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We work with completed adelic structures on an arithmetic surface and justify that the construction under consideration is compatible with Arakelov geometry. The ring of completed adeles is algebraically and topologically self-dual and fundamental adelic subspaces are self orthogonal with respect to a natural differential pairing. We show that the Arakelov intersection pairing can be lifted to an idelic intersection pairing. [ABSTRACT FROM AUTHOR]

In this note, we establish a duality result under the residue paring between certain two-dimensional adelic spaces, which are associated to a closed point on an arithmetic surface. [ABSTRACT FROM AUTHOR]

Abstract: We explicitly study Katoʼs residue homomorphisms in Milnor K-theory, which are closely related to Contou-Carrère symbols. As applications we establish several reciprocity laws for certain locally defined maps on K-groups that are associated to arithmetic surfaces. [Copyright &y& Elsevier]