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Hindry, Marc

Autour d’une conjecture de Serge Lang. (Around a conjecture by Serge Lang). (English) Zbl 0638.14026

Invent. Math. 94, No. 3, 575-603 (1988).
We study a conjecture of Serge Lang describing the intersection of an algebraic subvariety of an algebraic group \(G\) with a subgroup \(\Gamma\) of finite rank in \(G({\mathbb{C}})\). We prove the conjecture when \(\Gamma\) is the torsion subgroup of \(G\), extending previous result of Raynaud (case when \(G\) is an abelian variety) and Laurent (case when \(G\) is a linear torus). The method gives explicit results in terms of some Galois theoretic constant. We also prove some results and formulate some remarks on the general conjecture.
Reviewer: M.Hindry

Cited in 7 Reviews
Cited in 74 Documents

MSC:

14L10 Group varieties
20G15 Linear algebraic groups over arbitrary fields

Keywords:

intersection with a subgroup; algebraic subvariety of an algebraic group

Citations:

Zbl 0528.14013
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Full Text: DOI EuDML

References:

[1] Bertrand, D.: Minimal heights and polarizations on abelian varieties. Preprint MSRI, Berkeley juin 87 (à paraître) · Zbl 0847.11036
[2] Bertrand, D.: Galois representations and transcendental numbers. Proceedings of Durham conference 1986. (New advances in Transcendance theory, Baler A. ed. Cambridge University Press (à paraître)
[3] Bertrand D., Philippon, P.: Sous-groupes de groupes algébriques commutatifs Ill. J. Math.32, 263-280 (1988) · Zbl 0618.14020
[4] Bogomolov, F.: Points of finite order on an abelian variety. Math. USSR Izv.17, 55-72 (1981) · Zbl 0466.14015 · doi:10.1070/IM1981v017n01ABEH001329
[5] Brown, M.: Schémas en groupes et lemmes de zéros. Publ. Paris6, problèmes diophantiens73, exposé no 3, 17 pages (1984-85)
[6] Chabauty, C.: Sur les points rationnels des variétés algébriques dont l’irrégularité est supérieure à la dimension. C. R. Acad. Sci. Paris, Ser. A212 (1941) p 1022-1024 · JFM 67.0105.02
[7] Coleman, R.: Torsion points on curves andp-adic abelian inegrals. Ann. Math.121, 111-168 (1985) · Zbl 0578.14038 · doi:10.2307/1971194
[8] Coleman, R.: Ramified points on curves. Duke Math. J.54, 615-40 (1987) · Zbl 0626.14022 · doi:10.1215/S0012-7094-87-05425-1
[9] Coleman, R.: Effective Chabauty. Duke Math. J.52, 765-770 (1985) · Zbl 0588.14015 · doi:10.1215/S0012-7094-85-05240-8
[10] Faltings, G.: Endlichkeitssätze für abelsche Varietaten über Zahlkörpern. Invent. Math.73, 349-366 (1983) · Zbl 0588.14026 · doi:10.1007/BF01388432
[11] Frey, G.: A remark about isogenies of elliptic curves over quadratic fields. Compos. Math.58, 133-134 (1986) · Zbl 0596.14021
[12] Gross, B., Rohrlich, D.: Some result on the Mordell-Weil group of the Jacobian of the Fermat curve. Invent. Math.44, 201-224 (1978) · Zbl 0369.14011 · doi:10.1007/BF01403161
[13] Hartshorne, R.: Algebraic geometry. Berlin-Heidelberg-New York: Springer 1977 · Zbl 0367.14001
[14] Hindry M.: Points de torsion sur les sous-variétés de variétés abéliennes. C.R. Acad. Sci., Paris Ser. A-304,12, 311-314 (1987) · Zbl 0621.14026
[15] Hindry, M.: Points de torsion sur les sous-variétés de groupes algébriques. Séminaire de théorie des nombres de Bordeaux (1986-87) exposé no 18
[16] Hindry, M.: Thèse de doctorat. (Université Paris 6, mai 1987)
[17] Hindry, M.: Points quadratiques sur les courbes. C.R. Acad. Sci. Paris, Ser. A-305, 219-221 (1987) · Zbl 0677.14005
[18] Katz, N., Lang, S.: Finiteness in geometric classfield theory. Enseignement Math.27, 285-319 (1981) · Zbl 0495.14011
[19] Koblitz, N., Rohrlich, D.: Simple factors in the jacobian of the Fermat curve. Can. J. Math.XXX, 1183-1205 (1978) · Zbl 0399.14023 · doi:10.4153/CJM-1978-099-6
[20] Lang, S.: Fundamentals of diophantine geometry. Berlin-Heidelberg-New York: Springer 1983 · Zbl 0528.14013
[21] Lang, S.: Division points on curves. Ann. Mat. Pura Appl.LXX, 229-234 (1965) · Zbl 0151.27401
[22] Lang, S.: Complex multiplication. Berlin-Heidelberg-New York: Springer 1983 · Zbl 0536.14029
[23] Lange, H.: Translations de groupes algébriques commutatifs. C.R. Acad. Sci. Paris, Ser. A,300, 255-258 (1985) · Zbl 0579.14040
[24] Laurent, M.: Equations diophantiennes exponentielles. Invent. Math.78, 299-327 (1984) · Zbl 0554.10009 · doi:10.1007/BF01388597
[25] Manin, Y.: Thep-torsion of elliptic curves is uniformly bounded. Izv. Akad. Nauk. USSR33, 433-438 (1969) AMS Transl. · Zbl 0205.25002 · doi:10.1070/IM1969v003n03ABEH000787
[26] Masser, D.: Small values of the quadratic part of the Néron-Tate height. Compos. Math.53, 153-170 (1986)
[27] Mazur, B.: Modular curves and the Eisenstein ideal. Publ. Math., Inst. Hautes Etud. Sci.47, 33-186 (1977) · Zbl 0394.14008 · doi:10.1007/BF02684339
[28] Mumford, D.: A remark on Mordell’s conjecture. Amer. J. Math.LXXXVII, 1007-1016 (1965) · Zbl 0151.27301 · doi:10.2307/2373258
[29] Philippon, P.: Lemme de zéros dans les groupes algébriques commutatifs. Bull. Soc. Math. France114, 355-383 (1986) · Zbl 0617.14001
[30] Raynaud, M.: Courbe sur une variété abélienne et points de torsion. Invent. Math.71, 207-233 (1983) · Zbl 0564.14020 · doi:10.1007/BF01393342
[31] Raynaud, M.: Sous-variété d’une variété abélienne et points de torsion. Arithmetic and geometry (dédié à Shafarevic) Vol. 1, pp 327-352). Boston: Birkhäuser 1983
[32] Raynaud, M.: Around the Mordell conjecture for function fields and a conjecture of Serge Lang. Proceedings of algebraic geometry of Tokyo (1982). (Lect. Notes Math., Vol. 1016). Berlin-Heidelberg-New York: Springer 1983
[33] Ribet, K.: Kummer theory on extension of abelian varieties by tori. Duke Math. J.46, 745-761 (1979) · Zbl 0428.14018 · doi:10.1215/S0012-7094-79-04638-6
[34] Voir aussi: Deficient points on extension of abelian varieties byG m . (en collaboration avec O. Jacquinot) J. Number Theory25, 133-151 (1987) · Zbl 0667.14021 · doi:10.1016/0022-314X(87)90020-5
[35] Ribet, K.: Division fields of abelian varieties with complex multiplication. Mém. Soc. Math. France2, 75-94 (1980) · Zbl 0452.14009
[36] Serre, J.-P.: Groupes algébriques et corps de classes. Paris: Hermann 1959 · Zbl 0097.35604
[37] Serre, J.-P.: Quelques propriétés des groupes algébriques commutatifs. Appendice dans Astérisque69-70, 191-202 (1979)
[38] Serre, J.-P.: Résumé des cours au collège de France, (1985-86). Ann. Collège France (1986) 95-99
[39] Shimura, G., Taniyama, Y.: Complex multiplication of abelian varieties and its applications to number theory. Publ. Math. Soc. Japan6, Tokyo (1961) · Zbl 0112.03502
[40] Silverberg, A.: Points de torsion des variétés abéliennes de type CM. Publ. Univ. Paris 6, no 79 problèmes diophantiens 1985-86
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