Théorème de Baker et modules de Drinfeld. (Baker's theorem and Drinfeld modules)
In Duke Math. J. 58, 559-575 (1989; Zbl 0687.12008) \textit{Jing Yu} established the following result: Let \(\mathbb{F}_ q\) be the field of \(q\)- elements, \({\mathcal C}\) a smooth projective geometrically irreducible curve over \(\mathbb{F}_ q\) and \(\infty\) a fixed close point. We set \(A\) to be the affine ring of \({\mathcal C}-\infty\). Let \(k\) be the function field of \({\mathcal C}\) with completion \(k_ \infty\) at \(\infty\). Let \(\overline{k}_ \infty\) be a fixed algebraic closure of \(k_ \infty\) with \(\overline{k}\subset k_ \infty\) being the algebraic closure of \(k\). Let \(\varphi\) be a Drinfeld module defined over \(\overline{k}\) associated to the \(A\)-lattice \(L\subset\overline{k}_ \infty\). Finally, we let \(e_ L(z)\) be the exponential function of \(\varphi\) and \(k_ L\) the field of multiplications of \(\varphi\). Theorem: Let \(\alpha_ 1,\dots,\alpha_ n\) be elements of \(\overline{k}_ \infty\) such that \(e_ L(\alpha_ i)\in\overline{k}\) for all \(i\). If \(\alpha_ 1,\dots,\alpha_ n\) are linearly independent over \(k_ L\), then any linear combination of \(\alpha_ 1,\dots,\alpha_ n\), with coefficients not all zero from the separable closure \(k_ L^ s\) of \(k_ L\), is transcendental over \(k\). Yu's result is an analog of the qualitative version of Baker's theorem. It is refined in the present paper when \(\varphi\) has complex multiplication. The author then strengthens the theorem by replacing \(k_ L^ s\) with \(\overline{k}\).
scientific article; zbMATH DE number 166069