This is asurvey paper about Serre's conjectures that was based on lectures Ken Ribet gave at the Park City Mathematics Institute.

Here is the pdf file.

This is a nicely written survey article on the conjectures
in the title of the paper. The conjectures of Serre in question are about
the modularity of mod\,$p$, 2-dimensional, continuous, odd, absolutely
irreducible representations of the absolute Galois group $G_ Q$ of $ Q$.
There is a more refined version which also predicts certain minimal modular
invariants from which these Galois representations arise. While the conjectures
in their qualitative form are still wide open there has been considerable
progress in proving that the qualitative form of the conjecture implies
the refined form. It is this implication, which is a consequence of deep
work of many mathematicians, that this paper surveys in the main. The
paper also has useful exercises that will be of help to someone wishing
to learn about this area, and two appendices by K. Buzzard and B. Conrad
on mod $ l$ multiplicity one principles and constructions of Galois representations
attached to weight 2 newforms. Reviewed by Chandrashekhar Khare |