Mordell-Weil Torsion in the Mirror of Multi-Sections

TL;DR

This paper explores the mirror duality between genus-one fibers with multi-sections and those with Mordell-Weil torsion, providing combinatorial criteria and confirming conjectures through explicit calculations in toric hypersurfaces.

Contribution

It introduces a combinatorial criterion to identify Mordell-Weil torsion from polytopes and confirms the conjecture across all 3134 complete intersection genus-one curves in toric spaces.

Findings

Confirmed the conjecture for all 3134 models through explicit calculation.

Identified 1024 inequivalent models after relabeling coefficients.

Discovered fibers with both multi-sections and Mordell-Weil torsion in the Jacobian.

Abstract

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.