Semi-Abelian Schemes and Heights of Cycles in Moduli Spaces of Abelian Varieties

We study extension properties of Barsotti-Tate groups, and we establish diophantine inequalities involving heights of cycles with respect to logarithmically singular hermitian line bundles on arithmetic varieties. We apply these results to bound heights of cycles on moduli spaces of abelian varieties, induced by quotients of abelian schemes by levels of Barsotti-Tate subgroups, over function fields over number fields. To achieve this aim, we combine our results with Rumely’s theorem on integral points on possibly open arithmetic surfaces and with Faltings’ theorems on heights of abelian schemes in isogeny classes.