  In the present paper, we study a purely inseparable counterpart of
Abhyankar's conjecture for the affine line in positive characteristic, and
prove its validity for all the finite local non-abelian simple group schemes in
characteristic $p>5$. The crucial point is how to deal with finite local group
schemes which cannot be realized as the Frobenius kernel of a smooth algebraic
group. Such group schemes appear as the ones associated with Cartan type Lie
algebras. We settle the problem for such Lie algebras by making use of natural
gradations or filtrations on them.

Algebraic Geometry;Number Theory;Representation Theory