Zeta Morphisms for Rank Two Universal Deformations

In his work on the Iwasawa main conjecture for elliptic Hecke eigen cusp newforms, Kazuya Kato constructed a map which we call the zeta morphism whose target is the Iwasawa cohomology of the associated p-adic Galois representations. Combining Fukaya-Kato’s idea for constructing the zeta morphisms for Hida families with many deep results in the p-adic (local and global) Langlands correspondence for GL_2/Q, we extend this map for the universal deformations of odd absolutely irreducible mod p Galois representations of rank two. As an application, we prove a theorem, which roughly says that, under some mu= 0 assumption, the Iwasawa main conjecture (without p-adic L- function) for one modular form implies the same conjecture for arbitrary congruent modular forms.