2024/10/18 by Suluyer, Hamide, Kuru, Hamide, Mohammad Sadek +1
Mathematics · #Advanced Algebra and Geometry #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2410.14454
openalex publication_date 2024/10/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
It was conjectured by Flynn that there exists a constant κ such that, for any integer g ≥ 2, any m ≤ κg, there exists a hyperelliptic curve of genus g over \mathbb Q with a rational m-torsion point on its Jacobian. Leprévost proved this conjecture with κ=3. In this work we prove that given an integer N in the interval [3g,4g+1], g≥ 3, satisfying certain partition conditions, there exist parametric families of hyperelliptic Jacobian varieties with a rational torsion point of order N. In particular, we establish the existence of such varieties for N=4g+1 when g is odd and for N=4g-1 when g is even. A few explicit applications of this result produce the first known infinite examples of torsion 13 when g=3, torsion 15 when g=4, and torsion 17,18,21 when g=5. In fact, we show that infinitely many of the latter abelian varieties are absolutely simple.