Torsion points of order $\boldsymbol {2g+1}$ on odd degree hyperelliptic curves of genus $\boldsymbol {g}$

Abstract:

Let $K$ be an algebraically closed field of characteristic different from $2$, let $g$ be a positive integer, let $f(x)\in K[x]$ be a degree $2g+1$ monic polynomial without multiple roots, let $\mathcal {C}_f: y^2=f(x)$ be the corresponding genus $g$ hyperelliptic curve over $K$, and let $J$ be the Jacobian of $\mathcal {C}_f$. We identify $\mathcal {C}_f$ with the image of its canonical embedding into $J$ (the infinite point of $\mathcal {C}_f$ goes to the zero of the group law on $J$). It is known [Izv. Math. 83 (2019), pp. 501–520] that if $g\ge 2$, then $\mathcal {C}_f(K)$ contains no points of orders lying between $3$ and $2g$.

In this paper we study torsion points of order $2g+1$ on $\mathcal {C}_f(K)$. Despite the striking difference between the cases of $g=1$ and $g\ge 2$, 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 $\operatorname {char}(K)$, 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 $\mathcal {C}_f$. If $f(x)$ has rational coefficients and $g\le 51$, then there are at most two rational points of order $2g+1$ on $\mathcal {C}_f$. (However, there exist odd degree genus $52$ hyperelliptic curves over $\mathbb {Q}$ that have at least four rational points of order 105.)