Isogenous hyperelliptic and non-hyperelliptic Jacobians with maximal complex multiplication

TL;DR

This paper investigates complex multiplication in genus 3 Jacobians, identifying specific sextic CM fields where both hyperelliptic and non-hyperelliptic Jacobians with CM by the same field coexist, combining theoretical and numerical methods.

Contribution

It classifies sextic CM fields with Jacobians of both hyperelliptic and non-hyperelliptic types, providing explicit examples and invariants for these curves.

Findings

14 such sextic CM fields found among 547,156 in the LMFDB

Explicit equations provided for some Jacobians

Invariants of the corresponding curves determined

Abstract

We analyze complex multiplication for Jacobians of curves of genus 3, as well as the resulting Shimura class groups and their subgroups corresponding to Galois conjugation over the reflex field. We combine our results with numerical methods to find CM fields $K$ for which there exist both hyperelliptic and non-hyperelliptic curves whose Jacobian has complex multiplication by $\mathbb{Z}_K$. More precisely, we find all sextic CM fields $K$ in the LMFDB for which (heuristically) Jacobians of both types with CM by $\mathbb{Z}_K$ exist. There turn out to be 14 such fields among the 547,156 sextic CM fields that the LMFDB contains. We determine invariants of the corresponding curves, and in the simplest case we also give an explicit defining equation.