  We present a systematic study of elliptic fibrations for F-theory
realizations of gauge theories with two U(1) factors. In particular, we
determine a new class of SU(5) x U(1)^2 fibrations, which can be used to
engineer Grand Unified Theories, with multiple, differently charged, 10 matter
representations. To determine these models we apply Tate's algorithm to
elliptic fibrations with two U(1) symmetries, which are realized in terms of a
cubic in P^2. In the process, we find fibers which are not characterized solely
in terms of vanishing orders, but with some additional specialization, which
plays a key role in the construction of these novel SU(5) models with multiple
10 matter. We also determine a table of Tate-like forms for Kodaira fibers with
two U(1)s.

High Energy Physics - Theory