# On different expressions for invariants of hyperelliptic curves of genus 3

## TL;DR

This paper establishes formulas linking different invariants of genus 3 hyperelliptic curves, enabling modular expressions and analysis of reduction types, which are useful for computational and theoretical applications in algebraic geometry.

## Contribution

It provides explicit passage formulas between Tsuyumine and Shioda invariants for genus 3 hyperelliptic curves, and describes Shioda invariants in terms of roots differences, aiding in reduction type analysis.

## Key findings

- Derived formulas connecting Tsuyumine and Shioda invariants.
- Expressed Shioda invariants via roots differences.
- Provided criteria for bad reduction types.

## Abstract

In this paper we give a passage formula between different invariants of genus 3 hyperelliptic curves: in particular between Tsuyumine and Shioda invariants. This is needed to get modular expressions for Shioda invariants, that is, for example, useful for proving the correctness of numerically computed equations of CM genus 3 hyperelliptic curves. On the other hand, we also get Shioda invariants described in terms of differences of roots of the equation defining the hyperelliptic curve, that has applications for studying the reduction type of the curve. Under certain conditions on its jacobian, we give a criterion for determining the type of bad reduction of a genus 3 hyperelliptic curve.