Higher Chow groups and higher dimensional class field theory

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Higher Chow groups of varieties are important objects in arithmetic geometry in general. When the base field is of arithmetic nature, higher Chow groups are known to be related with arithmetic objects such as the fundamental group and Brauer group. Over a local field, such relation has been studied for smooth projective varieties. We extended it to non-projective varieties. Besides, in a joint work with Bruno Kahn, we described the tensor product in the triangulated category of motives in terms of Somekawa K-groups, which provides us with a new method to compute the higher Chow groups of a product of varieties.