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Zbl 1111.11038
Gaudron, Éric
Linear forms of effective logaritms on abelian varieties. (Formes linéaires de logarithmes effectives sur les variétés abéliennes.) (French. English summary)
[J] Ann. Sci. Éc. Norm. Supér. (4) 39, No. 5, 699-773 (2006). ISSN 0012-9593

The main theorem of this remarkable paper is the first completely explicit lower bound for a non-vanishing linear form in logarithms of algebraic numbers on an abelian variety. The estimate involves the heights and degrees of the coefficients of the linear form and of the algebraic points on the variety. These are classically considered as the main parameters: previous estimates were explicit only as far as the dependence on these data is concerned. In this paper the author also provides an explicit dependence on the natural invariants of the abelian variety: dimension, Falting's height and degree of a polarisation. Also the numerical constant is computed; it is not a huge one, as one might be afraid of. \par The proof rests on a far-reaching development of methods arising from Arakelov's theory [{\it J.-B. Bost},`` Périodes et isogénies des variétés abéliennes sur les corps de nombres [d'après D. Masser et G. Wüstholz]". Séminaire Bourbaki. Volume 1994/95. Exposés 790-804. Paris: Société Mathématique de France, Astérisque. 237, 115--161, Exp. No. 795 (1996; Zbl 0936.11042); "Algebraic leaves of algebraic foliations over number fields." Publ. Math., Inst. Hautes Étud. Sci. 93, 161--221 (2001; Zbl 1034.14010)]. \par The author shows how to include in a geometrical setting within Bost's slope method a number of previous techniques, including those arising from works by A.~Baker (extrapolation technique), Chudnovsky (change of variables) and N.~Hirata-Kohno (redundant variables). The estimate is sharp: in term of an upper bound $B$ for the height of the coefficients of the linear form, it is of the best possible shape $B\sp{-C}$, where $C$ is a positive constant which depends on all other parameters apart from $B$. Such a sharp estimate is new for abelian varieties of dimension $\ge 2$: it was known for elliptic curves only, after the work of {\it S. David} and {\it N. Hirata-Kohno} on the one hand [Recent progress on linear forms in elliptic logarithms. Wüstholz, Gisbert (ed.), A panorama in number theory or The view from Baker's garden. Based on a conference in honor of Alan Baker's 60th birthday, Zürich, Switzerland, 1999. Cambridge: Cambridge University Press, 26--37 (2002; Zbl 1041.11053)], and of the author on the other [Mesures d'indépendance linéaire de logarithmes dans un groupe algébrique commutatif. Invent. Math. 162, No. 1, 137--188 (2005; Zbl 1120.11031)].
[Michel Waldschmidt (Paris)]

MSC 2000:: *11J86 Linear forms in logarithms; Baker's method
11G10 Abelian varieties of dimension $>1$
14G40 Arithmetic varieties and schemes

Keywords: Lower bounds for linear forms; abelian logarithms; Baker's method; Arakelov theory; slope method

Citations: Zbl 0936.11042; Zbl 1034.14010; Zbl 1041.11053; Zbl 1120.11031

Cited in: Zbl 1145.11004

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