TORSION POINTS OF ORDER 2g+1 ON ODD DEGREE HYPERELLIPTIC CURVES OF GENUS g

Let K be an algebraically closed field of characteristic different from 2, let g be a positive integer, let f(x) is an element of K[x] be a degree 2g + 1 monic polynomial without multiple roots, let C-f : y(2) = f(x) be the corresponding genus g hyperelliptic curve over K, and let J be 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 [Izv. Math. 83 (2019), pp. 501-520] 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.)