  Using Buium's theory of arithmetic differential characters, we construct a
filtered $F$-isocrystal ${\bf H}(A)_K$ associated to an abelian scheme $A$ over
a $p$-adically complete discrete valuation ring with perfect residue field. As
a filtered vector space, ${\bf H}(A)_K$ admits a natural map to the usual de
Rham cohomology of $A$, but the Frobenius operator comes from arithmetic
differential theory and is not the same as the usual crystalline one. When $A$
is an elliptic curve, we show that ${\bf H}(A)_K$ has a natural integral model
${\bf H}(A)$, which implies an integral refinement of a result of Buium's on
arithmetic differential characters. The weak admissibility of ${\bf H}(A)_K$
depends on the invertibility of an arithmetic-differential modular parameter.
Thus the Fontaine functor associates to suitably generic $A$ a local Galois
representation of an apparently new kind.
