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Eine Klassenzahlformel für singuläre Moduln der Picardschen Modulgruppen

In: Algebraic Geometry

Author

Listed:
  • Jan Feustel

    (Akademie der Wissenschaften der DDR, Karl-Weierstraß-Institut für Mathematik)

Abstract

Es ist altbekannt, daß die elliptische Invariante J(τ) den Körper der Modulfunktionen auf der oberen Halbebene bzgl. S12(Z) erzeugt und auf den Punkten τ ∈ H, die imaginär-quadratische Zahlkörper erzeugen (und damit Fixpunkte von Elementen aus Gl 2 + (Z) sind), algebraische Werte annimmt. Durch Zuordnung dieser Punkte mit Q(τ) = K zu Idealen in Ordnungen des imaginär-quadratischen Zahlkörpers K erhalten wir mit der ‘Klassengleichung’ eine algebraische Gleichung für J(τ) mit rationalen Koeffizienten. In diesem Zusammenhang sei erinnert, daß J(τ) zusammen mit der Weierstraßschen ℊ-(Funktion der zugehörigen elliptischen Kurve E τ alle Abelschen Erweiterungen von Q(τ) erzeugt (‘Kroneckers Jugendtraum’).

Suggested Citation

  • Jan Feustel, 1990. "Eine Klassenzahlformel für singuläre Moduln der Picardschen Modulgruppen," Springer Books, in: H. Kurke & J. H. M. Steenbrink (ed.), Algebraic Geometry, pages 87-100, Springer.
    • Handle: RePEc:spr:sprchp:978-94-009-0685-3_6
    DOI: 10.1007/978-94-009-0685-3_6
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