An important question in modern number theory is whether a given representation of a Galois group into GLn(Fp), for p a prime, can be obtained as the reduction modulo p
of a representation with coefficients of characteristic zero, and if so what structure can be put on these liftings?
Deformation theory provides a general answer to these kind of questions by producing a "universal" lifting from which all other lifts are obtained.
In this course we will develop the basics of this theory.
The main goal will be to prove the existence of universal deformations.
We will then discuss tools which can be used to compute these universal deformations and consider instances in which they have explicit descriptions.