Equations and Automorphic Forms

方程和自守形式

• 批准号: 0758379
• 负责人: Christopher Skinner
• 金额: $ 43.12万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-05-01 至 2014-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0758379&HistoricalAwards=false
• 关键词: Equations; Automorphic; Forms

ABSTRACTPrincipal Investigator: Wiles, Andrew J. Proposal Number: DMS - 0758379Institution: Princeton UniversityTitle: Equations and Automorphic FormsThis project aims to study the problem of solving polynomial equations. The formula for the solution of the quadratic equation, at least in special cases, was known to the Babylonians and the solutions to cubic and quartic equations were developed by Italian mathematicians in the 16th century. In the 19th century it was shown that a general equation of degree greater than or equal to five has no formula of the same kind. However it is possible that equations in more than one variable might always have simple solutions obtained by extracting roots, just as exist for the quadratic, cubic and quartic equations. Our first goal is to try to find families of equations for which there are such solutions. Our second goal is to try to describe the solutions in many cases in terms of functions with symmetries called modular forms. These symmetries are more complicated than, but are related to, the kinds of symmetries satisfied by the trigonometric functions. The particular set of curves which will be investigated are the curves of genus one. These curves have received a lot of attention in recent years. The PI hopes to extend his research into whether these curves always have solvable points. His second and principal goal is to try to prove cases of non-solvable base change. This would enable one to describe suitable representations of Galois groups in terms of automorphic forms. This is a crucial part of the Langlands program and seems to be a key stumbling block in the way of proving general results about functoriality.

摘要项目负责人：Wiles， Andrew J.项目编号：DMS - 0758379机构：普林斯顿大学题目：方程组和自同构形式本项目旨在研究多项式方程的求解问题。二次方程的解公式，至少在特殊情况下，是巴比伦人知道的，而三次方程和四次方程的解是16世纪意大利数学家发明的。在19世纪，人们发现大于或等于5次的一般方程没有相同类型的公式。然而，有多个变量的方程可能总是有简单的解，通过拔根得到，就像二次方程、三次方程和四次方程一样。我们的第一个目标是找到有这样的解的方程组。我们的第二个目标是，在很多情况下，用对称的函数来描述解，称为模形式。这些对称比三角函数所满足的对称更复杂，但又与之相关。我们要研究的一组特殊的曲线是属1的曲线。近年来，这些曲线受到了很多关注。PI希望将他的研究扩展到这些曲线是否总是有可解点。他的第二个也是最主要的目标是试图证明碱变化不可解的情况。这将使人们能够根据自同构形式描述伽罗瓦群的合适表示。这是朗兰兹纲领的关键部分，似乎是证明关于函数性的一般结果的关键绊脚石。