2d local non-archimedean local fields have two integral structures, of rank
1 and of rank 2.
Correspondingly, there are two 2d adelic structures on elliptic surfaces: a
geometric one and an analytic one.
The geometric additive structure is self-dual and its topological
properties imply a new short proof of the Riemann-Roch theorem,
while its adelic view of Arakelov geometry should imply a new proof of the
Faltings-Riemann-Roch theorem.
The zeta integral of the surface is an integral over the product of two
copies of the analytic multiplicative structure.
Using 2d Iwasawa-Tate theory (2010) one can compute its pole at the central
point by using an interaction between the two adelic structures originating
from explicit 2d class field theory, and reformulate the BSD conjecture as
a property closely related to the discreteness of rational functions in
full geometric adeles.