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

2023/12/06 by Kühne, Lukas, Roulleau, Xavier
#14G35 #14J25 #14J27 #14N20 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2312.03470

Abstract

For an integer n≥ 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 Ξ₁(n) over the modular curve X₁(n). In this way, we obtain a model of Ξ₁(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.

Related

• Dynamical systems on some elliptic modular surfaces via operators on line arrangements
• Modular curves X₁(n) as moduli spaces of point arrangements and applications
• Explicit moduli spaces for congruences of elliptic curves
• Some explicit elliptic modular surfaces
• Rings of Hilbert modular forms, computations on Hilbert modular surfaces, and the Oda-Hamahata conjecture