On the Rank of Jacobian Varieties of the Curves $y^s=ax^r+b$

TL;DR

This paper investigates the ranks of Jacobian varieties of a family of algebraic curves defined by specific equations, proving under Lang's conjecture that these ranks are uniformly bounded across the family.

Contribution

It establishes a uniform bound on the Mordell-Weil ranks of Jacobians for a broad family of curves assuming Lang's conjecture, linking geometric properties to arithmetic rank bounds.

Findings

Ranks are uniformly bounded under Lang's conjecture.

Constructs a parameter space with fibers of increasing genus.

Analyzes the geometry of the parameter space to derive bounds.

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.