Some remarks on the Picard curves over a finite field

In this paper we study the Newton polygon of the L-polynomial L(t) associate to the Picard curves y(3) = x(4)-1, y(3) = x(4) - x defined over a finite field F-P. In the former case we get a complete classification. In the latter case we obtained a partial result. As a consequence of our result we obtain a criterion to find a supersingular Picard curves for the above two cases. Our main results are stated in Theorems 3.1 and 4.1. Curves over finite fields. (c) 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.