The Galois representations associated to a Drinfeld module in special characteristic. II: Openness
This paper is the second in a series of three. For the other two see [\textit{R. Pink}, J. Number Theory 116, No. 2, 324--347 (2006; Zbl 1173.11036) and \textit{R. Pink} and \textit{M. Traulsen}, J. Number Theory 116, No. 2, 373--395 (2006; Zbl 1173.11038)]. Let \(k\) be a finite field of \(q\) elements, \(X/k\) a smooth and projective curve with exact field of constants \(k\), \(\infty\) a closed point on \(X\). Denote the function field of \(X\) by \(F\) and the ring of regular functions on \(X - \infty\) by \(A\subset F\). Also, let \(K\) be a finitely generated field over \(k\) and \(K^s\) a separable closure of \(K\). Let \(\phi\) be a Drinfeld \(A\)-module of rank \(r\) over \(K\). Let \(\lambda\subset A\) be a maximal ideal different from the characteristic of \(\phi\). Denote by \(T_\lambda(\phi)\) the \(\lambda\)-adic Tate module of \(\phi\) and by \(V_\lambda(\phi)\) its extension of scalars from \(A_\lambda\) to \(F_\lambda\). Then \(V_\lambda(\phi)\) is an \(F_\lambda\)-vector space of dimension \(r\) on which \(\text{Gal}(K^s/K)\) acts continuously. Denote the action by \(\rho_\lambda : \text{Gal}( K^s/K ) \to \Aut(V_\lambda(\phi)) \cong \text{GL}_r(F_\lambda).\) Let \(P\) be a finite set of maximal ideals of \(A\), different from the characteristic of \(\phi\) and denote by \(V_P(\phi)\) the product of the \(V_\lambda(\phi)\) over all \(\lambda\) in \(P\). Denote the action of \(\text{Gal}(K^s/K)\) on \(V_P(\phi)\) by \(\rho_P\). \textit{R. Pink} [Publ. Res. Inst. Math. Sci. 33, No. 3, 393--425 (1997; Zbl 0895.11025)] had previously shown: Theorem. If \(\text{End}(\phi_{K^s}) = A\) and \(\phi\) is of generic characteristic then the image of \(\rho_P\) is an open subgroup of \(\text{GL}(V_P(\phi))\). The present paper considers \(\phi\) of finite characteristic. In that case the image of \(\rho_P\) cannot be open. To see this in the rank \(1\) case, note that there is a finite extension of \(K\) over which \(\phi\) becomes isomorphic to a module defined over a finite field. So up to a finite group the image of \(\rho_\lambda\) is pro-cyclic and hence cannot be open in \(F_\lambda^\times\). For higher rank consider the determinant to which the same argument applies. The main results of this paper give a detailed description of the possibilities for the image of the geometric Galois group intersected with the special linear group, i.e. for \(\rho_P(\text{Gal}( K^s / K\bar{k})) \cap \text{SL}(V_\lambda)\) up to commensurability. Criteria to decide this group are given in terms of the endomorphism algebra of \(\phi\) and in terms of the characteristic polynomials of Frobenius elements.
scientific article; zbMATH DE number 5015056