  We analyze general F-theory compactifications with U(1) x U(1) x U(1) Abelian
gauge symmetry by constructing the general elliptically fibered Calabi-Yau
manifolds with a rank three Mordell-Weil group of rational sections. The
general elliptic fiber is shown to be a complete intersection of two
non-generic quadrics in P^3 and resolved elliptic fibrations are obtained by
embedding the fiber as the generic Calabi-Yau complete intersection into Bl_3
P^3, the blow-up of P^3 at three points. For a fixed base B, there are finitely
many Calabi-Yau elliptic fibrations. Thus, F-theory compactifications on these
Calabi-Yau manifolds are shown to be labeled by integral points in reflexive
polytopes constructed from the nef-partition of Bl_3 P^3. We determine all 14
massless matter representations to six and four dimensions by an explicit study
of the codimension two singularities of the elliptic fibration. We obtain three
matter representations charged under all three U(1)-factors, most notably a
tri-fundamental representation. The existence of these representations, which
are not present in generic perturbative Type II compactifications, signifies an
intriguing universal structure of codimension two singularities of the elliptic
fibrations with higher rank Mordell-Weil groups. We also compute explicitly the
corresponding 14 multiplicities of massless hypermultiplets of a
six-dimensional F-theory compactification for a general base B.
