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发生与其最极端的表现进行比较。 例如，人们不能通过只考虑可见光来理解光的基本性质。 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谱上的极端相反位置。 在这个项目中，PI试图研究模块形式，出现在数学的多个分支中的基本数学对象。PI的目标是将他的研究集中在这个对象的一个新发现的极值实例上，以加深我们对经典模形式的理解，就像理解x射线和无线电波可以加深我们对可见光的理解一样。更准确地说，Andreatta，Iovita和Pilloni最近的工作产生了扩展的本征变量，其中包含位于特征p权重上的额外点。 这些额外的点可以被看作是在权空间边界附近的经典模形式的极限。 在这个项目中，PI将系统地研究这些扩展的本征簇，重点是研究他们的岩泽理论。 也就是说，他试图关联到这些特点p自守形式都塞尔默群和p进L-功能，并制定了一个主要的猜想有关这两个对象。 此外，他的目的是表明，积分结构给予扩展本征簇周围这些特征p点允许岩泽理论信息之间的流动来回特征0和特征p世界。 例如，他的目标是从特征0主猜想推导出特征p主猜想，并在良好的情况下表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.