(semi-)Stable models of modular curves and the Spectral Halo

摘要：针对colemi的DMS-0401594奖，我提出确定模曲线的半稳定模型。现在我们知道，每一条在有理数上定义的椭圆曲线都是模曲线的商。如果奇点仅在它的约化上并且是普通的双点，则称它是半稳定的。我们知道，多亏了Mum-ford，每条曲线都有一个半稳定的模型，在基础扩展之后，但对于大多数(例如。（质数立方级）1不知道1。我打算弥补这个缺陷。我建议确定模曲线的半稳定模型。模曲线在当代代数数论中占有重要地位。例如，它们在怀尔斯证明费马大定理的过程中起到了至关重要的作用。人们现在知道，每一条由有理数（即有理数）定义的椭圆曲线（即由只有有理数系数的方程定义的曲线）都是模曲线的商，这已经被用来为Birch-Swinnerton-Dyer猜想提供关于这种曲线上有理数点集结构的大部分理论证据。模曲线最初是定义在复数上的光滑曲线。志村证明了它们可以用理性来定义。由有理数定义的曲线模型本质上是一组系数为整数的方程。如果包含在方程中的奇异点是最简单的一类，就说它是半稳定的。多亏了David Mumford，我们知道，每条曲线都有一个半稳定模型，但对于大多数模曲线，我们不知道。我打算弥补这个缺陷。

Abstract for award DMS-0401594 of ColemanI propose to determine semi-stable models of the modular curves. One now knows that every elliptic curve defined over the rationals is a quotient of a modular curve. It is said to be semi-stable if the singularities are only on its reductions and are ordinary double points. One knows, thanks to Mum-ford, that every curve has a semi-stable model, after a base extension, but for most (eg., those of prime cubed level) one doesn't know one. I plan to remedy this defect.I propose to determine semi-stable models of the modular curves. Modular curves play a prominent role in contemporary algebraic number theory. For example, they were crucial in Wiles' proof of Fermat's last theorem. One now knows that every elliptic curve defined over the rational numbers (a.k.a. the rationals) (i.e., defined by equations with only rational coefficients) is a quotient of a modular curve and this has been used to provide most of the theoretical evidence for the Birch-Swinnerton-Dyer conjecture on the structure of the set of rational points on such a curve. Modular curves first arose as smooth curves defined over the complex numbers. Shimura proved they could be defined over he rationals. A model for a curve defined the rationals is essentially a set of equations with integer coefficients. It is said to be semi-stable if its singularities incorporated in the equations are of the simplest sort. One knows, thanks to David Mumford, that every curve has a semi-stable model but for most modular curves one doesn't know one. I plan to remedy this defect.