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 Γ ⊂ Aut(Ω). We prove the following generalization of Nadel's result: for any non-negative integer g, there exists a finite étale cover Xg = Ω/ Γ(g) of X determined by a subgroup Γ(g) ⊂ Γ 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 Xg, the image f (R) lies in the boundary ∂Xg:= X*g\ Xg. 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.