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

Abstract

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 étale 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.