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On polarised class groups of orders in quartic CM-fields. (English) Zbl 1417.11114
Summary: 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 N. Murabayashi and A. Umegaki [J. Algebra 235, No. 1, 267–274 (2001; Zbl 0974.14015)] to a list of abelian surfaces over the rationals with complex multiplication by arbitrary orders.

11G15 Complex multiplication and moduli of abelian varieties
14K15 Arithmetic ground fields for abelian varieties
11R65 Class groups and Picard groups of orders
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