Greenberg's conjecture and Iwasawa module of Real biquadratic fields II
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866908759156064256 |
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| author | Chems-Eddin, Mohamed Mahmoud Mamry, Hamza El |
| author_facet | Chems-Eddin, Mohamed Mahmoud Mamry, Hamza El |
| contents | In this paper we are interested in the stability of the $2$-rank of the class group in the cyclotomic $\mathbb{Z}_2$-extension of real biquadratic fields. In fact, we give several families of real biquadratic fields $K$ such that $ rank(A(K)) =rank(A_\infty(K))$ and $rank(A(K))\leq 3$, where $A(K)$ and $A_\infty(K)$ are the $2$-class group and the $2$-Iwasawa module of $K$ respectively. Moreover, Greenberg's conjecture is verified for some new families of number fields; in particular, we determine the complete list of all real biquadratic fields with trivial $2$-Iwasawa module. This work is a continuation of M. M. Chems-Eddin, Greenberg's conjecture and Iwasawa module of real biquadratic fields I, J. Number Theory, 281 (2026), 224-266. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_07067 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Greenberg's conjecture and Iwasawa module of Real biquadratic fields II Chems-Eddin, Mohamed Mahmoud Mamry, Hamza El Number Theory In this paper we are interested in the stability of the $2$-rank of the class group in the cyclotomic $\mathbb{Z}_2$-extension of real biquadratic fields. In fact, we give several families of real biquadratic fields $K$ such that $ rank(A(K)) =rank(A_\infty(K))$ and $rank(A(K))\leq 3$, where $A(K)$ and $A_\infty(K)$ are the $2$-class group and the $2$-Iwasawa module of $K$ respectively. Moreover, Greenberg's conjecture is verified for some new families of number fields; in particular, we determine the complete list of all real biquadratic fields with trivial $2$-Iwasawa module. This work is a continuation of M. M. Chems-Eddin, Greenberg's conjecture and Iwasawa module of real biquadratic fields I, J. Number Theory, 281 (2026), 224-266. |
| title | Greenberg's conjecture and Iwasawa module of Real biquadratic fields II |
| topic | Number Theory |
| url | https://arxiv.org/abs/2601.07067 |