The Langlands program relates number theory to the harmonic analysis on p-adic groups. An instance of this is the modularity theorem proving Fermat's Last Theorem as a corollary. In classical harmonic analysis, one is interested in irreducible unitary representations of p-adic groups. However, to gain a deeper understanding of arithmetic, it becomes necessary to study representations of p-adic groups on p-adic vector spaces rather than on complex vector spaces. In this talk, I will present my work on p-adic representations of p-adic groups and talk about its significance to the problems in number theory.
