I would like to go over the theory of p-adic modular forms in as elementary way as possible. The theory is the current frontier of algebraic number theory, and played significant role in the proof of modularity of elliptic curves. The idea is to look at a family of modular forms on the upper half space, and vary the weight p-adically. One would then like to construct a space parametrizing such families. This space is called the eigenvariety, as the eigenvalues under the action of the Hecke algebra are those which are parametrized. There are applications to automorphic representations too.
I intend to start with basic material on p-adic distributions and integration, the structure of the Iwasawa algebra Z_p[[X]] and modules over it, p-adic L-functions, as presented e.g. in Lang's book Cyclotomic Fields, Chapters 2, 5, 4. Then I'll explain an old beautiful article of Serre (Publ. Math. IHES 1962) on completely continuous endomorphisms of p-adic Banach spaces. Beginning from essentially nothing I'll explain what are Banach spaces, Fredholm determinant and resolvant, Riesz theory.
In the 2nd part of this course I'll explain Coleman's Invent. 1997 paper on p-adic spaces of families of modular forms, in particular why p-adic modular forms of small slope are classical. The construction of the eigencurve by Coleman and Mazur will be next. If we get this far, there are generalizations/extensions by Buzzard, and possible applications to automorphic forms.