Almost every previous method to deal with aspects of the BSD conjecture relies on a local study. I will explain a new method to study the BSD conjecture (and several other problems) which uses global geometric structures. We work with (minimal) regular models of elliptic curves over global fields. On one hand, we extend the classical unscrewing technique of Kodaira, Shioda and Tate in the setting of adelic geometry to give a two-dimensional adelic interpretation of the Picard rank of the surface. On the other hand, its analytic rank is related to invariants of the second adelic structure of analytic nature which is involved in two-dimensional zeta integrals method. The multiplicative groups of the two adelic structures are related a via pairing into Milnor K_2 which come from explicit two-dimensional class field theory. This allows to reformulate the rank part of the BSD conjecture as the absence of zeros and poles at s=1 of a certain auxiliary adelic integral over the image of the function field of the curve. The latter property in positive characteristic appears to be related with Riemann-Roch type theorems. In positive characteristic it follows from discreteness of global functions which is a byproduct of an adelic proof of the Riemann-Roch theorem on surface via using geometric adelic duality. In characteristic zero it is expected to follow from an adelic proof of the Faltings-Riemann-Roch theorem.