Remark on height functions
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913476580999168 |
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| author | Nikolaev, Igor V. |
| author_facet | Nikolaev, Igor V. |
| contents | Let $k$ be a number field and $V(k)$ an $n$-dimensional projective variety over $k$. We use the $K$-theory of a $C^*$-algebra $A_V$ associated to $V(k)$ to define a height of points of $V(k)$. The corresponding counting function is calculated and we show that it coincides with the known formulas for $n=1$. As an application, it is proved that the set $V(k)$ is finite, whenever the sum of the odd Betti numbers of $V(k)$ exceeds $n+1$. Our construction depends on the $n$-dimensional Minkowski question-mark function studied by Panti and others. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_12020 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Remark on height functions Nikolaev, Igor V. Number Theory Operator Algebras 11G50, 46L85 Let $k$ be a number field and $V(k)$ an $n$-dimensional projective variety over $k$. We use the $K$-theory of a $C^*$-algebra $A_V$ associated to $V(k)$ to define a height of points of $V(k)$. The corresponding counting function is calculated and we show that it coincides with the known formulas for $n=1$. As an application, it is proved that the set $V(k)$ is finite, whenever the sum of the odd Betti numbers of $V(k)$ exceeds $n+1$. Our construction depends on the $n$-dimensional Minkowski question-mark function studied by Panti and others. |
| title | Remark on height functions |
| topic | Number Theory Operator Algebras 11G50, 46L85 |
| url | https://arxiv.org/abs/2408.12020 |