Derivatives of a Drinfeld module and transcendence
Let \({\mathbf k}= \mathbb{F}_q(T)\) and let \({\mathbf A}= \mathbb{F}_q[T]\). Let \({\mathbf K}= \mathbb{F}_q((1/ T))\) equipped with the derivative \(':= d/dT\). One knows that this derivation uniquely extends to the separable closure \({\mathbf K}^{\text{sep}}\) of \({\mathbf K}\) inside a fixed algebraic closure of \({\mathbf K}\). In the paper being reviewed, the author continues his study of the use of \('\) in the transcendency theory of Drinfeld modules. In particular, the following very interesting results are established. Theorem: 1. Let \(\phi\) be a Drinfeld module which is defined over the algebraic closure of \({\mathbf k}\). Let \(0\neq \alpha\) be such that \(e(\alpha)\) is separable over \({\mathbf k}\), where \(e(z)\) is the exponential function of \(\phi\). Then \(1\), \(\alpha\), \(\alpha'\) are linearly independent over the algebraic closure of \({\mathbf k}\). 2. Let \(\pi\) be the element in \({\mathbf K}\) such that the period of the Carlitz module is \((T- T^q)^{1/(q- 1)}\cdot \pi\). Then \(1,\pi,\dots, \pi^{(p- 1)}\) are linearly independent over the algebraic closure of \({\mathbf k}\). 3. Let \(0\neq \alpha\in {\mathbf K}^{\text{sep}}\). Then \(\pi \alpha^p\) and \(\pi+ \alpha^p\) are transcendental over \({\mathbf k}\). The author also gives a proof of a theorem of Baker type on logarithms of elements in the algebraic closure of \({\mathbf k}\).
scientific article; zbMATH DE number 850798
scientific article