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

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 prove the necessary congruence $30\mid h$. After symmetrization, the associated discriminant equation defines, for each fixed nonzero slice parameter $h$, a genus-one curve over $\mathbb{Q}(S)$; to study Mordell-Weil rank, one must pass to its Jacobian fibration $E_h/\mathbb{Q}(S)$. We show that $E_h$ carries a global rational $2$-torsion section and never has full rational $2$-torsion over $\mathbb{Q}(S)$. We also prove that no nonsingular rational specialization acquires additional rational $2$-torsion: by homogeneity, the relevant square condition reduces to rational points on a universal genus-two hyperelliptic curve, whose rational points are determined via a verified Magma computation using a rank-$0$ bound and the Chabauty-Coleman method. We further show that, after the normalization $x=S/h$, the Jacobian fibrations for all $h\neq 0$ become isomorphic over a rational function field. For the representative slice $h=30$, we compute the classical invariants of the associated binary quartic, obtain an explicit Weierstrass model, and apply the Gusić-Tadić injectivity criterion together with verified specialized-rank computations to prove the uniform bound $\mathrm{rank}\,E_h(\mathbb{Q}(S))\le 1$ for all $h\neq 0$. We then construct an explicit rational section on the universal Jacobian model and, specializing at $S=12$ on the slice $h=30$, show via injective specialization that this section has infinite order. Consequently, $\mathrm{rank}\,E_h(\mathbb{Q}(S))=1$ for every $h\in\mathbb{Q}^\times$. We conclude by recording the additional integrality, parity, and size conditions required to recover integer solutions from the genus-one/Jacobian framework.