Completed cohomology and Iwasawa theory

We compare two different constructions of cyclotomic p-adic L-functions for modular forms and their relationship to Galois cohomology: one using Kato’s Euler system and the other using Emerton’s p-adically completed cohomology of modular curves. At a more technical level, we prove the equality of two elements of a local Iwasawa cohomology group, one arising from Kato’s Euler system, and the other from the theory of modular symbols and p-adic local Langlands correspondence for GL2(Qp). We show that this equality holds even in the cases when the construction of p-adic L-functions is still unknown (i.e. when the modular form f is supercuspidal at p). Thus, we are able to give some representation-theoretic descriptions of Kato’s Euler system. We also compare two different constructions of anti-cyclotomic p-adic L-functions for modular forms on quaternion algebras: one defined by Bertolini and Darmon in [3] and the other using Emerton’s p-adically completed cohomology of Shimura sets