Higher Chow groups of varieties over local fields

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Higher Chow groups of varieties is a wide generalization of the ideal class group in the algebraic number theory, and are regarded as a fundamental research theme. When varieties are defined over a local field, it naturally appears in arithmetic context, such as the higher dimensional class field theory and Brauer-Manin pairing. We developed the class field theory for a product of curves, and the Brauer-Manin pairing for open varieties (which was previously considered only for complete varieties).