Mordell-Weil torsion in the mirror of multi-sections

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.