Smooth curves having a large automorphism p-group in characteristic p>0

Let k be an algebraically closed field of characteristic p>0 and C a connected nonsingular projective curve over k with genus g>1. Let G be a p-subgroup of the k-automorphism group of C such that |G| > 2pg/(p-1). Then, C -->C/G is an étale cover of the affine line Spec k[X] totally ramified at infinity. To study such actions, we focus on the second ramification group G_2 of G at infinity, knowing that G_2 actually coincides with the derived group of G. We first display realizations of such actions with G_2 abelian of arbitrary large exponent . Our examples come from the construction of curves with many rational points using ray class field theory for global function fields. Then, considering additive polynomials of k[X], we obtain a structure theorem for the functions parametrizing the Artin-Schreier cover C --> C/G_2, in the case of a p-elementary abelian G_2. We finally emphasize the link between the curves obtained in this last case and supersingular curves (i.e. curves whose Jacobian is isogeneous to a product of supersingular elliptic curves).