Logarithms of rational points of abelian varieties. (Logarithmes des points rationnels des variétés abéliennes.) (French) Zbl 1445.11069

Let \(k\) be a number field of degree \(D\) and \(A\) an abelian variety of dimension \(g\) with a polarisation \(L\) over \(k\). Let \(\sigma_0:k\to{\mathbb{C}}\) be a complex embedding, \(\exp_{A_{\sigma_0}}:t_{A_{\sigma_0}}\to A_{\sigma_0}\) the associated exponential map of the complex abelian variety \(A_{\sigma_0}\). For \(u\in t_{A_{\sigma_0}}\) and \(p\in A(k)\) such that \(\exp_{A_{\sigma_0}}(u)=\sigma_0(p)\), let \(A_u\) be the smallest abelian subvariety of \(A_{\sigma_0}\), the tangent space at the origin of which contains \(u\). The main result of the paper under review is an explicit upper bound for the degree of \(A_u\) with respect to the polarisation \(L\). The upper bound depends on \(g\), \(D\), the dimension of \(A_u\), the Hermitian form of \(L_{\sigma_0}\), the Néron-Tate height \(\hat{\mathrm{h}}_L(p)\) relative to \(L\) of the point \(p\) and the stable Faltings height \({\mathrm{h}}_L(A)\) of \(A\). The first result in this direction, due to [D. Masser and G. Wüstholz, Ann. Math. (2) 137, No. 2, 407–458 (1993; Zbl 0796.11023)], dealt with the special case \(p=0\), and is known as the Period Theorem; it was effective but not explicit. An improved and explicit version of the Period Theorem was obtained by É. Gaudron and G. Rémond [Comment. Math. Helv. 89, No. 2, 343–403 (2014; Zbl 1297.11058)], using the Arakelovian approach of J.-B. Bost [in: Séminaire Bourbaki. Volume 1994/95. Exposés 790-804. Paris: Société Mathématique de France. 115–161, Exp. No. 795 (1996; Zbl 0936.11042)] and a result of S. David (unpublished). The generalisation to a nontorsion point is called by the author the Generalized Period theorem. The authors deduce several important results. The first one is an explicit lower bound for \(\log\Vert u\Vert_{L,\sigma_0}\) – see [F. Pellarin, J. Number Theory 88, No. 2, 241–262 (2001; Zbl 1032.11031)] and [É. Gaudron, Ann. Sci. Éc. Norm. Supér. (4) 39, No. 5, 699–773 (2006; Zbl 1111.11038)]. Another consequence is an explicit lower bound for the Néron-Tate height of a nontorsion point; to get an optimal dependence on \(D\) is the Generalized Lehmer Problem, on the height of \(A\) is the conjecture of Lang-Silverman. The third consequence is an explicit version of a theorem of D. Bertrand [Duke Math. J. 80, No. 1, 223–250 (1995; Zbl 0847.11036)] giving an upper bound for \(({\mathrm{deg}}_L A_p)^{1/\dim A_p}\) which is linear in terms of the height of \(p\).
The Generalized Period Theorem is a lower bound for a linear form in one logarithm of an algebraic point. For an extension to linear forms in several logarithms, see [F. Ballaÿ, Diss. Math. 543, 1–78 (2019; Zbl 1436.11088)].


11J86 Linear forms in logarithms; Baker’s method
11J95 Results involving abelian varieties
11G10 Abelian varieties of dimension \(> 1\)
Full Text: DOI


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