For any variety X/k, we consider the Beilinson–Huber adeles AX as a differ- ential graded k-algebra and examine the category Moddg of differential graded A-modules. We characterize the modules associated to certain quasi-coherent sheaves and define an adelic Chern class c(M) for modules which are graded free of rank 1. We study the intersection pairing in terms of a cup product and prove a version of the Bloch–Quillen formula that respects this cup product.

Fesenko [6] proved Serre duality and the Riemann–Roch theorem for surfaces using a topological duality on the adeles. On the other hand, Mattuck–Tate [18] and Grothendieck [12] provided proofs of the Riemann hypothesis for curves using the Riemann–Roch theorem for surfaces by studying the graph of the Frobenius morphism on the surface S = C × C . Therefore the combined results of Fesenko and Mattuck–Tate–Grothendieck can be said to provide an adelic proof of the Riemann hypothesis for a curve C over a finite field. We apply the results of this thesis to the adelic intersection pairing, and state a version of the Hodge index theorem which implies the Riemann hypothesis for curves.