Subject Area: Modular forms, Galois representations
A central area in number theory is the study of special values of L-functions of automorphic forms, which are analytic objects. Many of the problems in number theory can be studied in terms of the L-function of certain automorphic forms. One fruitful way of studying the special values of these L-functions is through the p-adic interpolation of these values, for a prime p. This is carried out through the Bloch-Kato Tamagawa Number Conjecture and the Main Conjecture of Iwasawa theory. These conjectures relate the p-adic interpolation of special values of L-function which are analytic objects with arithmetic objects known as Selmer groups.
In a vast generalization, by considering an infinite extension of a number field whose Galois group is a p-adic Lie group, many deep and beautiful conjectures were formulated relating objects of arithmetic nature, again typified by a Selmer group of Galois representations and p-adic nature of their corresponding L-functions.
We have carried out a study of an important invariant that tells us about the structure of these Selmer groups. We are also interested in studying the p-adic nature of representations of Galois groups which are fundamental in understanding the Selmer groups.