Adelic interpretation of the Euler characteristic for one-dimensional global fields
| Autor: | Czerniawska, Weronika |
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| Jazyk: | angličtina |
| Rok vydání: | 2022 |
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| Popis: | Tate's 'Fourier analysis in Number Fields and Hecke's zeta function', preceded by work of Iwasawa, showed in that adelic language can be extremely useful for arithmetic purposes. In fact one can see this work as one of the main impulses for development of arithmetic geometry. Indeed, it developed a theory in which number fields and function fields of algebraic curves are seen as objects of the same kind. In this paper we generalize the work of Tate. By using certain natural renormalization of Haar measure on adeles we obtain the relative Riemann-Roch theorem out of the adelic Poisson summation formulae. We express the relative Euler characteristics using purely adelic language. The formulas not only cover absolute and relative case, but also both the case of an arithmetic curve and a projective curve over a finite field. These results are inspired by work of A.Borisov [1] that views the size of cohomology as a dimension of so-called ghost space i.e. a locally compact abelian group enriched with an appropriate structure of convolution of Haar measures on it. 9 pages, comments welcome |
| Databáze: | OpenAIRE |
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