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

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.