Hyperelliptic curves and their invariants: geometric, arithmetic and algorithmic aspects

TL;DR

This paper explores the classification and reconstruction of genus 3 hyperelliptic curves using invariant theory, providing explicit methods for identifying isomorphism classes and reconstructing models from invariants.

Contribution

It introduces algorithms for reconstructing hyperelliptic models from invariants and applies classical invariant theory to classify genus 3 hyperelliptic curves.

Findings

Explicit characterization of isomorphism classes of genus 3 hyperelliptic curves

Algorithms for reconstructing hyperelliptic models from invariants

Analysis of geometric and arithmetic properties of these curves

Abstract

We apply classical invariant theory of binary forms to explicitly characterize isomorphism classes of hyperelliptic curves of small genus and, conversely, propose algorithms for reconstructing hyperelliptic models from given invariants. We focus on genus 3 hyperelliptic curves. Both geometric and arithmetic aspects are considered.