On the Galois structure of units in totally real $p$-rational number fields
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866914073297289216 |
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| author | Bouazzaoui, Zakariae Lim, Donghyeok |
| author_facet | Bouazzaoui, Zakariae Lim, Donghyeok |
| contents | The theory of factor-equivalence of integral lattices establishes a far-reaching relationship between the Galois module structure of the unit group of the ring of integers of a number field and its arithmetic. For a number field $K$ that is Galois over $\mathbb{Q}$ or an imaginary quadratic field, we prove a necessary and sufficient condition on the quotients of class numbers of subfields of $K$, for the quotient $E_{K}$ of the unit group of the ring of integers of $K$ modulo the subgroup of roots of unity to be factor equivalent to the standard cyclic Galois module. Using strong arithmetic properties of totally real $p$-rational number fields, we prove that the non-abelian $p$-rational $p$-extensions of $\mathbb{Q}$ do not admit Minkowski units, thereby extending a result of Burns to non-abelian number fields. We also study the relative Galois module structure of $E_{L}$ for varying Galois extensions $L/F$ of totally real $p$-rational number fields whose Galois groups are isomorphic to a fixed finite group $G$. In that case, we prove that there exists a finite set $Ω$ of $\mathbb{Z}_p[G]$-lattices such that for every $L$, $\mathbb{Z}_{p} \otimes_{\mathbb{Z}} E_{L}$ is factor equivalent to $\mathbb{Z}_{p}[G]^{n} \oplus X$ as $\mathbb{Z}_p[G]$-lattices for some $X \in Ω$ and an integer $n \geq 0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_13525 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the Galois structure of units in totally real $p$-rational number fields Bouazzaoui, Zakariae Lim, Donghyeok Number Theory 11R33, 11R80 The theory of factor-equivalence of integral lattices establishes a far-reaching relationship between the Galois module structure of the unit group of the ring of integers of a number field and its arithmetic. For a number field $K$ that is Galois over $\mathbb{Q}$ or an imaginary quadratic field, we prove a necessary and sufficient condition on the quotients of class numbers of subfields of $K$, for the quotient $E_{K}$ of the unit group of the ring of integers of $K$ modulo the subgroup of roots of unity to be factor equivalent to the standard cyclic Galois module. Using strong arithmetic properties of totally real $p$-rational number fields, we prove that the non-abelian $p$-rational $p$-extensions of $\mathbb{Q}$ do not admit Minkowski units, thereby extending a result of Burns to non-abelian number fields. We also study the relative Galois module structure of $E_{L}$ for varying Galois extensions $L/F$ of totally real $p$-rational number fields whose Galois groups are isomorphic to a fixed finite group $G$. In that case, we prove that there exists a finite set $Ω$ of $\mathbb{Z}_p[G]$-lattices such that for every $L$, $\mathbb{Z}_{p} \otimes_{\mathbb{Z}} E_{L}$ is factor equivalent to $\mathbb{Z}_{p}[G]^{n} \oplus X$ as $\mathbb{Z}_p[G]$-lattices for some $X \in Ω$ and an integer $n \geq 0$. |
| title | On the Galois structure of units in totally real $p$-rational number fields |
| topic | Number Theory 11R33, 11R80 |
| url | https://arxiv.org/abs/2311.13525 |