Transcendence and Drinfeld modules.(English)Zbl 0586.12010

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 $$k$$.

MSC:

 11J93 Transcendence theory of Drinfel’d and $$t$$-modules 11G09 Drinfel’d modules; higher-dimensional motives, etc. 11R58 Arithmetic theory of algebraic function fields 14L05 Formal groups, $$p$$-divisible groups
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References:

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