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

TL;DR

The paper investigates the uniform boundedness of the Mordell–Weil rank of Jacobians for curves defined by $C: y^s = a x^r + b$ with fixed $r,s \ge 2$, conditional on the strong Lang conjecture. It develops a parameter-space framework that produces a family of complete-intersection fibers with controlled genus and gonality, and connects rational points on these fibers to the rank via a Dimitrov–Gao–Habegger-type bound. The key contributions are (i) generalizing Yamagishi's geometric strategy to a two-parameter family, (ii) establishing finiteness and, under Lang-type hypotheses, uniform bounds on rational points of the fiber curves, and (iii) a concrete illustration using Elkies’ rank-$17$ elliptic curve, where the complexity of a high-genus fiber encodes information about many rational points. Together, these results provide conditional evidence for rank boundedness in families of Jacobians through a geometric mechanism that converts point abundance into genus and gonality constraints.

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.

Theorems & Definitions (33)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 2.1: Faltings, Faltings1986
• Conjecture 2.2: Lang, Lang1986Lang1991
• Theorem 2.3: Uniformity I, Caporaso1997
• Theorem 2.4: Uniformity II, Caporaso1997
• Theorem 2.5: Correlation Theorem, Caporaso1997
• Theorem 2.6: Dimitrov2021
• Theorem 2.7: Lazarsfeld, Lazarsfeld1997