The nonexistence of certain level structures on abelian varieties over complex function fields. (English) Zbl 0675.14018

Nadel, Alan Michael

Ann. Math. (2) 129, No. 1, 161-178 (1989).

Let X(n) the quotient of the g-Siegel upper-half plane by the congruence subgroup \(\Gamma(n)\); the author proves the following result: for \(n\geq \max (g(g+1)/2,28)\) the image of any non constant holomorphic map from \({\mathbb{C}}\) to a compactification of X(n) must be contained in the complement of X(n). He deduces a result similar to the famous Mazur theorem concerning elliptic curves over \({\mathbb{Q}}:\) Over a complex function field of genus \(g\geq 1\) it is impossible to find a non constant principally polarized abelian variety of dimension g with a level n- structure (n as above). The proof of the first result uses the notion of “big divisors” as well as Baily-Borel and toroidal compactifications of bounded symmetric domains.

Reviewer: R.Gillard

MSC:

14K20	Analytic theory of abelian varieties; abelian integrals and differentials
32M15	Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)

Keywords:

nonexistence of certain level structures on abelian varieties; g-Siegel upper-half plane; complex function field; compactifications of bounded symmetric domains