Two-dimensional local non-archimedean local fields arising from two-dimensional arithmetic geometry, e.g. formal power series over p-adic numbers, have two distinct integral structures: of rank 1 and of rank 2.
Correspondingly, there are two distinct two-dimensional adelic structures on elliptic surfaces.
Interestingly, they have a number of similarities with two symmetries of IUT.
My talk will explain how an interaction between the two adelic structures on proper models of elliptic curves over global fields helps us to understand the meaning of the classical BSD conjecture and produce its equivalent reformulation in purely adelic terms.
Part of this work is joint work with W. Czerniawska and P. Dolce.