Families p-Modular Forms

从卡茨和塞尔的工作开始，由希达发展起来，人们已经看到，模数形式自然地生活在家庭中，这种观点有着广泛的算术应用。 特别是，这些想法已经被怀尔斯和泰勒应用于证明费马大定理，并被Buzzard和泰勒应用于阿廷猜想的新情况。 他们煽动马祖尔的理论变形空间的剩余交涉这是关键的上述工作泰勒，怀尔斯。 我们已经证明了与马祖尔，有一个自然的曲线称为特征曲线的有限斜率形式，其性质已经揭示了一些以前的主题。 这条曲线映射到上面的变形空间，但它的图像仍然非常神秘。 我们和威廉·斯坦因一起证明了在变形空间中与无限斜率的模形式相关联的一些点是在特征曲线的像的拓扑闭包中，而另一些则不是。 我们的研究旨在更好地理解特征曲线的图像，这是数学领域中的一个提议，称为算术代数几何;这是数论的技术和问题与代数几何的技术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”，它与不断变化的深度和重要的数论数据集密切直接相关。 理解本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，这反过来又为大多数现代密码学和数字安全提供了基础。

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.