The Fontaine-Mazur conjecture via p-adic modular forms

通过 p-adic 模形式的 Fontaine-Mazur 猜想

• 批准号: 0400666
• 负责人: Mark Kisin
• 金额: $ 10.73万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2007-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0400666&HistoricalAwards=false
• 关键词: Fontaine; Mazur; conjecture; via; adic

Abstract for award of Kisin DMS-0400666The Fontaine-Mazur conjecture via p-adic modular forms.One of the most fundamental objects of arithmetic is the absolute Galois group of a number field. A rich source of representations for such groups is the p-adic cohomology of algebraic varieties. The Fontaine-Mazur conjecture predicts precisely which p-adic Galois representations ought to arise in this way. What is remarkable about the conjecture is that the most subtle condition in its formulation involves only the restriction of the Galois representation to the decomposition groups of primes above p. In certain situations Fontaine-Mazur combine their philosophy with that of Langlands, and predict which Galois representations come from modular eigenforms. A few years ago Coleman and Mazur, building on work of Hida, discovered that modular eigenforms tend to move in p-adic families. The aim of the project is to approach the Fontaine-Mazur conjecture using the geometry of these families, and corresponding families of Galois representations: The modularity of some Galois representations can be proved following ideas of Wiles and Taylor-Wiles, and then one hopes to deduce the modularity of the rest by p-adic interpolation style arguments.Almost ten years ago Wiles proved Fermat's Last Theorem. He did this by relating elliptic curves to modular forms. The latter are complex functions which admit an incredibly large number of symmetries. Wiles' breakthrough involved the use of the p-adic Galois representation attached to an elliptic curve. His result can be viewed as a special case of a more general philosophy due to Fontaine and Mazur, which predicts that a certain class of p-adic Galois representations always arise from modular forms. My aim is to approach this conjecture by exploiting the fact discovered by Hida and Coleman-Mazur - that modular forms tend to move in p-adic families. This is rather surprising given that they are complex valued functions. It serves as a striking illustration that although modular forms are defined as analytic objects, they are of a profoundly arithmetic nature.

Kisin 获奖摘要 DMS-0400666通过 p 进模形式的 Fontaine-Mazur 猜想。算术最基本的对象之一是数域的绝对伽罗瓦群。此类群的表示的丰富来源是代数簇的 p 进上同调。方丹-马祖尔猜想准确地预测了哪些 p 进伽罗瓦表示应该以这种方式出现。该猜想的显着之处在于，其表述中最微妙的条件仅涉及伽罗瓦表示对 p 以上素数的分解群的限制。在某些情况下，方丹-马祖尔将他们的哲学与朗兰兹的哲学结合起来，并预测哪些伽罗瓦表示来自模本征形式。几年前，Coleman 和 Mazur 在 Hida 工作的基础上发现，模特征形倾向于在 p 进数族中移动。该项目的目的是使用这些族的几何学以及相应的伽罗瓦表示族来接近 Fontaine-Mazur 猜想：一些伽罗瓦表示的模块化可以按照怀尔斯和泰勒-怀尔斯的想法来证明，然后人们希望通过 p 进插值式论证推导其余部分的模块化。大约十年前，Wiles 证明了费马大定理。他通过将椭圆曲线与模形式联系起来来做到这一点。后者是复杂的函数，具有令人难以置信的大量对称性。怀尔斯的突破涉及使用附加到椭圆曲线的 p 进伽罗瓦表示。他的结果可以被视为 Fontaine 和 Mazur 提出的更一般哲学的一个特例，该哲学预测某一类 p 进伽罗瓦表示总是从模形式中产生。我的目标是通过利用 Hida 和 Coleman-Mazur 发现的事实来接近这个猜想——模块化形式倾向于在 p-adic 族中移动。鉴于它们是复杂的值函数，这是相当令人惊讶的。它作为一个引人注目的例证表明，尽管模形式被定义为分析对象，但它们具有深刻的算术性质。