On de Rham cohomology of Drinfeld modules of rank 2

Abstract

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 δ ( 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 dR ( 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 dR ( E ) ⊗ K . The above result can be viewed as a character theoretic interpretation of de Rham cohomology.