A major theme in number theory which has emerged over the last several decades is that one can better understand a given arithmetic object (be it a modular form, Galois representation, L-value) if one can put this object into a p-adic family. Modular forms fit into Hida/Coleman families; Galois representations are parametrized by universal deformation spaces; L-values are interpolated by p-adic L-functions. In this proposal, Pollack seeks to study three different instances of p-adic variation. First, he aims to give a conjectural formula for the cyclotomic mu-invariant of a cuspidal eigenform. His method is to transfer information from a p-adic family of Eisenstein series to the Hida family of the cuspform in order to calculate the relevant mu-invariant. Second, he aims to study the variation in p-adic families of Kato?s Euler system attached to a modular form. A novel aspect to his approach is to vary the Euler system over a Galois deformation space. As ordinary and non-ordinary forms are intertwined in this space, he hopes to transfer known Iwasawa theoretic information from the ordinary case to the non-ordinary case. Third, he proposes a number of computational projects relating to p-adic variation including the computation of Hida families and of weight one forms.
Number theory, one of the oldest fields of mathematics, has often borrowed methods from other mathematical fields to attack questions relating to the basic properties and patterns of numbers. For instance, calculus is the study of functions of a continuous variable, but nonetheless has had a profound impact on number theory despite its very discrete nature. This proposal explores the idea of trying to better understand certain number theory objects, namely modular forms, by putting them into a family and studying the family as a whole. Imagine the difference between studying the actions of an ant versus the role of an ant in its colony. The notion of a family in number theory is done through p-adic analysis. P-adic analysis is a generalization of modular arithmetic which itself gained great public fame in its role in public key cryptography, the backbone of internet commerce. Thus, by studying families of modular forms, the goal of the work proposed in this project is to deepen our understanding of any individual modular form on both a theoretical and computational level. Modular forms are intimately related to elliptic curves which are also very useful in cryptography and are commonly used to encrypt cellular transmissions. Thus any deepening of our understanding of modular forms could have potential cryptographic applications.
