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

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)
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