Uniform boundedness of level structures on abelian varieties over complex function fields

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• 作者：Jun-Muk Hwang and Wing-Keung To
• 关键词：Mathematics Subject Classification (2000) 14K10 ; 32M15 ; 32N10
• 刊名：Mathematische Annalen
• 出版年：2006
• 出版时间：June 2006
• 年：2006
• 卷：335
• 期：2
• 页码：363-377
• 全文大小：175 KB

文摘

Let X = Ω/Γ be a smooth quotient of a bounded symmetric domain Ω by an arithmetic subgroup . We prove the following generalization of Nadel's result: for any non-negative integer g, there exists a finite ¨¦tale cover X_g = Ω/Γ(g) of X determined by a subgroup depending only on g, such that for any compact Riemann surface R of genus g and any non-constant holomorphic map f : R → X_g* from R into the Satake-Baily-Borel compactification X_g* of X_g, the image f(R) lies in the boundary ∂X_g: = X*_g\X_g. Nadel proved it for g = 0 or 1. Moreover, for any positive integer n and any non-negative integer g≥0, we show that there exists a positive number a(n,g) depending only on n and g with the following property: a principally polarized non-isotrivial n-dimensional abelian variety over a complex function field of genus g does not have a level-N structure for N≥a(n,g). This was proved by Nadel for g = 0 or 1, and by Noguchi for arbitrary g under the additional hypothesis that the abelian variety has non-empty singular fibers.