Lacunary Wronskians on Genus One Curves

Let X be a nonsingular projective curve of genus one defined over an algebraically closed field of characteristic 0. Let D be a divisor ofX of degree n > 1 and let O be a (closed) point ofX. As is well known, there exists a unique morphism φD,O : X → X such that φD,O(P ) = Q if and only if the divisor nP −D−O+Q is principal. Our main result is a simple explicit description of the map φD,O in terms of Wronskians and certain Wronskian-like determinants lacunary in the sense that derivatives of some orders are skipped. Further, for n = 2, 3 we interpret our main result as a syzygy from classical invariant theory, thus reconciling our work with a circle of ideas treated in two papers by Weil and a recent paper by An, Kim, Marshall, Marshall, McCallum and Perlis.