Computing zeta functions of Artin–Schreier curves over finite fields

Abstract: The authors present a practical polynomial-time algorithm for computing the zeta function of certain Artin–Schreier curves over finite fields. This yields a method for computing the order of the Jacobian of an elliptic curve in characteristic 2, and more generally, any hyperelliptic curve in characteristic 2 whose affine equation is of a particular form. The algorithm is based upon an efficient reduction method for the Dwork cohomology of one-variable exponential sums.