Rank of Jacobian Varieties of Curves $y^s=x(ax^r+b)$
Sajad Salami
TL;DR
The paper conditional on the strong Lang conjecture establishes uniform bounds for the Mordell-Weil ranks of Jacobians J_C of curves C in the two-parameter family y^s = x(ax^r + b) over a number field k containing a primitive s-th root of unity. Building a geometric framework with parameter spaces W_n and a birational model X_n, it studies the fibers X_{a_n} to bound rational points via genus and gonality calculations, applying Faltings’ theorem and Lang-type uniformity. A key step is linking point counts to rank through the Dimitrov–Gao–Habegger result, yielding the main uniformity result and its genus-one corollaries, including conditional bounds for congruent-number and related elliptic curves. The work also provides concrete high-genus fiber examples (genus 20481 and 49) corresponding to known high-rank elliptic curves, illustrating the deep interplay between geometric constructions and arithmetic of rational points.
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.