Here is the homework for the final exam. The exercise is an adaptation of the proof of Theorem 1 in Ramakrishna's paper, Lifting Galois Representations, under some simplifying assumptions.

Below are my hand-written notes for the lectures. Most (but not all) of the material can be found in the references (see the Course Outline). The subdivision in the lecture notes does not correspond exactly to that of the classes.

Remarks and corrections are welcome.

Lecture 1 (Reminders on Galois groups, introduction to Galois representations)

Lecture 2 (Representable and pro-representable functors on categories of Artinian rings, statement of Schlessinger's criterion)

Lecture 3 (Proof of the existence of universal deformation pairs)

Lecture 4 (Interpretation of the dimension of deformation spaces in terms of group cohomology, relation with Leopoldt's conjecture)

Lecture 5 (Reminders on pro-p groups and their cohomology, representability of prime-to-p equivariant Hom functors from pro-p groups)

Lecture 6 (Explicit description of the universal deformation pair in some tame cases)

Lecture 7 (Relation between Vandiver's conjecture and the unobstructedness of the deformation problem in the reducible case)

Lecture 8 (Construction of the p-adic avatars of Hecke characters of a number field, of algebraic and non-algebraic weight)

Lecture 9 (A deformation-theoretic construction of families of p-adic Hecke characters of a number field)

Lecture 10 (Presentation of global deformation rings in terms of local ones)