Division by on odd-degree hyperelliptic curves and their Jacobians

Abstract

Let be an algebraically closed field of characteristic different from , a positive integer, a polynomial of degree with coefficients in and without multiple roots, the corresponding hyperelliptic curve of genus over , and its Jacobian. We identify with the image of its canonical embedding in (the infinite point of goes to the identity element of ). It is well known that for every there are exactly elements such that . Stoll constructed an algorithm that provides the Mumford representations of all such in terms of the Mumford representation of . The aim of this paper is to give explicit formulae for the Mumford representations of all such in terms of the coordinates , where is given by a point . We also prove that if , then does not contain torsion points of orders between and .