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 H(E) with a semilinear operator on it, and hence a canonical z-isocrystal H<inf>δ</inf>(E) was attached to any Drinfeld module E that depended on the invertibility of a differential modular parameter γ. In this paper, we prove that γ 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 H(E)∅K is isomorphic to H<inf>dR</inf>(E) ∅ K and the isomorphism preserve the canonical Hodge filtration. On the other hand, if E admits a lift of Frobenius, then H(E)∅K is isomorphic to the subobject Lie(E)^∗ ∅ K of H<inf>dR</inf>(E) ∅ K. The above result can be viewed as a character theoretic interpretation of de Rham cohomology.