Lacunary Wronskians on genus one curves

Abstract

Let X be a nonsingular projective curve of genus one defined over an algebraically closed field of characteristic 0. Let D be a divisor of X of degree n > 1 and let O be a (closed) point of X. As is well known, there exists a unique morphism φ D , O : X → X such that φ D , 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 φ D , 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.