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Journal of the Ramanujan Mathematical Society

Volume 34, Issue 4, December 2019  pp. 433–447.

Some explicit computations in Arakelov geometry of abelian varieties

Authors:  Éric Gaudron
Author institution:Université Clermont Auvergne, CNRS, LMBP, F-63000 Clermont-Ferrand, France

Summary:  Given a polarized complex abelian variety (A, L), a Gromov lemma makes a comparison between the sup and L2 norms of a global section of L. We give here an explicit bound which depends on the dimension, degree and injectivity diameter of (A, L). It rests on a more general estimate for the jet of a global section of L. As an application we deduce some estimates of the maximal slope of the tangent and cotangent spaces of a polarized abelian variety defined over a number field. These results are effective versions of previous works by Masser and Wüstholz on one hand and Bost on the other. They also improve some similar statements established by Graftieaux in 2000.

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