Genus-One Fibrations and the Jacobian of Linear Slices in the Quintic Equal-Sum Problem

Paper

Abstract

We study the Diophantine equation

subject to the linear constraint

. We establish the necessary modular condition

and focus on the first admissible slice

. The problem reduces to finding rational sections on a genus-one fibration over

, which need not admit a section. We explicitly construct the associated Jacobian elliptic surface

. By exploiting a global rational

-torsion point and applying the Gusić-Tadić injectivity criterion for the specialization homomorphism, we prove the upper bound

. This result severely restricts the possible structure of infinite families of integer solutions within the slice.