Remark on height functions
Abstract
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.
- Publication:
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arXiv e-prints
- Pub Date:
- August 2024
- DOI:
- 10.48550/arXiv.2408.12020
- arXiv:
- arXiv:2408.12020
- Bibcode:
- 2024arXiv240812020N
- Keywords:
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- Mathematics - Number Theory;
- Mathematics - Operator Algebras;
- 11G50;
- 46L85
- E-Print:
- 12 pages