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.
