On the torsion points of Drinfeld modules in abelian extensions (Q1861455)

Let \(A={\mathbb F}_q[T]\) and let \(\phi\) be a Drinfeld \(A\)-module of rank \(n\) on a finite extension \(K\) of \(k={\mathbb F}_q(T)\). The endomorphism ring \(\text{End}_{\overline{K}}(\phi)\) is an \(A\)-order in a finite extension of \(k\); and \(\phi\) is said to be of complex multiplication type if the degree of \(\text{End}_{\overline{K}}(\phi)\otimes_A k\) over \(k\) equals rank\((\phi)\), the maximal possible value. Let \(\phi(K^{\text{ab}})_{\text{tors}}\) be the \(A\)-module of torsion points of \(\phi\) in the maximal abelian extension \(K^{\text{ab}}\) of \(K\). The paper proves the following Theorem: Assume that \(K\) is large enough that all endomorphisms of \(\phi\) are defined over \(K\). Then \(\phi(K^{\text{ab}})_{\text{tors}}\) is infinite if and only if \(\phi\) is of complex multiplication type. Actually, the condition that all endomorphisms are defined over \(K\) is only mentioned before Lemma 2 as ``without loss of generality''. But in analogy with the corresponding results for abelian varieties over (the maximal abelian extension of) number fields, proved by \textit{Yu. Zarkhin} [Duke Math. J. 54, 131--145 (1987; Zbl 0632.14035)], it is to be expected that this condition is actually necessary for \(\phi(K^{\text{ab}})_{\text{tors}}\) to be infinite. For example the rank \(2\) Drinfeld module given by \(\phi_T(X)=X^{q^2}+TX\) has complex multiplication over \({\mathbb F}_{q^2}(T)\), but if \(K={\mathbb F}_q(T)\), then \(\phi(K^{\text{ab}})_{\text{tors}}=\{0\}\).