  The geometry of elliptic fibrations translates to the physics of gauge
theories in F-theory. We systematically develop the dictionary between
arithmetic structures on elliptic curves as well as desingularized elliptic
fibrations and symmetries of gauge theories on a circle. We show that the
Mordell-Weil group law matches integral large gauge transformations around the
circle in Abelian gauge theories and explain the significance of Mordell-Weil
torsion in this context. We also use Higgs transitions and circle large gauge
transformations to introduce a group law for genus-one fibrations with
multi-sections. Finally, we introduce a novel arithmetic structure on elliptic
fibrations with non-Abelian gauge groups in F-theory. It is defined on the set
of exceptional divisors resolving the singularities and divisor classes of
sections of the fibration. This group structure can be matched with certain
integral non-Abelian large gauge transformations around the circle when
studying the theory on the lower-dimensional Coulomb branch. Its existence is
required by consistency with Higgs transitions from the non-Abelian theory to
its Abelian phases in which it becomes the Mordell-Weil group. This hints
towards the existence of a new underlying geometric symmetry.
