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Convolution Structures and Arithmetic Cohomology

Published online by Cambridge University Press:  04 December 2007

Alexandr Borisov
Alexandr Borisov
Affiliation:
Department of Mathematics, Pennsylvania State University, University Park, PA 16802, U.S.A. e-mail: borisov@math.psu.edu
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Abstract

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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.

Keywords

Arakelov divisorsconvolution structuresnumber fieldsPontryagin dualityRiemann–RochSerre′s duality
Type
Research Article
Information
Compositio Mathematica , Volume 136 , Issue 3 , May 2003 , pp. 237 - 254
Copyright
© 2003 Kluwer Academic Publishers