)_4 \left(\frac{2}{q}\right)_4 \left(\frac{pq}{2}\right)_4=-1 $$ and $$ \left(\frac{2}{p}\right)_4=-1 \quad\text{or}\quad \left(\frac{2}{q}\right)_4=-1, $$ we prove that $\lambda_2(K)=0$. The proof combines Greenberg's criterion for the split prime case with a capitulation argument modeled on Kumakawa. The main new input is a square-class computation of the Hasse unit index of the biquadratic extension $K_2=\mathbb{Q}(\sqrt{pq}, \sqrt{2+\sqrt{2}})/\mathbf{Q}_1=\mathbb{Q}(\sqrt{2})$, showing that $q(K_2)\le 2$.