On de Rham cohomology of Drinfeld modules of rank 2

Previously, using the theory of delta characters for Drinfeld modules, one constructed a finite free R-module [Formula: see text] with a semilinear operator on it, and hence a canonical z-isocrystal [Formula: see text] was attached to any Drinfeld module E that depended on the invertibility of a differential modular parameter [Formula: see text]. In this paper, we prove that [Formula: see text] is invertible for a Drinfeld module of rank 2. As a consequence, if E does not admit a lift of Frobenius and K is the fraction field of the ring of definition, we show that [Formula: see text] is isomorphic to [Formula: see text] and the isomorphism preserve the canonical Hodge filtration. On the other hand, if E admits a lift of Frobenius, then [Formula: see text] is isomorphic to the subobject [Formula: see text] of [Formula: see text]. The above result can be viewed as a character theoretic interpretation of de Rham cohomology.