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One-parametric \(p\)-adic subgroups of a group variety. (Sous-groupes à un paramètre \(p\)-adique de variétés de groupe.) (French) Zbl 0358.10016
This paper establishes the \(p\)-adic analogues of classical results obtained by Schneider in 1937 on the transcendence of values of Weierstraß elliptic functions. These are deduced from a new transcendence criterion for functions satisfying a set of differential equations and a sequence of functional equations of a special nature. The method is extended to the study of the exponential map on group varieties, and provides a \(p\)-adic version of a result of S. Lang [Topology 1, 313–318 (1962; Zbl 0116.38105)] (as noted by the author [Transcendance et lois de groupes algébriques. Sémin. Delange-Pisot-Poitou, 18e Année 1976/77, Théor. des Nombres, Fasc. 1, Exposé 1, 10 p. (1977; Zbl 0374.14009)], the method also applies to the logarithm of certain formal groups).
The last part of this paper deals with measures of linear independence for “algebraic points” of the exponential and elliptic functions. The established lower bounds improve the corresponding archimedean results of N. I. Fel’dman [Tr. Mosk. Mat. Obshch. 18, 65–76 (1968; Zbl 0235.10019)] and J. Coates [Invent. Math. 11, 167–182 (1970; Zbl 0216.04403)]. A fundamental use is here made of recent results of Bashmakov and Ribet on the Galois groups attached to division points of rational points on elliptic curves and multiplicative groups; the proof of the (unpublished) result of Ribet is included in an Appendix. Sharper and more general bounds can be deduced from the zeros estimates for “elliptic polynomials” which have recently been established by D. Masser [Invent. Math. 45, 61–82 (1978; Zbl 0375.10022)].

11J61 Approximation in non-Archimedean valuations
11J89 Transcendence theory of elliptic and abelian functions
11J91 Transcendence theory of other special functions
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
14G05 Rational points
14L10 Group varieties
22E35 Analysis on \(p\)-adic Lie groups
33E05 Elliptic functions and integrals
Full Text: DOI EuDML
[1] Altman, A.: The size function on abelian varieties. Transactions Am. Math. Soc.164, 153-161 (1972) · Zbl 0244.14013
[2] Ba?makov, M.: The cohomology of abelian varieties over a number field. Russian Math. Surveys27 (6), 25-70 (1972) · Zbl 0271.14010 · doi:10.1070/RM1972v027n06ABEH001392
[3] Bertrand, D.: Algebraic values ofp-adic elliptic functions. Proc. Conf. Advances in Transcendence Theory, Cambridge (1976) (à paraître)
[4] Borel, A.: Linear algebraic groups. New York-Amsterdam: Benjamin 1969 · Zbl 0206.49801
[5] Coates, J.: An application of the division theory of elliptic functions to diophantine approximation. Inventiones math.11, 167-182 (1970) · Zbl 0216.04403 · doi:10.1007/BF01404611
[6] Coates, J., Lang, S.: Diophantine approximations on abelian varieties with complex multiplication. Inventiones math.34, 129-133 (1976) · Zbl 0342.10018 · doi:10.1007/BF01425479
[7] Fel’dman, N.I.: Approximation de certains nombres transcendants II. Izvestia Akad. Nauk SSSR, Ser. Mat.15, 153-176 (1951) (en russe)
[8] Fel’dman, N.I.: Un analogue elliptique d’une inégalité de A.O. Gel’fond. Trudy Moskov. Mat. Ob??.18, 65-76 (1968) (en russe)
[9] Gel’fond, A.O.: Sur la divisibilité de la différence des puissances de deux nombres entiers par une puissance d’un idéal premier. Mat. Sbornik N.S.7, 7-25 (1940)
[10] Lang, S.: Introduction to transcendental numbers. Reading: Addison Wesley 1966 · Zbl 0144.04101
[11] Lutz, E.: Sur l’équationy 2=x3?Ax?B dans les corpsP-adiques. J. reine ang. Math.177, 238-247 (1937) · JFM 63.0101.01 · doi:10.1515/crll.1937.177.238
[12] Mahler, K.: Ein Beweis der Transzendenz derP-adischen Exponentialfunktion. J. reine ang. Math.169, 61-66 (1932) · Zbl 0006.01101
[13] Mahler, K.: Über transzendenteP-adische Zahlen. Compositio Math.2, 238-247 (1935) · JFM 61.0187.01
[14] Masser, D.: Elliptic functions and transcendence. Lecture Notes in Math.437. Berlin-Heidelberg-New York Springer 1975 · Zbl 0312.10023
[15] Masser, D.: Linear forms in algebraic points of abelian functions. I, II. Math. Proc. Camb. Phil. Soc.77, 499-513 (1975) et79, 55-70 (1976) · Zbl 0306.14018 · doi:10.1017/S030500410005132X
[16] Ribet, K.: Manuscrit non publié (Mai 1976)
[17] Schneider, Th.: Einführung in die transzendenten Zahlen. Berlin: Springer 1957 · Zbl 0077.04703
[18] Serre, J.-P.: Lie algebras and Lie groups. New York-Amsterdam: Benjamin 1965 · Zbl 0132.27803
[19] Waldschmidt, M.: Propriétés arithmétiques des valeurs de fonctions méromorphes algébriquement indépendantes. Acta Arithm.23, 19-88 (1973)
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