Heights on algebraic curves

TL;DR

This paper explores the theory of heights on algebraic curves, defining minimal and moduli heights, establishing bounds for superelliptic curves, and explicitly determining constants for genus 2 and 3, including classifications of low-height hyperelliptic curves.

Contribution

It introduces bounds relating moduli height and minimal binary form height for algebraic curves, and provides explicit constants and classifications for genus 2 and 3 hyperelliptic curves.

Findings

Proved that the moduli height of superelliptic curves is bounded by a constant times the minimal height.

Explicitly determined the constant for genus 2 and 3 curves.

Computed complete lists of hyperelliptic curves of genus 2 and 3 with height 1.

Abstract

In these lectures we cover basics of the theory of heights starting with the heights in the projective space, heights of polynomials, and heights of the algebraic curves. We define the minimal height of binary forms and moduli height for algebraic curves and prove that the moduli height of superelliptic curves $\mathcal H (f) \leq c_0 \tilde H (f)$ where $c_0$ is a constant and $\tilde H$ the minimal height of the corresponding binary form. For genus $g=2$ and 3 such constant is explicitly determined. Furthermore, complete lists of curves of genus 2 and genus 3 hyperelliptic curves with height 1 are computed.