2024/11/05 by Boris M. Bekker, Bekker, Boris M., Yuri G. Zarhin +1
Mathematics · #11G10 #11G30 #14G27 #14H40 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.2411.03508
openalex publication_date 2024/11/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let d≥ 2 be a positive integer, K an algebraically closed field of characteristic not dividing d, n≥ d+1 a positive integer that is prime to d, f(x)∈ K[x] a degree n monic polynomial without multiple roots, Cf,d: yd=f(x) the corresponding smooth plane affine curve over K, Cf,d a smooth projective model of Cf,d and J(Cf,d) the Jacobian of Cf,d . We identify Cf,d with the image of its canonical embedding into J(Cf,d) (such that the infinite point of Cf,d goes to the zero of the group law on J(Cf,d)). Earlier the second named author proved that if d=2 and n=2g+1 ≥ 5 then the genus g hyperelliptic curve Cf,2 contains no points of orders lying between 3 and n-1=2g. In the present paper we generalize this result to the case of arbitrary d. Namely, we prove that if P is a point of order m>1 on Cf,d, then either m=d or m≥ n. We also describe all curves Cf,d having a point of order n.