In this course, we first study classical Iwasawa theory, which is about the relation between class group of cyclotomic fields and Lfunctions. We then study recent conjectures of Romyar Sharifi, which are in some sense a refinement of the Iwasawa main conjecture. It relates the homology of modular curves modulo some Eisenstein ideal to the arithmetic of cyclotomic fields. Many results have been proved towards this conjecture. If times permits, we will explain some of these results.

Algebraic number theory, some algebraic topology (singular homology), basic knowledge on modular forms.

1. A first course in modular forms, Diamond and Shurman, Springer.
2. Cyclotomic fields and zeta values, Coates and Sujatha, Springer.
3. Modular curves and cyclotomic fields, lecture notes by Romyar Sharifi.