Mean-periodicity and zeta functions

Ivan Fesenko[1]; Guillaume Ricotta[2]; Masatoshi Suzuki[3]

  • [1] University of Nottingham School of Math Sciences University Park Nottingham NG7 2RD (England)
  • [2] Université Bordeaux 1 Institut de Mathématiques de Bordeaux 351, cours de la Liberation 33405 Talence cedex (France) ETH Zürich Forschungsinstitut für Mathematik HG J 16.2 Rämistrasse 101 8092 Zürich (Switzerland )
  • [3] The University of Tokyo Graduate School of Mathematical Sciences 3-8-1 Komaba Meguro-ku Tokyo 153-8914 Japan)

Annales de l’institut Fourier (2012)

  • Volume: 62, Issue: 5, page 1819-1887
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Fesenko, Ivan, Ricotta, Guillaume, and Suzuki, Masatoshi. "Mean-periodicity and zeta functions." Annales de l’institut Fourier 62.5 (2012): 1819-1887. <http://eudml.org/doc/251099>.

@article{Fesenko2012,
abstract = {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.},
affiliation = {University of Nottingham School of Math Sciences University Park Nottingham NG7 2RD (England); Université Bordeaux 1 Institut de Mathématiques de Bordeaux 351, cours de la Liberation 33405 Talence cedex (France) ETH Zürich Forschungsinstitut für Mathematik HG J 16.2 Rämistrasse 101 8092 Zürich (Switzerland ); The University of Tokyo Graduate School of Mathematical Sciences 3-8-1 Komaba Meguro-ku Tokyo 153-8914 Japan)},
author = {Fesenko, Ivan, Ricotta, Guillaume, Suzuki, Masatoshi},
journal = {Annales de l’institut Fourier},
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; Hasse-Weil -functions of curves over global fields; zeta functions of elliptic curves over number fields; higher adelic analysis},
language = {eng},
number = {5},
pages = {1819-1887},
publisher = {Association des Annales de l’institut Fourier},
title = {Mean-periodicity and zeta functions},
url = {http://eudml.org/doc/251099},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Fesenko, Ivan
AU - Ricotta, Guillaume
AU - Suzuki, Masatoshi
TI - Mean-periodicity and zeta functions
JO - Annales de l’institut Fourier
PY - 2012
PB - Association des Annales de l’institut Fourier
VL - 62
IS - 5
SP - 1819
EP - 1887
AB - 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.
LA - eng
KW - 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; Hasse-Weil -functions of curves over global fields; zeta functions of elliptic curves over number fields; higher adelic analysis
UR - http://eudml.org/doc/251099
ER -

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