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Tate Trees for Elliptic Fibrations with Rank one Mordell-Weil group

U(1) symmetries play a central role in constructing phenomenologically viable F-theory compactifications that realize Grand Unified Theories (GUTs). In F-theory, gauge symmetries with abelian gauge factors are modeled by singular elliptic fibrations with additional rational sections, i.e. a non-trivial Mordell-Weil rank. To determine the full scope of possible low energy theories with abelian gauge factors, which allow for an F-theory realization, it is central to obtain a comprehensive list of all singular elliptic fibrations with extra sections. We answer this question for the case of one abelian factor by applying Tate's algorithm to the elliptic fiber realized as a quartic in the weighted projective space P^{(1,1,2)}, which guarantees, in addition to the zero section, the existence of an additional rational section. The algorithm gives rise to a tree-like enhancement structure, where each fiber is characterized by a Kodaira fiber type, that governs the non-abelian gauge factor, and the separation of the two sections. We determine Tate-like forms for elliptic fibrations with one extra section for all Kodaira fiber types. In addition to standard Tate forms that are determined by the vanishing order of the coefficient sections in the quartic (so-called canonical models),the algorithm also gives rise to fibrations that require non-trivial relations among the coefficient sections. Such non-canonical models have phenomenologically interesting properties, as they allow for a richer charged matter content, and thus codimension two fiber structure, than the canonical models that have been considered thus far in the literature. As an application we determine the complete set of codimension one fibers types, matter spectra, both canonical and non-canonical, for SU(5) x U(1) models.