Rational Torsion on Hyperelliptic Jacobian Varieties
Abstract
ABSTRACT It was conjectured by Flynn that there exists a constant such that, for any integer , any , there exists a hyperelliptic curve of genus over with a rational ‐torsion point on its Jacobian. Leprévost proved this conjecture with . In this work, we prove that given an integer in the interval , , satisfying certain partition conditions, there exist parametric families of hyperelliptic Jacobian varieties with a rational torsion point of order . In particular, we establish the existence of such varieties for when is odd and for when is even. A few explicit applications of this result produce the first known infinite examples of torsion 13 when , torsion 15 when , and torsion 17, 18, 21 when . In fact, we show that infinitely many of the latter abelian varieties are absolutely simple.
Fields of Study & Subjects
Advanced Differential Equations and Dynamical Systems
Algebraic Geometry and Number Theory
Commutative Algebra and Its Applications
Publication details
- Journal / source
- Mathematische Nachrichten
- Publisher
- Wiley
- Volume
- 299
- Issue
- 5
- Pages
- 1241–1248
- Publication date
- 1 April 2026
- Language
- EN