  We present a systematic construction of F-theory compactifications with
Abelian gauge symmetries in addition to a non-Abelian gauge group G. The
formalism is generally applicable to models in global Tate form but we focus on
the phenomenologically interesting case of G=SU(5). The Abelian gauge factors
arise due to extra global sections resulting from a specific factorisation of
the Tate polynomial which describes the elliptic fibration. These
constructions, which accommodate up to four different U(1) factors, are worked
out in detail for the two possible embeddings of a single U(1) factor into E8,
usually denoted SU(5) x U(1)_X and SU(5) x U(1)_PQ. The resolved models can be
understood either patchwise via a small resolution or in terms of a
P_{1,1,2}[4] description of the elliptic fibration. We derive the U(1) charges
of the fields from the geometry, construct the U(1) gauge fluxes and exemplify
the structure of the Yukawa interaction points. A particularly interesting
result is that the global SU(5) x U(1)_PQ model exhibits extra SU(5)-singlet
states which are incompatible with a single global decomposition of the 248 of
E8. The states in turn lead to new Yukawa type couplings which have not been
considered in local model building.
