The Galois representations associated to a Drinfeld module in special characteristic. III: Image of the group ring
This paper is the third 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 J. Number Theory 116, No. 2, 348--372 (2006; Zbl 1173.11037)]. 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 \(A\subset F\). Let \(K\) be a finitely generated field over \(k\), of transcendence degree \(1\) and let \( K^s \) be 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 \(\phi\). This is an \(A_\lambda \)-module free of rank \(r\) on which \(G := \text{Gal}(K^s/K)\) acts continuously. Assume that \( \phi \) has finite characteristic. Denote the ring of endomorphisms of \(\phi\) (over \(K\)) by \(E\). The following two theorems are shown in the present paper: Theorem A. Assume that \(E = A\). Then for almost all maximal ideals \(\lambda \) of \(A\) the \( A/\Lambda[ G ]\)-module \( \phi[\lambda] = T_\lambda( \phi ) \otimes A/\lambda\) is absolutely irreducible. Theorem B. For almost all maximal ideals \(\lambda \) of \(A\) the ring \(E\) and the image of \(A_\lambda[G]\) in \(\text{End}_{A_\lambda}(T_\lambda(\phi))\) are commutants of each other. These theorems have now be shown to hold for Drinfeld modules of arbitrary characteristic over finitely generated fields \(K\) of arbitrary transcendence degree by \textit{R. Pink} and \textit{E. Rütsche} [J. Number Theory 129, No. 4, 866--881 (2009; Zbl 1246.11121)].
scientific article; zbMATH DE number 5015057
scientific article