An Approximation of Local Antiderivatives of Relative Differential on Arithmetic Surface

TL;DR

This paper constructs rational functions on an arithmetic surface to approximate local antiderivatives of a relative differential, establishing a link between the height of rational curves and the norm of differentials, leading to height bounds.

Contribution

It introduces a method to approximate local antiderivatives on arithmetic surfaces and relates these to height bounds of rational curves, a novel approach in arithmetic geometry.

Findings

Constructed rational functions approximating local antiderivatives

Established a relation between curve height and differential norm

Provided an upper bound for the height of rational curves

Abstract

Let $\omega$ be a relative differential on aithmetic surface $X$. We construct a family of rational functions $G_x$ on $X\otimes\Bbb{C}$, which can approximate local antiderivatives of $\omega$ over an open set on $X\otimes\Bbb{C}$. From this family of functions, we construct a rational function $G_2$ on $X$. The function $G_2$ can generate an element in the ring of integers of a number field, which can approximate an inner product produced by $\omega$ and the conjugate of $\omega$ over an open set on $X\otimes\Bbb{C}$. This will give a relation between the height of a rational curve $E_P$ on $X$ and the canonical norm of $\omega$ on $X\otimes\Bbb{C}$. This relation will give an upper bound for the height of $E_P$ under a few assumptions.