  In F-theory compactifications, the abelian gauge sector is encoded in global
structures of the internal geometry. These structures lie at the intersection
of algebraic and arithmetic description of elliptic fibrations: While the
Mordell--Weil lattice is related to the continuous abelian sector, the
Tate--Shafarevich group is conjectured to encode discrete abelian symmetries in
F-theory. In these notes we review both subjects with a focus on recent
findings such as the global gauge group and gauge enhancements. We then
highlight the application to F-theory model building.
