Iwasawa theory for modular forms

Abstract

In this paper we give an overview of some aspects of Iwasawa theory for modular forms. We start with the classical formulation in terms of $p$-adic $L$-functions in the ordinary case and the $\pm$-formulation for supersingular elliptic curves. Then we discuss some recent progresses in the proof of the corresponding Iwasawa main conjectures formulated by Kato (Conjecture 4.1), which relates the index of his zeta element to the characteristic ideal of the strict Selmer groups.