Study of the moduli of Galois representations of number fields and function fields

We have constructed a moduli scheme of Galois representations and studied its properties, and obtained some basic results. We have also obtained several related results, such as: (1) a vanishing theorem of the Galois-fixed subspace of a Galois representation of a rather general type of complete discrete valuation field (a generalization of a theorem of Imai) and its application to Iwasawa theory, (2) a result on the congruence of Galois representations and its application to non-existence theorems a la Rasmussen-Tamagawa, (3) proof of the fact that the Hecke field of a geometric Galois represntation is often (say, with density 1 primes, in certain cases) generated by the trace of the Frobenius for a single finite prime, (4) an upper bound of the number of the connected components of the Zariski closure of the image of a Galois representation.