  We construct the general form of an F-theory compactification with two U(1)
factors based on a general elliptically fibered Calabi-Yau manifold with
Mordell-Weil group of rank two. This construction produces broad classes of
models with diverse matter spectra, including many that are not realized in
earlier F-theory constructions with U(1)xU(1) gauge symmetry. Generic U(1)xU(1)
models can be related to a Higgsed non-Abelian model with gauge group
SU(2)xSU(2)xSU(3), SU(2)^3xSU(3), or a subgroup thereof. The nonlocal
horizontal divisors of the Mordell-Weil group are replaced with local vertical
divisors associated with the Cartan generators of non-Abelian gauge groups from
Kodaira singularities. We give a global resolution of codimension two
singularities of the Abelian model; we identify the full anomaly free matter
content, and match it to the unHiggsed non-Abelian model. The non-Abelian
Weierstrass model exhibits a new algebraic description of the singularities in
the fibration that results in the first explicit construction of matter in the
symmetric representation of SU(3). This matter is realized on double point
singularities of the discriminant locus. The construction suggests a
generalization to U(1)^k factors with k>2, which can be studied by Higgsing
theories with larger non-Abelian gauge groups.
