On the rank of Jacobian varieties of the curves ys = axr + b
Abstract
We study the family of algebraic curves of genus $\geq 1$ defined by the affine equations $y^s=ax^r+b$ over a number field $k$, where $r \geq 2$ and $s\geq 2$ are fixed integers. Assuming the strong version of Lang's conjecture on varieties of general type, we prove that the Mordell-Weil rank of the Jacobian varieties of these curves is uniformly bounded. The proof proceeds by constructing a parameter space for curves in the family with a given number of rational points and analyzing the geometry of its fibers, which are shown to be complete intersection curves of increasing genus.
Fields of Study & Subjects
Algebraic Geometry and Number Theory
Advanced Differential Equations and Dynamical Systems
Polynomial and algebraic computation
Publication details
- Journal / source
- Journal of Number Theory
- Volume
- 290
- Pages
- 331–346
- Publication date
- 16 June 2026
- Language
- EN