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 ƒ(x) \in K[x] be a degree 2g+1 monic polynomial without multiple roots, let Cƒ: y2 = ƒ(x) be the corresponding genus g hyperelliptic curve over K, and let J be the Jacobian of Cƒ. We identify Cƒ with the image of its canonical embedding into J (the infinite point of Cƒ 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 Cƒ(K) contains no points of orders lying between 3 and 2g. In this paper we study torsion points of order 2g + 1 on Cƒ(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 ƒ(x) has real coefficients, then there are at most two real points of order 2g + 1 on Cƒ. If ƒ(x) has rational coefficients and g ≤ 51, then there are at most two rational points of order 2g+1 on Cƒ. (However, there exist odd degree genus 52 hyperelliptic curves over Q that have at least four rational points of order 105.) [ABSTRACT FROM AUTHOR]