The Iwasawa main conjecture asserts a relationship between certain

-adic

-functions and characteristic polynomials associated with the

-part of the class group of the cyclotomic

-extension of an abelian.extension of

. The main conjecture over abelian extensions of

was first proved by Mazur and Wiles using

-dimensional Galois representations attached to cusp forms that are congruent to ordinary Eisenstein series. Wiles generalized the method of Mazur-Wiles to the setting of Hilbert modular forms and proved the main conjecture over totally real fields. A few years later, Ohta gave a refinement of Wiles's proof of the main conjecture over abelian extensions of

by constructing Galois representations attached to cusp forms using the action of

on the cohomology of modular curves. One of the key steps in Ohta's proof is to compute the congruence modules related to Eisenstein series.

In the first talk, we will talk about how to generalize Ohta's work to the setting of Hilbert modular forms. In the second talk, we will introduce some ingredients used in the setting of Hilbert modular forms.