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Remark on height functions
- Publication Year :
- 2024
-
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.<br />Comment: 12 pages
- Subjects :
- Mathematics - Number Theory
Mathematics - Operator Algebras
11G50, 46L85
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2408.12020
- Document Type :
- Working Paper