Automorphic and Galois representations

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I have studied the Iwasawa theory and Langlands conjecture over number fields, motivated by A.Wiles' work. Here the identification of deformation rings of Galois representations and Hecke algebras (called Mazur conjecture) plays a central role. I have shown that the Hecke algebra of GL (2) over a totally real field is a local complete intersection ring, and is identified with a universal deformation ring of a mod p modular representation.The essential step in the work is, that the freeness property of a cohomology group of a modular curve over a Hecke ring (used essentially in Taylor-Wiles work) is a consequence of axioms which are easier to verify. I have named the axioms Taylor-Wiles system, in analogy with Euler systems (I should note that a similar idea was found independently by F.Diamond). By constructing a Taylor-Wiles system by Shimura curves, the Mazur conjecture over general totally real fields is proved. By combining the result with a level optimization argument (the even degree case of the Mazur principle is most difficult), many two dimensional 1-adic Galois representations correspond to automorphic representations, thus verifying the Langlands correspondence in these cases. Especially, a generalization of Taniyama-Shimura conjecture is shown fairly generally. There is a report on this work, including recent results. The detail is distributed as a preprint (Deformation rings and Hecke algebras in the totally real case), submitted to a journal, and 2 other articles are under preparation.Even in case of reducible residual representations, it is shown that there are infinitely many reducible representations which satisfy the Mazur conjecture, when the totally real field is fixed.