  We study the global structure of the gauge group $G$ of F-theory compactified
on an elliptic fibration $Y$. The global properties of $G$ are encoded in the
torsion subgroup of the Mordell-Weil group of rational sections of $Y$.
Generalising the Shioda map to torsional sections we construct a specific
integer divisor class on $Y$ as a fractional linear combination of the
resolution divisors associated with the Cartan subalgebra of $G$. This divisor
class can be interpreted as an element of the refined coweight lattice of the
gauge group. As a result, the spectrum of admissible matter representations is
strongly constrained and the gauge group is non-simply connected. We exemplify
our results by a detailed analysis of the general elliptic fibration with
Mordell-Weil group $\mathbb Z_2$ and $\mathbb Z_3$ as well as a further
specialization to $\mathbb Z \oplus \mathbb Z_2$. Our analysis exploits the
representation of these fibrations as hypersurfaces in toric geometry.
