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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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