Genus-One Fibrations and the Jacobian of Linear Slices in the Quintic Equal-Sum Problem
Authors
Valery Asiryan
Abstract
We study the Diophantine equation $a^5+b^5=c^5+d^5$ under the linear slicing constraint $(c+d)-(a+b)=h$. We first establish the necessary modular constraint $30 \mid h$. For any nonzero slice $h$, the problem reduces to finding rational points on a genus-one fibration over $\mathbb{Q}(S)$. Passing to the Jacobian fibration $E_h/\mathbb{Q}(S)$, we identify a global rational $2$-torsion section and prove that $E_h$ never admits full rational $2$-torsion. This is achieved by reducing the splitting condition of the $2$-division field to finding rational points on a universal genus-two hyperelliptic curve, for which we rigorously verify the set of rational points using the method of Chabauty and Coleman. We further show that the Jacobian fibrations for all $h \neq 0$ are isomorphic over a rational function field. Focusing on the representative slice $h=30$, we compute the explicit invariants of the elliptic surface and apply the Gusić--Tadić injectivity criterion for the specialization homomorphism. Based on verified computations of specialized ranks, we establish the uniform upper bound $\mathrm{rank}\,E_h(\mathbb{Q}(S)) \le 1$ for all $h \neq 0$. Finally, we discuss the additional inequality and parity constraints required to recover integer solutions from the fibration.
