Extended Eigenvarieties and Their Iwasawa Theory

研究任何科学现象的一个基本方法是将该现象的标准出现与其最极端的表现进行比较。例如，如果只考虑可见光，就不能理解光的基本性质。随着光的波长变化，我们看到了一系列可能的行为，可见光在中心，x射线和无线电波在波长光谱上的极端相反位置。在这个项目中，PI试图研究模块化形式，这是数学多个分支中出现的基本数学对象。PI的目标是将他的研究重点放在这个物体的一个新发现的极端实例上，以加深我们对经典模块形式的理解，类似于理解x射线和无线电波如何使我们加深对可见光的理解。更准确地说，Andreatta， Iovita和Pilloni最近的工作已经产生了扩展特征变量，其中包含位于特征p权值之上的附加点。这些附加点可以看作是权空间边界附近经典模形式的极限。在这个项目中，PI将对这些扩展特征变量进行系统的研究，重点是研究它们的Iwasawa理论。也就是说，他试图将塞尔默群和p进l函数与这些特征p自同构形式联系起来，并形成一个关于这两个对象的主要猜想。此外，他的目标是证明在这些特征p点周围给出的扩展特征变分的积分结构允许Iwasawa理论信息在特征0和特征p世界之间来回流动。例如，他旨在从特征0主猜想推导出特征p主猜想，并在良好的情况下证明这些猜想实际上是等价的。

A fundamental approach to studying any scientific phenomena is to compare standard occurrences of that phenomena with its most extreme manifestations. For instance, one cannot understand the fundamental nature of light by considering only visible light. As the wavelength of light varies, we see a whole family of possible behaviors with visible light in the center and with x-rays and radio waves at extreme opposite positions on the wavelength spectrum. In this project, the PI seeks to study modular forms, fundamental mathematical objects which appear in multiple branches of mathematics. The PI aims to focus his study on a newly discovered extremal instance of this object in an effort to deepen our understanding of classical modular forms akin to how understanding x-rays and radio waves allows us to deepen our understanding of visible light.More precisely, the recent work of Andreatta, Iovita, and Pilloni has produced extended eigenvarities which contain additional points sitting over characteristic p weights. These additional points can be viewed as limits of classical modular forms near the boundary of weight space. In this project, the PI will make a systematic study of these extended eigenvarieties with a large emphasis going towards studying their Iwasawa theory. Namely, he seeks to associate to these characteristic p automorphic forms both Selmer groups and p-adic L-functions, and to formulate a main conjecture relating these two objects. Additionally, he aims to show that the integral structure given to the extended eigenvariety around these characteristic p points allows for Iwasawa theoretic information to flow back and forth between the characteristic 0 and characteristic p world. For instance, he aims to deduce a characteristic p main conjecture from the characteristic 0 main conjecture, and in good situations show that these conjectures are actually equivalent.