Elliptic matroids and modular curves

Abstract

For $n\ge 4$, let $T_n$ be the rank-3 matroid on $\mathbb{Z}/n\mathbb{Z}$ whose bases are the three-element non-zero-sum subsets. Let $X_1(n{)}^{\circ }$ denote the open subscheme of the modular curve $X_1(n)$ obtained by removing the cusps corresponding to reducible Néron polygons. For $n\ge 10$, we give a purely algebraic and incidence-theoretic proof that, for every field $k$ with $\mathrm{char}(k)$ not dividing $n$, there is a natural bijection between $X_1(n{)}^{\circ }(k)$ and rescaling classes of $k$-realizations of $T_n$. For $k=\mathbb{C}$, this recovers a theorem of Borisov and Roulleau. We then upgrade the field-valued correspondence to an isomorphism of schemes over $\mathbb{Z}[1/n]$. The main new ingredient is a deformation-theoretic argument which allows us to verify the isomorphism on points valued in Artinian local rings. As consequences, the modular curve $X_1(n{)}^{\circ }$ acquires a natural model over $\mathbb{Z}[1/n]$ as a matroid realization space, and, for primes $p\ge 11$, the non-representability of $T_p$ over $\mathbb{Q}$ is equivalent to the prime-order case of Mazur's celebrated theorem on rational torsion points of elliptic curves. In an appendix, we explain how to upgrade the realization space of a matroid from an affine scheme over $\mathbb{Z}$ to an affine band scheme (in the sense of Baker-Jin-Lorscheid) over ${\mathbb{F}}_1^{\pm }$.

41 pages