椭圆拟阵与模曲线
Elliptic matroids and modular curves
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中文总结 AI 辅助
针对n≥4的秩3拟阵Tₙ，研究其与模曲线X₁(n)°的对应关系，通过代数与关联理论证明特征不整除n的域k上存在自然双射，将域值对应升级为概形同构，还得到相关推论并在附录讨论拟阵实现空间的升级。
中文摘要 AI 辅助
对于n≥4，设Tₙ是定义在ℤ/nℤ上的秩3拟阵，其基为三元非零和子集。设X₁(n)°表示模曲线X₁(n)的开子概形，通过移除对应可约内隆多边形的尖点得到。对于n≥10，我们给出纯代数与关联理论的证明：对每个特征不整除n的域k，X₁(n)°(k)与Tₙ的k实现的缩放类之间存在自然双射。当k=ℂ时，这重现了Borisov和Roulleau的定理。我们随后将域值对应升级为ℤ[1/n]上的概形同构。主要新要素是形变理论论证，它使我们能验证在阿廷局部环上取值的点的同构。作为推论，模曲线X₁(n)°作为ℤ[1/n]上的拟阵实现空间获得自然模型；对于素数p≥11，Tₙ在ℚ上不可表示等价于Mazur关于椭圆曲线有理挠点的著名定理的素数阶情形。在附录中，我们解释如何将拟阵的实现空间从ℤ上的仿射概形升级为𝔽₁^±上的仿射带形概形（按Baker-Jin-Lorscheid的定义）。
英文摘要
For $n\geq 4$, 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 $n \geq 10$, 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 $p \geq 11$, 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 $\mathbb{F}_1^{\pm}$.
