Division by on odd-degree hyperelliptic curves and their Jacobians

Let K be an algebraically closed field of characteristic different from 2, g a positive integer, f(x) a polynomial of degree 2g + 1 with coefficients in K and without multiple roots, C : Y2 = f(x) the corresponding hyperelliptic curve of genus g over K, and J its Jacobian. We identify C with the image of its canonical embedding in J (the infinite point of C goes to the identity element of J). It is well known that for every b ∈ J(K) there are exactly 22g elements a ∈ J(K) such that 2a = b. Stoll constructed an algorithm that provides the Mumford representations of all such a in terms of the Mumford representation of b. The aim of this paper is to give explicit formulae for the Mumford representations of all such a in terms of the coordinates a, b, where b ∈ J(K) is given by a point P = (a, b) ∈ C(K) ⊂ J(K). We also prove that if g > 1, then C(K) does not contain torsion points of orders between 3 and 2g.