Examples of CM curves of genus two defined over the reflex field

TL;DR

This paper extends van Wamelen's list of genus two curves with complex multiplication to include those defined over their reflex field's real quadratic subfield, introducing new methods and providing proofs for the list.

Contribution

It introduces a new height-reduction algorithm for hyperelliptic curves over totally real fields and proves the extended list of CM genus two curves.

Findings

Extended list of CM genus two curves over real quadratic fields

Development of a new height-reduction algorithm

Proof of the completeness of the list using denominator bounds

Abstract

In "Proving that a genus 2 curve has complex multiplication", van Wamelen lists 19 curves of genus two over $\mathbf{Q}$ with complex multiplication (CM). For each of the 19 curves, the CM-field turns out to be cyclic Galois over $\mathbf{Q}$. The generic case of non-Galois quartic CM-fields did not feature in this list, as the field of definition in that case always contains a real quadratic field, known as the real quadratic subfield of the reflex field. We extend van Wamelen's list to include curves of genus two defined over this real quadratic field. Our list therefore contains the smallest "generic" examples of CM curves of genus two. We explain our methods for obtaining this list, including a new height-reduction algorithm for arbitrary hyperelliptic curves over totally real number fields. Unlike Van Wamelen, we also give a proof of our list, which is made possible by our implementation of denominator bounds of Lauter and Viray for Igusa class polynomials.