  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 fibre both as a Bl^2P^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 utilise 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^1P^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 fibre Bl^1P_[1,1,2][4], corresponding to another class of
SU(5) x U(1) models.
