turkmath.org

Inspired by Hida's results on p-adic families of ordinary modular forms (and families of Galois representations attached to these objects by Deligne), Mazur proved the existence of a universal Galois representation attached to a mod p Galois representation $\bar{\rho}$, which in a sense parametrizes all possible liftings of the mod p Galois representation. The goal of this (mostly expository) talk is to describe Mazur's theory, along with its applications in arithmetic; such as Wiles' modularity lifting theorem (which yielded a proof of Fermat's Last Theorem), Taylor's proof of the (strong) Artin conjecture on the holomorphicity of Artin L-functions, and some results of myself on the Iwasawa theory of motives.