Greenberg's conjecture and Iwasawa module of Real biquadratic fields I
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908634762444800 |
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| author | Chems-Eddin, Mohamed Mahmoud |
| author_facet | Chems-Eddin, Mohamed Mahmoud |
| contents | The main aim of this paper is to investigate Greenberg's conjecture for real biquadratic fields. More precisely, we propose the following problem: What are real biquadratic number fields $k$ such that ${\rm rank}(A(k_\infty)) = {\rm rank}(A(k_1))$?, where $A(k_\infty)$ is the $2$-Iwasawa module of $k$ and $A(k_1)$ is the $2$-class group of $k_1$ the first layer of the cyclotomic $\mathbb Z_2$-extension of $k$. Moreover, we give several families of real biquadratic fields $k$ such that $A(k_\infty)$ is trivial or isomorphic to $\mathbb Z/2^{n} \mathbb Z$ or $\mathbb Z/2\mathbb Z \times\mathbb Z/2^n \mathbb Z$, where $n$ is a given positive integer. The reader can also find some results concerning the $2$-rank of the class group of certain real triquadratic fields. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_15727 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Greenberg's conjecture and Iwasawa module of Real biquadratic fields I Chems-Eddin, Mohamed Mahmoud Number Theory The main aim of this paper is to investigate Greenberg's conjecture for real biquadratic fields. More precisely, we propose the following problem: What are real biquadratic number fields $k$ such that ${\rm rank}(A(k_\infty)) = {\rm rank}(A(k_1))$?, where $A(k_\infty)$ is the $2$-Iwasawa module of $k$ and $A(k_1)$ is the $2$-class group of $k_1$ the first layer of the cyclotomic $\mathbb Z_2$-extension of $k$. Moreover, we give several families of real biquadratic fields $k$ such that $A(k_\infty)$ is trivial or isomorphic to $\mathbb Z/2^{n} \mathbb Z$ or $\mathbb Z/2\mathbb Z \times\mathbb Z/2^n \mathbb Z$, where $n$ is a given positive integer. The reader can also find some results concerning the $2$-rank of the class group of certain real triquadratic fields. |
| title | Greenberg's conjecture and Iwasawa module of Real biquadratic fields I |
| topic | Number Theory |
| url | https://arxiv.org/abs/2503.15727 |