The Unramified Fontaine-Mazur Conjecture

未分支的方丹-马祖尔猜想

• 批准号: 9970184
• 负责人: Nigel Boston
• 金额: $ 9万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-06-01 至 2004-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9970184&HistoricalAwards=false
• 关键词: Unramified; Fontaine; Mazur; Conjecture

This award supports research into one of the most important conjectures in number theory today, the Fontaine-Mazur conjecture. One part of this conjecture was resolved by Wiles in the course of his proof of Fermat's Last Theorem, but there is another part concerning unramified Galois representations that has barely been investigated. The conjecture says that all such should be trivial in the sense of having finite image. Motivated by group theory I will develop the evidence that there is a new kind of representation, namely certain Galois actions on rooted trees, for which the images are nontrivial. These representations are as natural to consider as the usual Galois representations and I will carry over as much of the theory as possible to the new case.My research is mostly in algebraic number theory, although I do bring tools from group theory and computational algebra to bear upon the problems I attack and I also have recently worked in coding theory and cryptography. Algebraic number theory involves the use of advanced algebra to attack very simply stated problems involving the integers, such as whether the sum of two nth powers can be another nth power (Fermat's Last Theorem, mentioned above). It has long been consideredvery pure mathematics but recent advances in coding theory, cryptography, and even finance have applied it to very good effect.

该奖项支持对当今数论中最重要的猜想之一——方丹-马祖尔猜想的研究。这个猜想的一部分是由怀尔斯在证明费马大定理的过程中解决的，但另一部分涉及未分支的伽罗瓦表示，几乎没有被研究过。该猜想表明，从具有有限图像的意义上来说，所有这些都应该是微不足道的。在群论的推动下，我将证明存在一种新的表示形式，即对有根树的某些伽罗瓦动作，对于这些树来说图像是不平凡的。这些表示与通常的伽罗瓦表示一样自然，我将尽可能多地继承理论到新的案例中。我的研究主要是代数数论，尽管我确实使用了群论和计算代数的工具来解决我所攻击的问题，而且我最近也从事编码理论和密码学方面的工作。代数数论涉及使用高级代数来解决涉及整数的非常简单的问题，例如两个 n 次方之和是否可以是另一个 n 次方（费马大定理，上面提到的）。长期以来，它一直被认为是非常纯粹的数学，但编码理论、密码学甚至金融领域的最新进展已将其应用到了非常好的效果。