Mathematical Research Letters

Volume 24 (2017)

Number 2

On polarised class groups of orders in quartic CM-fields

Pages: 247 – 270



Gaetan Bisson (Department of Mathematics, University of French Polynesia, Faaa, French Polynesia)

Marco Streng (Mathematisch Instituut, Universiteit Leiden, The Netherlands)


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.

Received 1 October 2013

Accepted 7 November 2015

Published 24 July 2017