Transcendence and Drinfeld modules (Q1071808)

Let \(k\) be a function field in one variable over a finite field, \(\infty\) a fixed prime of \(k\) and \(A\) the ring of functions regular away from \(\infty\). Let \(K=k_{\infty}\) be the completion of \(k\) at \(\infty\). This paper is the culmination of a long study by the author of the transcendence properties of Drinfeld \(A\)-modules. In it the author establishes an elegant function field version of the classical theorem of Schneider-Lang. He is then able to deduce, among other things, the following very important corollaries: 1) Let \(\phi\) be a Drinfeld module defined over the algebraic closure, \(k^{\text{ac}}\), of \(k\) and let \(M\) be the associated lattice. Let \(0\neq \omega \in M\). Then \(\omega\) is transcendental over \(k\). 2) Let \(\phi_ 1\) and \(\phi_ 2\) be two Drinfeld modules over \(k^{\text{ac}}\) with different ranks, and associated lattices \(M_ 1\) and \(M_ 2\). Let \(0\neq \omega_ i\in M_ i\), \(i=1,2\). Then the quotient of \(\omega_ 1\) by \(\omega_ 2\) is transcendental over \(k\). 3) \((A=\mathbb F_ q[T])\) Let \(\tau \in k^{\text{ac}}\cap (K^{\text{ac}}-K)\). Then the value of the modular function \(j\) (defined in analogy with classical elliptic theory) at \(\tau\) is transcendental over \(k\).