Semi-abelian schemes and heights of cycles in moduli spaces of abelian varieties
``Zarhin's trick'' is a famous argument in arithmetic geometry, usually used for people to establish finiteness results for isomorphism classes of abelian varieties not endowed with a polarization. This played a crucial role in Zarhin's proof of the Tate conjecture for abelian varieties in the function field case. Zarhin's main result can be described as follows. Let \(k\) be a field and \(l\) be a prime number different from the characteristic of \(k\). Assume that for every abelian variety \(A\) over \(k\) and any principal polarization \(\lambda\) over \(A\), the following property (denoted by \(Z(A,\lambda/k)\)) is satisfied: for any \(l\)-divisible group \(\{G_n\}_{n\geq0}\) with \(G_n\) a subgroup of \(A[l^n]\) Lagrangian with respect to the Weil paring attached to \(\lambda\), the quotients \(A/{G_n}\) belong to a finite family of \(k\)-isomorphism classes. Then the Tate conjecture holds for any abelian variety \(A\) over \(k\). The aim of the paper under review is to sketch a variant to various approaches to \(Z(A,\lambda/k)\) in the situation where \(k\) ia s function field of transcendence degree \(1\) over \(\mathbb{Q}\). The authors' approach avoids the Hodge theoretic arguments, but relies on higher dimensional Arakelov geometry. To achieve their aim, they study extension properties of Barsotti-Tate subgroups of abelian schemes with semi-stable reduction, over two dimensional regular bases, and they established Diophantine inequalities involving heights of cycles with respect to logarithmically singular hermitian line bundles, then to bound heights of cycles on moduli spaces of abelian varieties. The main tools the authors use are a theorem of Rumely as well as a refinement by Autissier, on the existence of integral points on open arithmetic surfaces, and Faltings' theorem on heights of quotients of an abelian scheme by the levels of a Barsotti-Tate subgroup. Moreover, these tools show that this paper may be of interest for Diophantine geometry more than the application to the Tate conjecture over function fields.
scientific article; zbMATH DE number 6158417
scientific article