One-parametric \(p\)-adic subgroups of a group variety
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 \textit{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 \textit{N. I. Fel'dman} [Tr. Mosk. Mat. Obshch. 18, 65--76 (1968; Zbl 0235.10019)] and \textit{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groups attached to division points of rational points on elliptic curves and multiplicative groups; the proof of the (unpublished) result of Ribet is included in an Appendix.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 \textit{D. Masser} [Invent. Math. 45, 61--82 (1978; Zbl 0375.10022)].
scientific article; zbMATH DE number 3557885
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