Torsion points of order 2𝑔+1 on odd degree hyperelliptic curves of genus 𝑔

Abstract

Let K K be an algebraically closed field of characteristic different from 2 2 , let g g be a positive integer, let f ( x ) ∈ K [ x ] f(x)\in K[x] be a degree 2 g + 1 2g+1 monic polynomial without multiple roots, let C f : y 2 = f ( x ) \mathcal {C}_f: y^2=f(x) be the corresponding genus g g hyperelliptic curve over K K , and let J J be the Jacobian of C f \mathcal {C}_f . We identify C f \mathcal {C}_f with the image of its canonical embedding into J J (the infinite point of C f \mathcal {C}_f goes to the zero of the group law on J J ). It is known [Izv. Math. 83 (2019), pp. 501–520] that if g ≥ 2 g\ge 2 , then C f ( K ) \mathcal {C}_f(K) contains no points of orders lying between 3 3 and 2 g 2g . In this paper we study torsion points of order 2 g + 1 2g+1 on C f ( K ) \mathcal {C}_f(K) . Despite the striking difference between the cases of g = 1 g=1 and g ≥ 2 g\ge 2 , some of our results may be viewed as a generalization of well-known results about points of order 3 3 on elliptic curves. E.g., if p = 2 g + 1 p=2g+1 is a prime that coincides with char ⁡ ( K ) \operatorname {char}(K) , then every odd degree genus g g hyperelliptic curve contains at most two points of order p p . If g g is odd and f ( x ) f(x) has real coefficients, then there are at most two real points of order 2 g + 1 2g+1 on C f \mathcal {C}_f . If f ( x ) f(x) has rational coefficients and g ≤ 51 g\le 51 , then there are at most two rational points of order 2 g + 1 2g+1 on C f \mathcal {C}_f . (However, there exist odd degree genus 52 52 hyperelliptic curves over Q \mathbb {Q} that have at least four rational points of order 105.)