Constructing genus 3 hyperelliptic Jacobians with CM

TL;DR

This paper presents an explicit algorithm for constructing genus 3 hyperelliptic curves over complex numbers with Jacobians that have complex multiplication by a given sextic CM field, enabling generation of such curves over finite fields with specified zeta functions.

Contribution

It introduces a comprehensive algorithm to compute Rosenhain invariants for genus 3 hyperelliptic Jacobians with CM by any sextic field, extending previous work and enabling practical curve construction.

Findings

Algorithm computes minimal polynomials of Rosenhain invariants.

Can generate curves over finite fields with specified zeta functions.

Works for any sextic CM field, generalizing prior methods.

Abstract

Given a sextic CM field $K$, we give an explicit method for finding all genus 3 hyperelliptic curves defined over $\mathbb{C}$ whose Jacobians are simple and have complex multiplication by the maximal order of this field, via an approximation of their Rosenhain invariants. Building on the work of Weng, we give an algorithm which works in complete generality, for any CM sextic field $K$, and computes minimal polynomials of the Rosenhain invariants for any period matrix of the Jacobian. This algorithm can be used to generate genus 3 hyperelliptic curves over a finite field $\mathbb{F}_p$ with a given zeta function by finding roots of the Rosenhain minimal polynomials modulo $p$.