Rank of Jacobian Varieties of Curves $y^s=x(ax^r+b)$

TL;DR

This paper proves that, assuming Lang's conjecture, the Mordell-Weil ranks of Jacobians of certain algebraic curves of the form y^s=x(ax^r+b) are uniformly bounded across the family, using geometric and number-theoretic methods.

Contribution

It establishes a uniform bound on ranks of Jacobians for a family of curves defined by specific equations, under the strong Lang conjecture, extending previous geometric approaches.

Findings

Ranks are uniformly bounded assuming Lang's conjecture.

Geometric analysis of parameter spaces and fibers informs the bound.

Results apply to genus one curves, specifically elliptic curves of the form y^2=x(x^2+B).

Abstract

Let $k$ be a number field. We investigate the Mordell-Weil ranks of Jacobian varieties $J_C$ associated with algebraic curves $C$ of genus $g \geq 1$ defined by affine equations of the form $y^s=x(ax^r+b)$, where $a, b \in k$ ($ab \neq 0$), and $r \geq 1, s \geq 2$ are fixed integers. Assuming the strong version of Lang's conjecture concerning rational points on varieties of general type, we establish that the ranks $r(J_C(k))$ are uniformly bounded as $C$ varies within this family. Our methodology builds upon the geometric approach employed by H. Yamagishi and subsequently adapted by the author for the family $y^s=ax^r+b$. We construct a parameter space $\mathcal{W}_n$ for curves possessing $n+1$ specified rational points and analyze its birational model $\mathcal{X}_n$, a complete intersection variety. The geometric properties of the fibers of $\Xc_n \to \text{Sym}^{n+1}(\mathbb{P}^1)$, specifically their genus and gonality, are studied. Combining these geometric insights with Faltings' theorem, uniformity conjectures stemming from Lang's work, and recent results connecting rank with the number of rational points, we deduce the main boundedness result. In the case of genus one curves $C$, it states that the rank of elliptic curves $y^2=x (x^2+B)$ is uniformly bounded subject to the strong version of Lang's conjecture.