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

2025/10/31 by Salami, Sajad · 1 citation
#11G50 #14G05 #14H40 #1G30 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2510.27109

Abstract

We study the family of algebraic curves of genus ≥ 1 defined by the affine equations y^s=ax^r+b over a number field k, where r ≥ 2 and s≥ 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.

Citations

• Uniformity in Mordell-Lang for curves
• Bounded ranks and Diophantine error terms
• The rational points on certain Abelian varieties over function fields
• A heuristic for boundedness of ranks of elliptic curves
• Trivial points on towers of curves
• Lectures on Linear Series

Cited by

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

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