In 1993, M. Kapranov asked a question: what should be a possible generalization of Langlands correspondence for two-dimesional local fields and for two-dimensional arithmetic schemes. Recently, in 2012, A.N. Parshin made a direct image conjecture on the connection between abelian two-dimensional Langlands correspondence and the classical one-dimensional Langlands correspondence. This conjecture is connected also with the analytic behaviour of L-functions of curves defined over global fields. In my talk, following an idea of Kapranov, I will explain the abelian case of the local two-dimensional Langlands correspondence. I will speak about my recent results: how to extend the construction from the above local case to the case of a global ring of Parshin?Beilinson adeles of two-dimensional arithmetic schemes. I will prove non-commutative reciprocity laws on these schemes. These reciprocity laws correspond to unramified and tamely ramified extensions. I will also give the categorical construction of analogs of unramified principal series representations (for general linear groups over two-dimensional local fields) and describe its main properties.