Differentiability of the arithmetic volume function

TL;DR

This paper introduces a positive intersection product in Arakelov geometry and proves the continuous differentiability of the arithmetic volume function, enabling the computation of various arithmetic invariants and distribution functions.

Contribution

It establishes the differentiability of the arithmetic volume function and introduces a positive intersection product in Arakelov geometry, advancing the understanding of arithmetic invariants.

Findings

Arithmetic volume function is continuously differentiable

Computed distribution function of the asymptotic measure

Derived formulas for several arithmetic invariants

Abstract

We introduce the positive intersection product in Arakelov geometry and prove that the arithmetic volume function is continuously differentiable. As applications, we compute the distribution function of the asymptotic measure of a Hermitian line bundle and several other arithmetic invariants.