By Eberhard Freitag (auth.)
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12 Hilfssatz. Jede Spitzenj(Jrm fE [rn' 0], n > 0, vom Gewicht 0 ver- schwindet identisch. 12 1 Folgerung. Jede Modulform vom Gewicht 0 ist konstant. Beweis. 11 besitzt f(Z) ein Maximum in IHn und ist daher konstant. Die Konstante muß 0 sein, denn es gilt limf(itE) =0. t~(j) Die Folgerung ergibt sich nun durch Induktion nach n folgendermaßen: Sei f E[f,'" 0] eine Modulform, so daß fl
M- 1; M E%, M E%} ist kompakt, da sie stetiges Bild der kompakten Menge % x % unter der Abbildung (M, M) --+ M . M- 1 ist. 11 Satz. :=Sp(n, Z) operiert auf IHn eigentlich diskontinuierlich. 12 Folgerung. , M 6 Hilfssatz. Wenn das letzte Diagonalelement tnn einer semipositiven Matrix T verschwindet, so T= gilt (~l ~), Tl = Ti n - l )~O. Beweis. Man nutze aus, daß jede zweireihige Unterdeterminante von T nicht negativ ist. 7 Bemerkung. Der