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Zbl 0732.14016
Feustel, Jan
Eine Klassenzahlformel für singuläre Moduln der Picardschen Modulgruppen. (A class number formula for singular moduli of Picard modular groups). (German)
[J] Compos. Math. 76, No. 1-2, 87-100 (1990). ISSN 0010-437X; ISSN 1570-5846/e

It is known that the ring of modular forms of the Picard modular group $\Gamma\sb K\subset U\sb K$ of an imaginary quadratic field K which operates on the 1-ball $B\subset {\bbfC}\sp 2$ is generated by three elements and that the values of certain quotients of these generators, i.e. modular functions, on so-called K-singular moduli $\tau\in B$, i.e. fixed points of $U\sb K$, are algebraic. \par As in the classical cases of the elliptic and Hilbert modular group these singular moduli correspond to ideals in orders of K (which in the classical cases at least furnish an algebraic equation with rational coefficients for the generators of the field of modular functions). Having this final result in mind the author first characterises all K- singular moduli by arithmetic data of K, secondly shows that a certain canonically given jacobian $Jac(C\sb{\Phi}\sp{-1}{}\sb{(\tau)})$ is simple as long as $\tau\in B$ is K-singular and finally expresses the number of K-singular moduli in terms of class-numbers of CM-extension, $K\subset L$ of degree 3 following essentially the ideas of Hecke.
[F.W.Knoeller (Marburg)]

MSC 2000:: *14C22 Picard groups
14D20 Algebraic moduli problems
14G35 Modular and Shimura varieties
11R11 Quadratic extensions
14K30 Picard schemes, higher Jacobians
11R29 Class numbers, class groups, discriminants

Keywords: elliptic modular group; Picard modular group; imaginary quadratic field; K-singular moduli; Hilbert modular group; jacobian; class-numbers of CM-extension

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