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arXiv · 2410.17458

Iwasawa module of the cyclotomic $\mathbb{Z}_{2}$-extension of certain real quadratic fields

Josue Avila
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Abstract

For a real quadratic field $K=\mathbb{Q}(\sqrt{D})$, let $K_{\infty}$ denote the cyclotomic $\mathbb{Z}_{p}$-extension of $K$. Greenberg conjectured that the corresponding Iwasawa module $X_{\infty}$ is finite. Building on the work of Mouhib and Movahhedi, we provide new examples of real quadratic fields for which the conjecture holds, when $X_{\infty}$ is cyclic and the prime is $p=2$. Furthermore, we find a fundamental system of units for certain biquadratic fields of the form $\mathbb{Q}(\sqrt{2}, \sqrt{D})$ and show how to use it to calculate the order of $X_{\infty}$.

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Josue Avila. 2024-10-22. Iwasawa module of the cyclotomic $\mathbb{Z}_{2}$-extension of certain real quadratic fields. https://arxiv.org/abs/2410.17458

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