
Version 4 Summary: This preprint presents a revised and self-contained resolution of a question concerning the existence of elliptic curves over the rational function field $\mathbb{Q}(t)$ with torsion subgroup $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/8\mathbb{Z}$ and positive Mordell–Weil rank. Earlier work by the author investigated this problem using a specific parametrized family of elliptic curves, motivated by geometric and spectral considerations. It was subsequently brought to the author’s attention that this family did not, in fact, furnish a genuine rational parametrization of the modular curve $X_1(2,8)$, and that some conclusions drawn from it required reassessment. In particular, certain examples previously highlighted were already known over $\mathbb{Q}$ and did not address the function-field problem under consideration. The present work corrects this oversight by reframing the problem at the level of the universal elliptic surface associated with $X_1(2,8)$ and analyzing it using arithmetic surface theory. The main result establishes that **no elliptic curve over $\mathbb{Q}(t)$ with torsion subgroup $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/8\mathbb{Z}$ can have positive Mordell–Weil rank**. The obstruction is shown to be intrinsic and arithmetic in nature, arising from persistent non-split multiplicative fibers with quadratic splitting fields whose Galois action forces all rational sections to be torsion. This obstruction is proved to persist under all rational base changes, including ramified base change. The argument combines:- explicit analysis of the singular fibers of the universal surface,- descent constraints from the Galois action on component groups,- a Shioda–Tate computation confirming rank zero,- and supporting computational verification using the Magma algebra system. While this revision supersedes the earlier parametrized approach, it does not invalidate all observations made there. In particular, certain geometric and spectral phenomena previously reported—such as structured projection behavior and covariance patterns—appear to be independent of the specific parametrization and may reflect more general features of elliptic surfaces or modular families. These aspects are not claimed as results here, but are noted as potential directions for future investigation once placed in a corrected arithmetic framework. This preprint is released on Zenodo to provide a transparent and citable record of the corrected argument, the definitive negative resolution of the rank-one question for torsion structure $(2,8)$ over $\mathbb{Q}(t)$, and the lessons learned in the process. DOI version note: This Version 4 supersedes previous preprints following a structural reassessment of the underlying modular family. The present version provides a complete and rigorous proof that elliptic curves over $\mathbb{Q}(t)$ with torsion subgroup $\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/8\mathbb{Z}$ admit no positive Mordell–Weil rank.
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