Convolution structures are group-like objects that were extensively studied by harmonic analysts. We use them to define H0 and H1 for Arakelov divisors over number fields. We prove the analogs of the Riemann–Roch and Serre duality theorems. This brings more structure to the works of Tate and van der Geer and Schoof. The H1 is defined by a procedure very similar to the usual Ĉech cohomology. Serre′s duality becomes Pontryagin duality of convolution structures. The whole theory is parallel to the geometric case.