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

摘要

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 $ gge 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 $ gge 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 $ gle 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.).