On the Iwasawa $λ$-invariant of the cyclotomic $\mathbb{Z}_2$-extension of a family of real quadratic fields in which $2$ splits
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866913107328106496 |
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| author | Ávila, Josué Chnaras, Foivos |
| author_facet | Ávila, Josué Chnaras, Foivos |
| contents | 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 $λ_2(K)=0$. 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 $q(K_2)\le 2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_09111 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the Iwasawa $λ$-invariant of the cyclotomic $\mathbb{Z}_2$-extension of a family of real quadratic fields in which $2$ splits Ávila, Josué Chnaras, Foivos Number Theory 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 $λ_2(K)=0$. 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 $q(K_2)\le 2$. |
| title | On the Iwasawa $λ$-invariant of the cyclotomic $\mathbb{Z}_2$-extension of a family of real quadratic fields in which $2$ splits |
| topic | Number Theory |
| url | https://arxiv.org/abs/2605.09111 |