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We analyze general F-theory compactifications with U(1) x U(1) x U(1) Abeliangauge symmetry by constructing the general elliptically fibered Calabi-Yaumanifolds with a rank three Mordell-Weil group of rational sections. Thegeneral elliptic fiber is shown to be a complete intersection of twonon-generic quadrics in P^3 and resolved elliptic fibrations are obtained byembedding the fiber as the generic Calabi-Yau complete intersection into Bl_3P^3, the blow-up of P^3 at three points. For a fixed base B, there are finitelymany Calabi-Yau elliptic fibrations. Thus, F-theory compactifications on theseCalabi-Yau manifolds are shown to be labeled by integral points in reflexivepolytopes constructed from the nef-partition of Bl_3 P^3. We determine all 14massless matter representations to six and four dimensions by an explicit studyof the codimension two singularities of the elliptic fibration. We obtain threematter representations charged under all three U(1)-factors, most notably atri-fundamental representation. The existence of these representations, whichare not present in generic perturbative Type II compactifications, signifies anintriguing universal structure of codimension two singularities of the ellipticfibrations with higher rank Mordell-Weil groups. We also compute explicitly thecorresponding 14 multiplicities of massless hypermultiplets of asix-dimensional F-theory compactification for a general base B.

Elliptic Fibrations with Rank Three Mordell-Weil Group: F-theory with U(1) x U(1) x U(1) Gauge Symmetry