Remark on height functions

2024/08/21 by Igor V. Nikolaev, Nikolaev, Igor V.
Mathematics · #11G50 #46L85 #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #FOS: Mathematics #Mathematics #Number Theory (math.NT) #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.2408.12020

openalex publication_date 2024/08/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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