On topics in Iwasawa theory and p-adic Langlands program.

Abstract: Recent developments in the p-adic Langlands program allow us to
revisit classical conjectures in Iwasawa Theory which help in
understanding the arithmetic of Galois representations arising from
elliptic curves, modular forms and automorphic forms.

This talk will cover various aspects of my research domain and will be
expository in nature. We will present results in Iwasawa Theory and p-adic
Hodge Theory and combine them to frame the signed Iwasawa main conjectures
for non-ordinary automorphic representations. Alongside, we will also
discuss Greenberg’s p-rationality conjecture and use it to construct
Galois representations with big open images in reductive groups.
Furthermore, we will discuss results in p-adic functional analysis and
show the existence of rigid analytic vectors in the crystalline
representations arising from the p-adic Langlands program. These works are
a culmination of several projects with various collaborators.