Remark on height functions

TL;DR

This paper introduces a new height function for points on algebraic varieties over number fields using $K$-theory of associated $C^*$-algebras, and establishes finiteness results based on Betti numbers.

Contribution

It defines a novel height function via $K$-theory and proves finiteness of rational points under certain topological conditions.

Findings

The height function coincides with classical formulas for curves ($n=1$).

The set of rational points is finite if the sum of odd Betti numbers exceeds $n+1$.

The construction utilizes the Minkowski question-mark function.

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.