Genus 3 hyperelliptic curves with CM via Shimura Reciprocity

TL;DR

This paper develops an algorithm to compute invariants and class polynomials for genus 3 hyperelliptic curves with complex multiplication, leveraging Shimura reciprocity law to classify and analyze their Galois conjugates.

Contribution

It introduces a method to compute invariants and class polynomials for genus 3 hyperelliptic curves with CM using Shimura reciprocity law.

Findings

Algorithm for approximating invariants of hyperelliptic Jacobians with CM

Classification of Galois conjugates as hyperelliptic or not

Computation of class polynomials for genus 3 hyperelliptic curves

Abstract

Up to isomorphism over C, every simple principally polarized abelian variety of dimension 3 is the Jacobian of a smooth projective curve of genus 3. Furthermore, this curve is either a hyperelliptic curve or a plane quartic. Given a sextic CM field K, we show that if there exists a hyperelliptic Jacobian with CM by K, then all principally polarized abelian varieties that are Galois conjugated to it are hyperelliptic. Using Shimura's reciprocity law, we give an algorithm for computing approximations of the invariants of the initial curve, as well as their Galois conjugates. This allows us ton define and compute class polynomials for genus 3 hyperelliptic curves with CM.