Halves of points of an odd degree hyperelliptic curve in its Jacobian

Let f(x)be a degree (2g + 1) monic polynomial with coefficients in an algebraically closed field K with char(K) ≠ 2 and without repeated roots. Let R ⊂ K be the (2g + 1 )-element set of roots of f(x). Let C: y2 = f(x) be an odd degree genus g hyperelliptic curve over K. Let J be the jacobian of C and J [2] ⊂ J(K) the (sub)group of points of order dividing 2. We identify C with the image of its canonical embedding into J (the infinite point of C goes to the identity element of J). Let P = (a, b) e C(K) ⊂ J(K) and which is J [2]-torsor. In a previous work we established an explicit bijection between the sets M1/2,P and The aim of this paper is to describe the induced action of J[2] on R1/2,P (i.e., how signs of square roots r(A) = √a - A should change).