The ring of adeles allows one to describe the Artin reciprocity law, which is a generalisation of quadratic reciprocity, and other reciprocity laws over finite fields. In addition, it is a classical theorem from Weil that G-bundles on an algebraic curve over a finite field can be described in terms of adeles for a reductive group G. Adeles are also connected with the adelic algebraic groups and adelic curves.
