The Galois representations associated to a Drinfeld module in special characteristic. I: Zariski density
This paper is the first in a series of three. For the other two see [\textit{R. Pink}, J. Number Theory 116, No. 2, 348--372 (2006; Zbl 1173.11037) 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 \(\rho_\lambda : \text{Gal}(K^s/K) \to \Aut(V_\lambda(\phi))\cong \text{GL}_r( F_\lambda).\) Theorem. If \(\text{End}(\phi_{K^s} ) = A\) then the image of \(\rho_\lambda\) is Zariski dense in \(\text{GL}_r \). In case \(\phi\) has generic characteristic this was already shown in \textit{R. Pink} [Publ. Res. Inst. Math. Sci. 33, No. 3, 393--425 (1997; Zbl 0895.11025)]. The current paper proves the theorem in the more difficult case of special characteristic. Note that one cannot hope for such a simple description of the Zariski closure of the Galois image in the analogous case of abelian varieties. Just knowing that the endomorphism ring of an abelian variety is \( \mathbb{Z} \) does not suffice to conclude that its Mumford-Tate group is the general symplectic group of rank \(2g\), see \textit{D. Mumford} [Math. Ann. 181, 345--351 (1969; Zbl 0169.23301)] for a counterexample.
scientific article; zbMATH DE number 5015055