# 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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