Families p-Modular Forms

从Katz和Serre的工作开始，并由Hida开发，我们已经看到模块化形式自然地存在于家庭中，并且这种观点具有广泛的算术应用。特别是，这些想法被Wiles和Taylor用于证明费马大定理，被Buzzard和Taylor用于证明Artin猜想的新情况。他们提出了Mazur的残馀表征的变形空间理论，这在前面提到的Taylor-Wiles的工作中是至关重要的。我们已经和Mazur一起证明了有一种叫做有限斜率形式的特征曲线的自然曲线，它的性质已经为前面的一些主题提供了启示。这条曲线映射到上述变形空间，但它的图像仍然非常神秘。与William Stein一起，我们证明了与无限斜率模形式相关的变形空间中的一些点在特征曲线像的拓扑闭包中，而另一些点不在特征曲线像的拓扑闭包中。我们的研究是为了更好地理解特征曲线的图像。这是数学领域的一个建议叫做算术代数几何；这就是数论的技巧和问题与代数几何的技巧和问题相结合的地方。本研究的主要重点是更好地理解一种特殊的代数几何曲线，称为“特征曲线”，它与不断变化的深刻而重要的数论数据密切而直接地联系在一起。了解特征曲线与其相应的数论数据之间的联系将推动数论领域的发展，从而为大多数现代密码学和数字安全提供基础。

Starting with the work of Katz and Serre and developed by Hida one has seen that modular forms naturally live in families and that this point of view has vast arithmetic applications. In particular, these ideas have been applied by Wiles and Taylor toward a proof Fermat's Last Theorem and by Buzzard and Taylor toward new cases of Artin's conjecture. They instigated Mazur's theory of deformation spaces of residual representations which was pivotal in the aforementioned work of Taylor-Wiles. We have shown with Mazur that there is a natural curve called the eigencurve of finite slope forms whose properties have already shed light on some of the previous topics. This curve maps into the above deformation space but its image is still very mysterious. With William Stein we have shown that some points in the deformation space associated to modular forms of infinite slope are in the topological closure of the image of the eigencurve and some are not. Our research is directed at a better understanding of the image of the eigencurve.This is a proposal in the area of mathematics called arithmetic algebraic geometry; this is where the techniques and questions of number theory merge with the techniques and questions of algebraic geometry. The main focus of this research is to better understand a particular special kind of algebraic geometric curve, called the "eigencurve," which is intimately and directly connected with a continuously varying collection of deep and important number theoretic data. Understanding the link between the eigencurve and its corresponding number theoretic data will advance the field of number theory, which in turn provides the underpinning for most of modern cryptography and digital security.