Torsion Points of order 2g+1 on odd degree hyperelliptic curves of genus g (original) (raw)

Let K be an algebraically closed field of characteristic different from 2, g a positive integer, f (x) ∈ K[x] a degree 2g + 1 monic polynomial without multiple roots, C f : y 2 = f (x) the corresponding genus g hyperelliptic curve over K, and J the Jacobian of C f. We identify C f with the image of its canonical embedding into J (the infinite point of C f goes to the zero of the group law on J). It is known [9] that if g ≥ 2, then C f (K) contains no points of orders lying between 3 and 2g. In this paper we study torsion points of order 2g + 1 on C f (K). Despite the striking difference between the cases of g = 1 and g ≥ 2, some of our results may be viewed as a generalization of well-known results about points of order 3 on elliptic curves. E.g., if p = 2g + 1 is a prime that coincides with char(K), then every odd degree genus g hyperelliptic curve contains at most two points of order p. If g is odd and f (x) has real coefficients, then there are at most two real points of order 2g + 1 on C f. If f (x) has rational coefficients and g ≤ 51, then there are at most two rational points of order 2g + 1 on C f. (However, there exist odd degree genus 52 hyperelliptic curves over Q that have at least four rational points of order 105.