Picard-Shimura class fields corresponding to a family of hyperelliptic curves

Abstract

We know explicit Picard modular functions, corresponding to a family of hyperelliptic curves, with the property that their values in CM points generate abelian extensions of the associated reflex fields (Matsumoto, Ann Sc Norm Sup Pisa 16(4):557–578, 1989, Riedel, In: Arithmetic and geometry around hypergeometric functions. Birkhauser, Basel, 2007, pp 273–285). In this note we study the number fields and their extensions occuring in this way. We show that every sextic CM field containing the fourth roots of unity is projectively generated by a singular modulus and appears as reflex field. In order to investigate the abelian extensions, we use the class field theoretic description of the field of moduli. In the unramified case we develop conditions that assure that the Picard-Shimura class field is equal to the reflex field or to the Hilbert class field. Finally, we determine these class fields for odd class numbers up to 11.