高度函数备注

arXiv - MATH - Operator Algebras Pub Date : 2024-08-21 DOI:arxiv-2408.12020

Igor V. Nikolaev

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引用次数: 0

摘要

让 $k$ 是一个数域，$V(k)$ 是 $k$ 上的 $n$ 维投影簇。我们利用与 $V(k)$ 相关联的 $C^*$-algebra $A_V$ 的 $K$ 理论来定义 $V(k)$ 的点高。我们计算了相应的计数函数，并证明它与已知的 $n=1$ 公式相吻合。作为应用，我们证明了只要 $V(k)$ 的奇数贝蒂数之和超过 $n+1$，集合 $V(k)$ 就是有限的。我们的构造依赖于潘提等人研究的 $n$ 维明考斯基问号函数。

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Remark on height functions

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.

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arXiv - MATH - Operator Algebras

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