Non-simply-connected gauge groups and rational points on elliptic curves

We consider the F-theory description of non-simply-connected gauge groups appearing in the E8 × E8 heterotic string. The analysis is closely tied to the arithmetic of torsion points on an elliptic curve. The general form of the corresponding elliptic fibration is given for all finite subgroups of E8 which are applicable in this context. We also study the closely-related question of point-like instantons on a K3 surface whose holonomy is a finite group. As an example we consider the case of the heterotic string on a K3 surface having the E8 gauge symmetry broken to SU(9)/3 or (E6 × SU(3))/3 by point-like instantons with 3 holonomy.