## Convolution structures and arithmetic cohomology.(English)Zbl 1158.11340

Summary: Convolution structures are group-like objects that were extensively studied by harmonic analysts. We use them to define $$H^0$$ and $$H^1$$ 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 $$H^1$$ is defined by a procedure very similar to the usual Čech cohomology. Serre’s duality becomes Pontryagin duality of convolution structures. The whole theory is parallel to the geometric case.

### MSC:

 11R04 Algebraic numbers; rings of algebraic integers 14G40 Arithmetic varieties and schemes; Arakelov theory; heights 43A35 Positive definite functions on groups, semigroups, etc. 43A40 Character groups and dual objects
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