AN ALGORITHM FOR CONSTRUCTING CERTAIN DIFFERENTIAL OPERATORS IN POSITIVE CHARACTERISTIC

Given a non-zero polynomial f in a polynomial ring R with coefficients in a finite field of prime characteristic p, we present an algorithm to compute a differential operator delta which raises 1/f to its pth power. For some specific families of polynomials, we also study the level of such a differential operator delta, i.e., the least integer e such that delta is R-pe -linear. In particular, we obtain a characterization of supersingular elliptic curves in terms of the level of the associated differential operator.