Drinfeld modules, introduced by Drinfeld himself, can be thought of as analogues of the multiplicative group and elliptic curves in the positive characteristic function field. One of the important differential arithmetic object that one can attach to a Drinfeld module is the group of differential characters. The elements of this group encode important diophantine information and are arithmetic analogues of Manin maps in the case of elliptic curves in differential algebra. We will classify the structure of this group of differential characters which also shows the existence of a family of interesting differential modular functions on the moduli of Drinfeld modules. But strikingly, this also leads to a finite rank $F$-crystal that can be canonically attached to any Drinfeld module using our arithmetic jet space theory. This $F$-crystal has a ‘Hodge-type’ filtration as well and also maps to the original Hodge sequence preserving the filtration but is intrinsically different from the crystalline cohomology module of a Drinfeld module. All the results can also be repeated for the elliptic curve over $p$-adic fields. This is joint work with Jim Borger.