Equations and Automorphic Forms

摘要主要研究者：Wiles，Andrew J.提案编号：DMS -0758379机构：普林斯顿大学标题：方程和自守形式这个项目旨在研究求解多项式方程的问题。公式的解决方案的二次方程，至少在特殊情况下，是已知的巴比伦和解决方案的三次和四次方程是由意大利数学家在16世纪。在19世纪，人们证明了大于或等于5的一般次数方程没有同类的公式。然而，它是可能的，方程在一个以上的变量可能总是有简单的解决方案，通过提取根，就像存在的二次，三次和四次方程。我们的第一个目标是试图找到有这样的解的方程族。我们的第二个目标是试图在许多情况下用具有对称性的函数（称为模形式）来描述解。这些对称性比三角函数所满足的对称性更复杂，但与三角函数所满足的对称性有关。 将被研究的曲线的特定集合是亏格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的点。他的第二个也是主要目标是试图证明不可解的碱基变化的情况。这将使人们能够用自守形式来描述伽罗瓦群的适当表示。这是朗兰兹纲领的一个关键部分，似乎是证明关于函性的一般结果的一个关键绊脚石。

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.