There is a lot of stuff written by Parshin, see for example his Invitation to higher local fields on arxiv. This also contains references to papers by people like Huber and Beilinson, some of which may be difficult to find.
arXiv:0909.1568 Igusa integrals and volume asymptotics in analytic and adelic geometry from arXiv Front: math.NT by Antoine Chambert-Loir, Yuri Tschinkel We establish asymptotic formulae for volumes of height balls in analytic varieties over local fields and in adelic points of algebraic varieties over number fields, relating the Mellin transforms of height functions to Igusa integrals and to global geometric invariants of the underlying variety. In the adelic setting, this involves the construction of general Tamagawa measures.
arXiv:0912.1577 Harmonic analysis on local fields and adelic spaces II from arXiv Front: math.NT by D. V. Osipov, A. N. Parshin This paper is the second part of arXiv:0707.1766. We develope harmonic analysis in some categories of filtered abelian groups and vector spaces over the fields R or C. These categories contain as objects local fields and adelic spaces arising from arithmetical surfaces. Some structure theorems are proven for quotients of the adelic groups of algebraic and arithmetical surfaces.
nLab page on Adelic geometry