Regular polygons, line operators, and elliptic modular surfaces as realization spaces of matroids

Abstract

For an integer $n\geq 7$, we investigate the matroid realization space of a specific deformation of the regular $n$-gon along with its lines of symmetry. It turns out that this particular realization space is birational to the elliptic modular surface $\Xi_{1}(n)$ over the modular curve $X_{1}(n)$. In this way, we obtain a model of $\Xi_{1}(n)$ defined over the rational numbers. Furthermore, a natural geometric operator acts on these matroid realizations. On the elliptic modular surface, this operator corresponds to the multiplication by $-2$ on the elliptic curves. This provides a new geometric approach to computing multiplication by $-2$ on elliptic curves.