An algorithm for constructing certain differential operators in positive characteristic

Abstract

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 δ which raises 1/ f to its pth power. For some specific families of polynomials, we also study the level of such a differential operator δ , i.e., the least integer e such that δ is R^{p^e} -linear. In particular, we obtain a characterization of supersingular elliptic curves in terms of the level of the associated differential operator.