Logarithmes des points rationnels des variétés abéliennes

Abstract : Let u be a logarithm of an algebraic point p of an abelian variety defined over a number field. Let Au be the smallest abelian subvariety whose tangent space at the origin contains u. We give a bound for the geometrical degree of Au which generalizes the famous period theorem by Masser and Wüstholz. From this statement, we also deduce some lower bounds for the norm of u and for the Néron-Tate height of p. The main new feature of these results is that the bounds are fully explicit in terms of the classical invariants of the problem, among which the dimension and the Faltings height of the abelian variety. The proofs rest on tools from Gel'fond-Baker theory of linear forms in logarithms and from Arakelov slope theory.
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Vincent Bosser, Eric Gaudron. Logarithmes des points rationnels des variétés abéliennes. Canadian Journal of Mathematics, University of Toronto Press, 2019, 71 (2), pp.247-298. ⟨https://www.cambridge.org/core/journals/canadian-journal-of-mathematics/article/logarithmes-des-points-rationnels-des-varietes-abeliennes/2B88A5E69C1547F9E49AB75D38918FC5⟩. ⟨10.4153/CJM-2018-005-7⟩. ⟨hal-02172182⟩

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