On polarised class groups of orders in quartic CM-fields

TL;DR

This paper establishes a necessary condition for pairs of orders in quartic CM-fields to share the same polarised class group, extending quadratic field results, with applications to abelian surface endomorphisms and CM classification.

Contribution

It introduces a new necessary condition for polarised class group equivalence in quartic CM-fields and applies it to endomorphism ring computations and CM abelian surface classification.

Findings

Derived a necessary condition for polarised class group equality in quartic CM-fields.

Extended classification of CM abelian surfaces over rationals.

Connected class group properties to endomorphism ring computations.

Abstract

We give an explicit necessary condition for pairs of orders in a quartic CM-field to have the same polarised class group. This generalises a simpler result for imaginary quadratic fields. We give an application of our results to computing endomorphism rings of abelian surfaces over finite fields, and we use our results to extend a completeness result of Murabayashi and Umegaki to a list of abelian surfaces over the rationals with complex multiplication by arbitrary orders.