Selective integration on higher adeles and the Euler characteristic of surfaces

The space of two-dimensional geometric adeles of a surface is far from being a locally compact space and there is no translation countably additive invariant nontrivial measure on it. At the same time, certain subquotients of the adeles are direct limits of compact subquotients or inverse limits of discrete subquotients, compatible in a special way. Using this fact, the paper defines a translation invariant measure and integration on certain subquotients of the geometric adeles of surfaces. This theory is considerably different from the theory of integration on analytic adeles of surfaces. After revising aspects of one-dimensional theory, the paper includes a full definition of two-dimensional geometric adeles. A number of their new topological properties are established. The new translation invariant measure and integration on selective subquotients of the geometric adeles is used for integrals of certain functions in a two-dimensional method describing the size of adelic cohomology groups of surfaces, without using standard adelic complexes. A formula for the Euler characteristic of the surface and its divisor in terms of integrals over subquotients of geometric adeles is proved. Using the Euler characteristic, a new two-dimensional adelic intersection number is introduced. For geometric surfaces it is a positive multiple of the standard intersection number. Several results in the previous study of geometric adeles are given new proofs.