2025/10/19 by Boris M. Bekker, Bekker, Boris, 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 #Number Theory (math.NT) #Tensor decomposition and applications
paper · pdf · doi:10.48550/arxiv.2510.16912
openalex publication_date 2025/10/19 · openalex created_date 2025/10/22 · openalex updated_date 2026/07/28
Let d≥ 2 be an integer, K0 a perfect field such that char(K0) does not divide d, n > d an integer prime to d, f(x)∈ K0[x] a degree n monic polynomial without repeated roots, and Cf,d a smooth projective model of the affine curve yd=f(x). Let J(Cf,d) be the Jacobian of the K0-curve Cf,d . We identify Cf,d with its canonical image in J(Cf,d) (such that the infinite point of Cf,d goes to the zero of the group law on J(Cf,d)). We say that an integer m>1 is (n,d)-reachable over K0 if there exists a polynomial f(x) as above such that Cf,d(K0) contains a torsion point of order m. Earlier we proved that if m is (n,d)-reachable, then either m=d or m ≥ n (in addition, both d and n are (n,d)-reachable). In the present paper we prove the following. If nn, then d⋅ [(n+d)/d] is (n,d)-reachable if and only if n-(d-1)⋅ [(n+d)/d]≥ 0. If char(K0)=0, then n+d is (n,d)-reachable if and only if d2-2d