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

摘要奖DMS-0401594 Coleman我建议确定半稳定模型的模块化曲线。现在我们知道，每一条定义在有理数上的椭圆曲线都是模曲线的商。如果奇点只在它的约化上，并且是普通的二重点，则称它是半稳定的。 人们知道，由于 芒福德，每一条曲线都有一个半稳定的模型，在一个基础扩展之后，但对于大多数（例如，一个人不知道一个人。 我计划弥补这个缺陷，我建议确定半稳定模型的模块化曲线。模曲线在当代代数数论中起着重要的作用。 例如，它们在怀尔斯证明费马最后定理时至关重要。人们现在知道，定义在有理数上的每一条椭圆曲线（也称为椭圆曲线）都是由一个有理数定义的。有理数）（即，由只有有理系数的方程定义）是模曲线的商，这已经被用来提供关于这种曲线上的有理点集的结构的Birch-Swinnerton-Dyer猜想的大部分理论证据。模曲线最初是作为复数上定义的光滑曲线出现的。志村证明了它们可以在有理数上定义。由有理数定义的曲线模型本质上是一组具有整数系数的方程。 如果方程中包含的奇点是最简单的，则称它是半稳定的。人们知道，由于 大卫芒福德，每一条曲线都有一个半稳定模型，但对于大多数模曲线，人们不知道一个。 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.