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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Factorial property of a ring of automorphic forms


Author: Shigeaki Tsuyumine
Journal: Trans. Amer. Math. Soc. 296 (1986), 111-123
MSC: Primary 11F03; Secondary 14H15, 32N05
DOI: https://doi.org/10.1090/S0002-9947-1986-0837801-9
MathSciNet review: 837801
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Abstract: A ring of automorphic forms is shown to be factorial under some conditions on the domain and on the Picard group. As an application, we show that any divisor on the moduli space $ {\mathfrak{M}_g}$ of curves of genus $ g \geqslant 3$ is defined by a single element, and that the Satake compactification of $ {\mathfrak{M}_g}$ is written as a projective spectrum of a factorial graded ring. We find a single element which defines the closure of $ {\mathfrak{M}\prime_4}$ in $ {\mathfrak{M}_4}$ where $ {\mathfrak{M}\prime_4}$ is the moduli of curves of genus four whose canonical curves are exhibited as complete intersections of quadric cones and of cubics in $ {\mathbb{P}^3}$.


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DOI: https://doi.org/10.1090/S0002-9947-1986-0837801-9
Keywords: Automorphic form, Picard group, Betti number, moduli of curves, theta constant, Schottky invariant
Article copyright: © Copyright 1986 American Mathematical Society