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

模曲线和光谱光环的（半）稳定模型

• 批准号: 0401594
• 负责人: Robert Coleman
• 金额: --
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0401594&HistoricalAwards=false
• 关键词: semi; Stable; models; modular; curves

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.

对于科尔曼的DMS-0401594奖，我建议确定模数曲线的半稳定模型。现在人们知道，定义在有理数上的每一条椭圆曲线都是模曲线的商。如果奇点只在它的约化上，并且是普通的双点，则称它是半稳定的。人们知道，多亏了Mum-Ford，在基本扩展之后，每条曲线都有一个半稳定模型，但对于大多数(例如，优质立方体水平的曲线)来说，人们不知道一个模型。我计划弥补这一缺陷，我建议确定模数曲线的半稳定模型。模曲线在当代代数数论中占有重要地位。例如，它们在威尔斯证明费马大定理的过程中起到了至关重要的作用。现在人们知道，定义在有理数上的每一条椭圆曲线(又称。有理数是模曲线的商，它被用来为Birch-Swinnerton-Dyer关于这种曲线上有理点集的结构的猜想提供了大部分的理论证据。模曲线最初是定义在复数上的光滑曲线。下村证明了他们可以被定义在他的理性之上。定义有理数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，则称其为半稳定的。人们知道，多亏了大卫·芒福德，每条曲线都有一个半稳定模型，但对于大多数模数曲线，人们不知道一个模型。我计划弥补这一缺陷。