Equations and Automorphic Forms

主要研究人员：Wiles，Andrew J.建议编号：DMS-0758379机构：普林斯顿大学标题：方程和自同构形式本项目旨在研究多项式方程的求解问题。至少在特殊情况下，二次方程的解的公式为巴比伦人所知，三次和四次方程的解是由意大利数学家在16世纪发展出来的。19世纪的研究表明，一般的大于或等于5次的方程式都没有相同的公式。然而，对于二次、三次和四次方程，存在这样一种可能性，即具有多个变量的方程可能总是具有通过求根而获得的简单解。我们的第一个目标是试图找到一族有这样解的方程式。我们的第二个目标是尝试在许多情况下用称为模形式的对称性的函数来描述解。这些对称性比三角函数所满足的对称性要复杂得多，但也与之相关。将要研究的特定曲线集是亏格一的曲线。近年来，这些曲线受到了极大的关注。PI希望将他的研究扩展到这些曲线是否总是有可解点。他的第二个也是主要的目标是试图证明不可解决的基础改变的情况。这将使人们能够用自同构的形式来描述伽罗瓦群的适当表示。这是朗兰兹计划的关键部分，似乎是证明关于功能性的一般结果的关键绊脚石。

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.