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Mean-Periodicity and Zeta Functions Tome 62, No 5 (2012), P R AN IE N R A U L E O S F D T E U L T I ’ I T N S ANNALES DE L’INSTITUT FOURIER Ivan FESENKO, Guillaume RICOTTA & Masatoshi SUZUKI Mean-periodicity and zeta functions Tome 62, no 5 (2012), p. 1819-1887. <http://aif.cedram.org/item?id=AIF_2012__62_5_1819_0> © Association des Annales de l’institut Fourier, 2012, tous droits réservés. L’accès aux articles de la revue « Annales de l’institut Fourier » (http://aif.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://aif.cedram.org/legal/). Toute re- production en tout ou partie de cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation à fin strictement per- sonnelle du copiste est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Ann. Inst. Fourier, Grenoble 62, 5 (2012) 1819-1887 MEAN-PERIODICITY AND ZETA FUNCTIONS by Ivan FESENKO, Guillaume RICOTTA & Masatoshi SUZUKI 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. Résumé. — 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 sché- mas 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 parti- culier, 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. Acknowledgements. — The authors thank A. Borichev for giving excel- lent lectures on mean-periodic functions in Nottingham and useful conver- sations on harmonic analysis issues, D. Goldfeld and J. Lagarias for their remarks and discussions, the referee for his interesting lengthy report, and many mathematicians for their remarks following talks given by the au- thors. This work was partially supported by ESPRC grant EP/E049109. G. Ricotta was partially financed by the ANR project “Aspects Arithmé- tiques des Matrices Aléatoires et du Chaos Quantique”. G. Ricotta is also Keywords: Zeta functions of elliptic curves over number fields, zeta functions of arith- metic schemes, mean-periodicity, boundary terms of zeta integrals, higher adelic analysis and geometry. Math. classification: 14G10, 42A75, 11G05, 11M41, 43A45. 1820 Ivan FESENKO, Guillaume RICOTTA & Masatoshi SUZUKI currently supported by a Marie Curie Intra European Fellowship within the 7th European Community Framework Programme (project ANER- AUTOHI, grant agreement number PIEF-GA-2009-25271). M. Suzuki was partially supported by Grant-in-Aid for JSPS Fellows No. 19-92 and Grant- in-Aid for Young Scientists (B) No. 21740004. 1. Introduction The Hecke-Weil correspondence between modular forms and Dirichlet se- ries plays a pivotal role in number theory. In this paper we propose a new correspondence between mean-periodic functions in several spaces of func- tions on the real line and a class of Dirichlet series which admit meromor- phic continuation and functional equation. This class includes arithmetic zeta functions and their products, quotients, derivatives. In the first approx- imation the new correspondence can be stated as this: the rescaled com- pleted zeta-function of arithmetic scheme S has meromorphic continuation of expected analytic shape and satisfies the functional equation s → 1 − s with sign ε if and only if the function f(exp(−t)) − ε exp(t)f(exp(t)) is a mean-periodic function in the space of smooth functions on the real line of exponential growth, where f(x) is the inverse Mellin transform of the product of an appropriate sufficiently large positive power of the completed Riemann zeta function and the rescaled completed zeta function. For pre- cise statements see Subsection 1.4 below and other subsections of this In- troduction. For the best of our knowledge the correspondence in this paper is the first correspondence which addresses the issue of functional analy- sis property of functions associated to the zeta functions which guarantees their meromorphic continuation and functional equation without asking for the stronger automorphic property of its L-factors. Whereas modular forms is a classical object investigated for more than 150 years, the theory of mean-periodic functions is a relatively recent part of functional analysis whose relations to arithmetic zeta functions is studied and described in this paper. The text presents the analytic aspects of the new correspondence in accessible and relatively self-contained form. As it is well known there is a large distance between the first important stage of stating the correspondence between L-functions and automorphic adelic representations and the second important stage of actually proving it even for some of the simplest notrivial cases. This work plays the role of the first stage for the new correspondence between the zeta functions in arithmetic geometry and mean-periodicity. ANNALES DE L’INSTITUT FOURIER MEAN-PERIODICITY AND ZETA FUNCTIONS 1821 1.1. Boundary terms in the classical one-dimensional case The completed zeta function of a number field K is defined on <e (s) > 1 by ζbK(s) := ζ∞,K(s)ζK (s) where ζK(s) is the classical Dedekind zeta function of K defined using the Euler product over all maximal ideal of the ring of integers of K and ζK,∞(s) is a finite product of Γ-factors defined in (5.6). It satisfies the integral representation (see [43]) Z s ζbK(s) = f(x) |x| µ × (x) × AK AK where f is an appropriately normalized function in the Schwartz-Bruhat space on and | | stands for the module on the ideles × of . An AK AK K application of analytic duality on K ⊂ AK leads to the decomposition ζbK(s) = ξ (f, s) + ξ f,b 1 − s + ωf (s) where fb is the Fourier transform of f, ξ(f, s) is an entire function and Z 1 s x ωf (s) = hf (x)x 0 x with Z Z −1 −1 hf (x) := − f(xγβ) − x fb x γβ µ(β)µ(γ). γ∈ 1 / × β∈∂ × AK K K Consider the weakest topology on AK, in which every character (namely continuous homomorphism to the unit circle) of AK is continuous. Then, the closure of K× is K and the boundary ∂K× of K× is K \ K× = {0}. The meromorphic continuation and the functional equation for ζbK(s) are equivalent to the meromorphic continuation and the functional equation for ωf (s). Let us remark that ωf (s) is the Laplace transform of Hf (t) := −t −t hf (e ) thanks to the change of variable x = e . The properties of the functions hf (x) and Hf (t), which are called the boundary terms for obvious reason, are crucial in order to have a better understanding of ωf (s). We have h (x) = −µ 1 × f(0) − x−1f(0) , f AK K b H (t) = −µ 1 × f(0) − etf(0) f AK K b since ∂K× is just the single point 0, with the appropriately normalized measure on the idele class group. As a consequence, ωf (s) is a rational TOME 62 (2012), FASCICULE 5 1822 Ivan FESENKO, Guillaume RICOTTA & Masatoshi SUZUKI function of s invariant with respect to f 7→ fb and s 7→ (1 − s). Thus, ζbK(s) admits a meromorphic continuation to C and satisfies a functional equation with respect to s 7→ (1 − s). 1.2. Mean-periodicity and analytic properties of Laplace transforms The previous discussion in the one-dimensional classical case naturally leads to the analytic study of Laplace transforms of specific functions. In particular, the meromorphic continuation and functional equation for × Mellin transforms of real-valued functions f on R+ of rapid decay at +∞ and polynomial order at 0+ is equivalent to the meromorphic continuation and functional equation of Z 1 Z +∞ s x −st ωf (s) := hf (x)x = Hf (t)e t 0 x 0 with −1 −1 hf (x) = f(x) − εx f x −t and Hf (t) = hf (e ), where ε = ±1 is the sign of the expected functional equation (see Section3). The functions hf (x) and Hf (t) are called the boundary terms by analogy. The main question is the following one. What property of hf (x) or Hf (t) implies the meromorphic continuation and functional equation for ωf (s)? One sufficient answer is mean-periodicity(1) . Mean-periodicity is an easy generalization of periodicity; a function g of a functional space X is X- mean-periodic if the space spanned by its translates is not dense in X. When the Hahn-Banach theorem is available in X, g is X-mean-periodic if and only if g satisfies a convolution equation g ∗ ϕ = 0 for some non- trivial element ϕ of the dual space X∗ and a suitable convolution ∗.
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