L-functions and equations

L 函数和方程

• 批准号: 0500202
• 负责人: Andrew Wiles
• 金额: --
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-05-01 至 2010-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0500202&HistoricalAwards=false
• 关键词: functions; equations

The first main theme of the proposal is the study of points on curves of genus 1, more particularly points which are defined over solvable extensions of the ground field. This is joint work with Ciperiani and aims to prove that for genus 1 curves defined over a totally real number field there are always points defined over a solvable extension of the base field. In the first instance we are aiming to prove the existence of such points while assuming that there exist points over all p-adic fields, but this condition should be easy to remove. We note that we do not wish to assume anything about the rank of the Jacobian of the curve. The second theme of the proposal is to continue a study of the problem of non-solvable base change. For the moment the intent is to restrict attention to the case of holomorphic forms on GL(2). There are two main problems being investigated in this project. The first problem is to try to extend the ideas used in the nineteenth century to understand equations. For equations of one variable it was shown by Abel and Galois that those of degree five or more do not have easy general formulas like the ones for quadratic equations. The reason for this was that the solutions to equations of degree four or less always live in what are called solvable extensions i.e. those extensions obtained by successively extracting square roots, cube roots etc. Surprisingly it is not known whether the same is true for equations with more than one variable. The first aim of this project is to show that many of the simpler kinds of equations in two variables do in fact have these simple kinds of solutions. Many constructions in number theory at the moment require the assumption that the equations involved do have such simple solutions. The second part of the project is an attempt to extend a particular and very important one of these constructions to the situationwhere the equations involved do not have any simple solutions.

该提案的第一个主题是研究亏格 1 曲线上的点，更具体地说是在地面场的可解扩展上定义的点。这是与 Ciperiani 的联合工作，旨在证明对于在全实数域上定义的 genus 1 曲线，总有在基域的可解扩展上定义的点。在第一个例子中，我们的目标是证明这些点的存在，同时假设所有 p 进场上都存在点，但这种条件应该很容易消除。我们注意到，我们不想对曲线的雅可比行列式的秩做出任何假设。该提案的第二个主题是继续研究不可解的基数变化问题。目前的目的是将注意力限制在 GL(2) 上的全纯形式的情况上。 该项目主要研究两个问题。第一个问题是尝试扩展十九世纪用于理解方程的思想。对于一个变量的方程，阿贝尔和伽罗瓦表明，五次或以上的方程没有像二次方程那样简单的通用公式。其原因是四次或更少的方程的解总是存在于所谓的可解扩展中，即通过连续提取平方根、立方根等而获得的扩展。令人惊讶的是，不知道对于具有多个变量的方程是否也是如此。该项目的第一个目标是证明许多更简单的二变量方程实际上具有这些简单的解。目前数论中的许多构造都需要假设所涉及的方程确实具有如此简单的解。该项目的第二部分是尝试将这些结构中的一个特定且非常重要的结构扩展到所涉及的方程没有任何简单解的情况。