The Fontaine-Mazur conjecture via p-adic modular forms

用p进模形式给出了Kisin dms -0400666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。这类群的一个丰富的表示来源是代数变体的p进上同调。方丹-马祖尔猜想准确地预言了哪一种p进伽罗瓦表示应该以这种方式出现。这个猜想值得注意的是，它的表述中最微妙的条件只涉及到伽罗瓦表示对p以上素数分解群的限制。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特征形式。几年前，Coleman和Mazur在Hida的研究基础上发现，模特征型倾向于在p进族中移动。该项目的目的是利用这些族的几何来接近Fontaine-Mazur猜想，以及相应的伽罗瓦表示族：一些伽罗瓦表示的模块化可以根据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导其余的模块化。大约十年前，怀尔斯证明了费马大定理。他将椭圆曲线与模形式联系起来。后者是复杂的函数，具有难以置信的大量对称性。怀尔斯的突破包括使用椭圆曲线上的p进伽罗瓦表示法。他的结果可以看作是由Fontaine和Mazur提出的一种更普遍的哲学的一个特例，这种哲学预测了某一类p进伽罗瓦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——模形式倾向于在p进族中移动——来接近这个猜想。考虑到它们是复值函数，这是相当令人惊讶的。它作为一个引人注目的例证，虽然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.