Higher adelic programme, adelic Riemann-Roch theorems and the BSD conjecture

The Birch and Swinnerton-Dyer conjecture is one the most important, still unsolved problem in mathematics, included in the list of the Millennium problems. Ivan Fesenko developed an adelic approach to the study of zeta-functions of elliptic curves in relation to the Conjecture. The method establishes relations between analysis and geometry that were not known before. This thesis presents some contributions to Fesenko's programme as well as places this results inside the whole context. The first part of the text introduces the geometric part of the theory. The central point of those considerations is the problem of discreteness of the field of rational functions of a surface inside the space of two-dimensional geometric Adeles, in both zero and positive characteristic. The second part presents the theory of analytic Adeles, adelic measure and integration on surfaces and two-dimensional zeta integral. We present how the rank part of the BSD conjecture is reduced to properties of certain boundary term of the zeta integral, and how it is related to the earlier mentioned discreteness of the function field