A note on Kummer theory of division points over singular Drinfeld modules (Q2748262)

Let \(A = \mathbb F_q[T]\) be an polynomial ring over a finite field \(\mathbb F_q\). A Drinfeld \(A\)-module \(\varphi\) of rank \(m\) is called singular or of CM-type if its endomorphism ring \({\mathcal O}\) is an order in a field extension of degree \(m\) over \(K = \mathbb F_q(T)\). NEWLINENEWLINENEWLINELet \(\varphi\) be defined over a field \(L/K\), \(\ell \in A\) a monic polynomial, \(\Lambda_{\ell}^{\varphi}\) the set of \(\ell\)-division points of \(\varphi\), \(\Gamma = \{a_1,\ldots,a_r\}\) a finite set of points of \(\varphi(L)\), \(\mathbb L_{\ell}\) the field extension generated over \(L(\Lambda_{\ell}^{\varphi})\) by the ``\(\ell\)-th roots'' of \(a_1,\ldots,a_r\), and \(H_{\Gamma}(\ell) = \text{Gal}(\mathbb L_{\ell}\mid L(\Lambda_{\ell}^{\varphi}))\). The author proves: If \(L\) is large enough, \(\varphi\) is singular, and \(a_1,\ldots,a_r \in L\) are \({\mathcal O}\)-linearly independent, then for almost all \(\ell\) (with an effectively determined set of exceptions), \(H_{\Gamma}(\ell)\) is isomorphic with \(\Lambda_{\ell}^{\varphi} \times \cdots \times \Lambda_{\ell}^{\varphi}\) (\(r\) copies). NEWLINENEWLINENEWLINEThis may be seen as the starting point for a Kummer theory of such Drinfeld modules.