Abstract: By a previous construction by J. Borger and the speaker, one attaches a canonical filtered isocrystal H(G) associated to the arithmetic jet spaces of a smooth commutative group scheme G. In this talk, using the characteristic polynomial of the Frobenius, we will show that whenever G is a semi-abelian scheme,  H(G) is weakly admissible. We will also comment on the relation of H(A) with the first crystalline cohomology when A is an abelian scheme. This is a joint work with L. Gurney and S. Pandit.