The arithmetic of elliptic fibrations in gauge theories on a circle

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.