The Fontaine-Mazur conjecture via p-adic modular forms

摘要Kisin DMS-0400666Fontaine-Mazur猜想的模形式。算术的一个最基本的对象是数域的绝对伽罗瓦群。这种群的一个丰富的表示来源是代数簇的p-进上同调。Fontaine-Mazur猜想准确地预测了应该以这种方式产生的p-进Galois表示。关于这个猜想，值得注意的是，其表述中最微妙的条件只涉及伽罗瓦表示仅限于p以上素数的分解群。在某些情况下，Fontaine-Mazur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，科尔曼和马祖尔在希达的工作的基础上，发现模本征形式往往在p-kar型家庭中移动。这个项目的目的是利用这些族和相应的Galois表示族的几何来逼近Fontaine-Mazur猜想：一些Galois表示的模性可以遵循Wiles和Taylor-Wiles的思想来证明，然后人们希望通过p-进插值风格的论证来推断其余的Galois表示的模性。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。Wiles的突破涉及到使用附加在椭圆曲线上的p进Galois表示。他的结果可以看作是Fontaine和Mazur提出的更一般哲学的一个特例，该哲学预言某类p-进Galois表示总是由模形式产生的。我的目的是通过利用Hida和Coleman-Mazur发现的事实来探讨这个猜想-模形式倾向于在p-Add家庭中移动。考虑到它们是复值函数，这相当令人惊讶。这是一个引人注目的例证，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。

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.