  We give further evidence that genus-one fibers with multi-sections are mirror
dual to fibers with Mordell-Weil torsion. In the physics of F-theory
compactifications this implies a relation between models with a non-simply
connected gauge group and those with discrete symmetries. We provide a
combinatorial explanation of this phenomenon for toric hypersurfaces. In
particular this leads to a criterion to deduce Mordell-Weil torsion directly
from the polytope. For all 3134 complete intersection genus-one curves in
three-dimensional toric ambient spaces we confirm the conjecture by explicit
calculation. We comment on several new features of these models: The
Weierstrass forms of many models can be identified by relabeling the
coefficient sections. This reduces the number of models to 1024 inequivalent
ones. We give an example of a fiber which contains only non-toric sections one
of which becomes toric when the fiber is realized in a different ambient space.
Similarly a singularity in codimension one can have a toric resolution in one
representation while it is non-toric in another. Finally we give a list of 24
inequivalent genus-one fibers that simultaneously exhibit multi-sections and
Mordell-Weil torsion in the Jacobian. We discuss a self-mirror example from
this list in detail.
