A classification of $\mathbb Q$-linear maps from $\overline{\mathbb Q}^\times/\overline{\mathbb Q}^\times_{\mathrm{tors}}$ to $\mathbb R$
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915558174228480 |
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| author | Samuels, Charles L. |
| author_facet | Samuels, Charles L. |
| contents | A 2009 article of Allcock and Vaaler explored the $\mathbb Q$-vector space $\mathcal G := \overline{\mathbb Q}^\times/{\overline{\mathbb Q}^\times_{\mathrm{tors}}}$, showing how to represent it as part of a function space on the places of $\overline{\mathbb Q}$. We establish a representation theorem for the $\mathbb R$-vector space of $\mathbb Q$-linear maps from $\mathcal G$ to $\mathbb R$, enabling us to classify extensions to $\mathcal G$ of completely additive arithmetic functions. We further outline a strategy to construct $\mathbb Q$-linear maps from $\mathcal G$ to $\mathbb Q$, i.e., elements of the algebraic dual of $\mathcal G$. Our results make heavy use of Dirichlet's $S$-unit Theorem as well as a measure-like object called a consistent map, first introduced by the author in previous work. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_09752 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A classification of $\mathbb Q$-linear maps from $\overline{\mathbb Q}^\times/\overline{\mathbb Q}^\times_{\mathrm{tors}}$ to $\mathbb R$ Samuels, Charles L. Number Theory Primary 11R04, 15A04, Secondary 11G50, 11R27, 46E15 A 2009 article of Allcock and Vaaler explored the $\mathbb Q$-vector space $\mathcal G := \overline{\mathbb Q}^\times/{\overline{\mathbb Q}^\times_{\mathrm{tors}}}$, showing how to represent it as part of a function space on the places of $\overline{\mathbb Q}$. We establish a representation theorem for the $\mathbb R$-vector space of $\mathbb Q$-linear maps from $\mathcal G$ to $\mathbb R$, enabling us to classify extensions to $\mathcal G$ of completely additive arithmetic functions. We further outline a strategy to construct $\mathbb Q$-linear maps from $\mathcal G$ to $\mathbb Q$, i.e., elements of the algebraic dual of $\mathcal G$. Our results make heavy use of Dirichlet's $S$-unit Theorem as well as a measure-like object called a consistent map, first introduced by the author in previous work. |
| title | A classification of $\mathbb Q$-linear maps from $\overline{\mathbb Q}^\times/\overline{\mathbb Q}^\times_{\mathrm{tors}}$ to $\mathbb R$ |
| topic | Number Theory Primary 11R04, 15A04, Secondary 11G50, 11R27, 46E15 |
| url | https://arxiv.org/abs/2503.09752 |