Elliptic fibrations for SU(5) x U(1) x U(1) F-theory vacua

Elliptic Calabi-Yau fibrations with Mordell-Weil group of rank two are constructed. Such geometries are the basis for F-theory compactifications with two Abelian gauge groups in addition to non-Abelian gauge symmetry. We present the elliptic fiber both as a Bl(2)P(2)[3] fibration and in the birationally equivalent Weierstrass form. The spectrum of charged singlets and their Yukawa interactions are worked out in generality. This framework can be combined with the toric construction of tops to implement additional non-Abelian gauge groups. We utilize the classification of tops to construct SU(5) x U(1) x U(1) gauge symmetries systematically and study the resulting geometries, presenting the defining equations, the matter curves and their charges, the Yukawa couplings and explaining the process in detail for an example. Brane recombination relates these geometries to a Bl(2)P(2)[3] fibration with a corresponding class of SU(5) x U(1) models. We also present the SU(5) tops based on the elliptic fiber Bl(1)P([1,1,2])[4], corresponding to another class of SU(5) x U(1) models.