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groups attached to division points of rational points on elliptic curves and multiplicative groups; the proof of the (unpublished) result of Ribet is included in an Appendix.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


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