Elliptic curves in two-dimensional abelian varieties and the algebraic independence of certain numbers (Q1095184)

The first elliptic analogue of the results of Brownawell, Shmelev and Waldschmidt on small transcendence degrees for numbers of the type \(\exp (u_ iv_ j)\) have been obtained by \textit{D. W. Masser} and \textit{G. Wüstholz} [Journées arithmétiques, Exeter 1980, Lond. Math. Soc. Lect. Note Ser. 56, 360-363 (1982; Zbl 0491.10025)] thanks to their zero estimates on group varieties. Their extension of these zero estimates to multiplicities in a given direction provides the author with the main tool for the study, given in the present article, of small transcendence degrees for the elliptic analogue of the \(\{u_ i,v_ j,\exp (u_ iv_ j)\}\) problem. For a generalization to arbitrary commutative algebraic groups, see the author's article [J. Number Theory 25, 279-307 (1987; Zbl 0608.10036)].