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Fonctions thêta et points de torsion des variétés abéliennes. (Theta functions and torsion points of abelian varieties). (French) Zbl 0741.14025
Let \(k\) be an arithmetic field of finite degree over \(\mathbb{Q}\) and \(A\) an abelian variety of dimension \(g\geq 1\) defined over \(k\). Given a torsion point of \(A\), \(e(\neq 0)\), let \(n(e)\) be the order of \(e\) and \(d(e)\) the degree, over \(k\), of the field of definition of \(e\). The main result of this paper is a lower bound for \(d(e)\) in terms of \(n(e)\), \(g\), the degree of \(k\) over \(\mathbb{Q}\), and the Faltings height of \(A\). With precision: There exists a constant \(c_ 1>0\) (depending on \(g\)) such that for any integer number \(\delta\geq 2\), any real number \(h\geq 1\), any field \(k\) of degree \(\leq\delta\) over \(\mathbb{Q}\) and any abelian variety \(A\) over \(k\) of dimension \(g\) with Faltings height \(h(A/k)\leq h\) — \(A\) simple and principally polarized —, the degree of the torsion points \(e\neq 0\) verify: \(d(e)\geq c_ 1h^{-3/2}\delta^{- 3/2}n(e)^{1/g}/(\log(\delta)\cdot\log(n(e))).\) The proof is transcendental and it is based on the study of the growth of the theta functions and the works of D. W. Masser [Compos. Math. 53, 153-170 (1984; Zbl 0551.14015) and in Diophantine approximations and transcendence theory, Semin., Bonn/FRG 1985, Lect. Notes Math. 1290, 109- 148 (1987; Zbl 0639.14025)].

MSC:
14K15 Arithmetic ground fields for abelian varieties
14K25 Theta functions and abelian varieties
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
14G99 Arithmetic problems in algebraic geometry; Diophantine geometry
11G10 Abelian varieties of dimension \(> 1\)
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References:
[1] A. Baker , On the periods of the Weierstrass p-function . Symposia Mathematicae, Indam, Rome, 1968/1969, IV, pp. 155-174, Academic Press, London, 1970. · Zbl 0223.10019
[2] D. Bertrand , Galois orbits on abelian varieties and zero estimates . J. Loxton et A. Van der Poorten (Eds.), Diophantine Analysis , LMS Lecture Notes, 109, pp. 21-35, Cambridge University Press, 1986. · Zbl 0597.10032
[3] P. Cohen , Explicit calculation of some effective constants in transcendence proofs . PhD thesis, University of Nottingham, 1985.
[4] S. David , Fonctions thêta et points de torsions des variétés abéliennes . Comptes rendus de l’Académie des Sciences, t. 305, pp. 211-214, 1987. · Zbl 0628.14035
[5] S. David , Théorie de Baker dans les familles de groupes algébriques commutatifs . En préparation, 1988.
[6] P. Deligne , Preuve des conjectures de Tate et de Shavarévitch [d’après G. Faltings] . Séminaire Bourbaki, 1983-84, pp. 616-01-616-17, Novembre 1983. · Zbl 0591.14026 · numdam:SB_1983-1984__26__25_0 · eudml:110028
[7] A. Faisant et G. Philibert , Aproximations simultanées de \tau et j(\tau ). Problèmes Diophantiens , n^\circ 66, Publications mathématiques de l’université Pierre et Marie Curie, 1983-1984.
[8] J. Igusa , Theta functions . Grundlehren Math. Wiss., n^\circ 194, Springer, 1972. · Zbl 0251.14016
[9] H. Lange , Families of translations of commutative algebraic groups . Journal of algebra, 109, pp. 260-265, 1987. · Zbl 0657.14026 · doi:10.1016/0021-8693(87)90174-8
[10] D. Masser , Small values of the quadratic part of the Néron-Tate height on an abelian variety . Compositio Math, 53, pp. 153-170, 1984. · Zbl 0551.14015 · numdam:CM_1984__53_2_153_0 · eudml:89682
[11] D. Masser , Small values of heights on families of abelian varieties . Proc. Conf. Bonn, Lecture Notes, 1290, pp. 109-148, Springer-Verlag, 1985. · Zbl 0639.14025
[12] D. Masser , Lettre à Daniel Bertrand , Novembre 1986.
[13] D. Masser et G. Wüstholz , Zero estimates on group varieties II . Inv. Math., 80, pp. 233-267, 1985. · Zbl 0564.10041 · doi:10.1007/BF01388605 · eudml:143229
[14] L. Moret-Bailly , Compactifications, hauteurs et finitude . Séminaire sur les pinceaux arithmétiques: la conjecture de Mordell, L. Szpiro, Astérique, 127, pp. 113-129, S.M.F., 1985. · Zbl 1182.14048
[15] D. Mumford , On the equations defining abelian varieties I . Inventiones Math, 1, pp. 287-354, 1966. · Zbl 0219.14024 · doi:10.1007/BF01389737 · eudml:141872
[16] D. Mumford , Abelian varieties . TIFR studies in mathematics, Oxford University Press, 1970. · Zbl 0223.14022
[17] D. Mumford , Tata Lectures on Theta I . Birkhäuser, Boston, 1983. · Zbl 0509.14049 · doi:10.1007/978-1-4899-2843-6
[18] I. Manin et G. Zarhin , Heights on families of abelian varieties . U.S.S.R. Math. Sbornik, 18, pp. 169-179, 1972. · Zbl 0263.14011 · doi:10.1070/SM1972v018n02ABEH001749
[19] P. Philippon , Lemmes de zéros dans les groupes algébriques commutatifs . Bull. S.M.F., 114, pp. 355-383, 1986. · Zbl 0617.14001 · doi:10.24033/bsmf.2060 · numdam:BSMF_1986__114__355_0 · eudml:87515
[20] P. Philippon et M. Waldschmidt , Formes linéaires de logarithmes dans les groupes algébriques commutatifs . Illinois Journal of Mathematics, 32(2), pp. 281-314, 1988. · Zbl 0651.10023
[21] M. Rosen, Abelian varieties over C. G. Cornell et J. Silverman (Eds.), Arithmetic Geometry , Springer-Verlag, Berlin-Heidelberg -New-York, 1986. · Zbl 0604.14032
[22] J.-P. Serre , Cours au collège de France , 1985 -1986.
[23] G. Shimura , On the derivatives of theta functions and modular forms . Duke Math. Journal, 44, pp 365-387, 1977. · Zbl 0371.14023 · doi:10.1215/S0012-7094-77-04416-7
[24] M. Waldschmidt , Transcendance et exponentielles en plusieurs variables . Inventiones Math., 63, pp 97-127, 1981. · Zbl 0454.10020 · doi:10.1007/BF01389195 · eudml:142792
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