Mean-periodicity and zeta functions
Annales de l'Institut Fourier, Volume 62 (2012) no. 5, pp. 1819-1887.

This paper establishes new bridges between zeta functions in number theory and modern harmonic analysis, namely between the class of complex functions, which contains the zeta functions of arithmetic schemes and closed with respect to product and quotient, and the class of mean-periodic functions in several spaces of functions on the real line. In particular, the meromorphic continuation and functional equation of the zeta function of an arithmetic scheme with its expected analytic shape is shown to correspond to mean-periodicity of a certain explicitly defined function associated to the zeta function. The case of elliptic curves over number fields and their regular models is treated in more details, and many other examples are included as well.

Cet article établit de nouveaux ponts entre les fonctions zeta en théorie des nombres et l’analyse harmonique moderne, c’est-à-dire entre la classe des fonctions de la variable complexe, qui contient les fonctions zeta des schémas arithmétiques et est stable par produit et quotient, et la classe des fonctions moyennes périodiques sur pluieurs espaces de fonctions de la droite réelle. En particulier, il est démontré que le prolongement méromorphe et l’équation fonctionnelle de la fonction zeta d’un schéma arithmétique correspond à la moyenne périodicité d’une fonction explicitement définie et associée à cette fonction zeta. Le cas des courbes elliptiques sur des corps de nombres et leurs modèles réguliers est traité en détails, et de nombreux exemples supplémentaires sont inclus.

DOI: 10.5802/aif.2737
Classification: 14G10, 42A75, 11G05, 11M41, 43A45
Keywords: Zeta functions of elliptic curves over number fields, zeta functions of arithmetic schemes, mean-periodicity, boundary terms of zeta integrals, higher adelic analysis and geometry
Fesenko, Ivan 1; Ricotta, Guillaume 2, 3; Suzuki, Masatoshi 4

1 University of Nottingham School of Math Sciences University Park Nottingham NG7 2RD (England)
2 ETH Zürich Forschungsinstitut für Mathematik HG J 16.2 Rämistrasse 101 8092 Zürich (Switzerland )
3 Université Bordeaux 1 Institut de Mathématiques de Bordeaux 351, cours de la Liberation 33405 Talence cedex (France)
4 The University of Tokyo Graduate School of Mathematical Sciences 3-8-1 Komaba Meguro-ku Tokyo 153-8914 Japan)
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Fesenko, Ivan; Ricotta, Guillaume; Suzuki, Masatoshi. Mean-periodicity and zeta functions. Annales de l'Institut Fourier, Volume 62 (2012) no. 5, pp. 1819-1887. doi : 10.5802/aif.2737.

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