Generalized fruit Diophantine equation and super elliptic curves

TL;DR

The paper investigates a general class of fruit Diophantine equations by linking arithmetic rigidity to the geometry of superelliptic curves. It proves nonexistence of integer solutions for a specific parameterized family via modular and parity arguments, then develops a Nagell–Lutz–style framework to control torsion on the Jacobians of the associated curves. By establishing discriminant-divisibility bounds that confine potential torsion to a finite set and applying reductions modulo primes of good reduction, it shows the corresponding Jacobians are torsion-free under the given hypotheses. The work culminates with explicit examples supporting the torsion-vanishing result and poses an open question on extending Nagell–Lutz analogues to all superelliptic curves, with potential implications for effectively determining rational points without rank constraints.

Abstract

In this article, we are interested in finding rational points on certain superelliptic curves.

Figures (2)

• Figure 1: Functoriality of the superelliptic covering map and its induced action on Jacobians. The vertical arrows denote the Abel--Jacobi embeddings, and the bottom arrow is the homomorphism induced by pushforward of divisor classes.
• Figure 2: Detection of rational torsion via reduction modulo primes of good reduction. A torsion element must survive reduction at every prime of good reduction. Coprimality of the group orders $\#\operatorname{Jac}(C_1^\ast)(\mathbb{F}_{p_1})$ and $\#\operatorname{Jac}(C_1^\ast)(\mathbb{F}_{p_2})$ forces the vanishing of rational torsion.

Theorems & Definitions (30)

• Theorem 2.1
• proof
• Corollary 2.2
• proof
• Corollary 2.3
• proof
• Remark 3.1
• Remark 3.2
• Theorem 3.3
• Remark 3.4