# The existence of supersingular curves of genus 4 in arbitrary characteristic

## TL;DR

This paper proves the existence of supersingular curves of genus 4 in any characteristic, expanding understanding of supersingular algebraic curves across different fields.

## Contribution

It demonstrates the existence of supersingular genus 4 curves in arbitrary characteristic and constructs such curves explicitly for p > 3.

## Key findings

- Existence of supersingular genus 4 curves in any characteristic.
- Construction of supersingular genus 4 curves via fiber products of supersingular elliptic curves for p > 3.
- Verification that the desingularization process preserves supersingularity.

## Abstract

We prove that there exists a supersingular nonsingular curve of genus $4$ in arbitrary characteristic $p$. For $p>3$ we shall prove that the desingularization of a certain fiber product over $\mathbb{P}^1$ of two supersingular elliptic curves is supersingular.