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UNIVERSITÉ MOHAMMED V

FACULTÉ DES SCIENCES

Rabat

Faculté des Sciences, 4 Avenue Ibn Battouta B.P. 1014 RP, Rabat – Maroc Tel +212 (0) 37 77 18 34/35/38, Fax : +212 (0) 37 77 42 61, http://www.fsr.ac.ma

N° d’ordre : 2871 THESE DE DOCTORAT

Présentée par

Mohammed SOUID El AININ

Titre

HOLOMORPHIC (Gamma, chi, nu)-THETA FUNCTION ASSOCIATED TO RANK r OF DISCRETE SUBGROUPS IN (C^g,+)

Discipline : Mathématiques

Laboratoire : Analyse et Applications

Spécialité : Analyse

Soutenance :le 05 Mai 2016

Devant le jury : Président :

Omar EL-FALLAH PES, Faculté des Sciences de Rabat

Examinateurs :

Ahmed INTISSAR PES, Faculté des Sciences de Rabat. Allal GHANMI PH, Faculté des Sciences de Rabat. El Hassan ZEROUALI PES, Faculté des Sciences de Rabat.

Nour eddine ASKOUR PES, Faculté des Sciences et Techniques Beni Mellal.

Invité :

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Remerciements

Cette thèse a été réalisée au département des mathématiques de la Faculté des Sciences , de l’Université Mohammed V de Rabat, au sein du Laboratoire d’Analyse et Applications avec l’équipe E.D.P et Géométrie spectrale, sous la direction de Mrs Ahmed INTISSAR et Allal GHANMI.

Je désire tout d’abord remercier mon directeur de thèse Monsieur Ahmed INTISSAR Pro-fesseur à la Faculté des Sciences de Rabat qui a accompagné mes activités de recherche et m’a fait partager sa grande culture mathématique. Je tiens à lui manifester ma gratitude pour l’aide qu’il ma apportée, sa patience, sa gentillesse et ses qualités humaines.

Je souhaite manifester ma reconnaissance particulièrement à mon co-encadrant Monsieur Allal GHANMI Professeur d’Habilité à la Faculté des Sciences de Rabat qui a assuré l’élaboration de cette thèse avec amabilité et compétence, avec un suivi constant et un in-térêt démontré tout au long de mon travail malgré ses multiples occupations. Je suis très heureux d’avoir pu profiter à sa grande culture mathématique. Son soutien, sa disponi-bilité et encouragements m’ont permis de bien mener à terme ce travail.

Je suis très reconnaissant envers Monsieur Omar EL-FALLAH Professeur à la Faculté des Sciences de Rabat d’avoir bien voulu s’intéresser à mon travail en acceptant d’être prési-dent du jury.

Monsieur El Hassan ZEROUALI Professeur à la Faculté des Sciences de Rabat m’a fait un très grand honneur en acceptant de rapporter cette thèse. Ainsi pour les remarques et les suggestions qu’il m’a faites.

Monsieur Nour Edinne ASKOUR Professeur d’étude à la Faculté des Sciences et Tech-niques de Beni Mellal a accepté d’être rapporteur de cette thèse. Je souhaite le remercier vivement ici pour le temps passé, pour son enthousiasme, ses remarques , ses perspectives intéressantes proposées pour mon sujet.

Je suis particulièrement touché que Monsieur Abdlhamid BOURASS Professeur à la Faculté des Sciences de Rabat, que j’ai eu la chance d’avoir comme enseignant, a accepté d’être invité à la soutenance de cette thèse.

Cette thèse s’est également inscrite dans un environnement humain particulièrement chaleureux. De nombreuses personnes m’ont aidé durant toutes les années de préparation. Je souhaite

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3 E.D.P et Géométrie Spectrale qui animent des séances de travail hebdomadaires dans notre séminaire de jeudi matin à 10h30 dans la salle des séminaires du département de mathématiques. Je tiens à remercier mes amis et mes collègues doctorants: Kamal DIKI, Aymen EL FARDI, Mohammed ZIYAT, Zakaryae MOUHCINE et Amal EL HAMYANI. Je remercie également Madame Rania INTISSAR pour son soutien et sa passion pour les mathématiques .

Je souhaiterais également exprimer toute ma reconnaissance à certains de mes professeurs, tant par leurs qualités d’enseignant que par leur gentillesse, ont marqué ma scolarité et pour eux j’ai une affection particulière.

Enfin, je remercie de tout mon coeur mes parents et ma famille qui m’ont soutenu et en-couragé durant toutes mes années d’études.

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4

Résumé

Nous donnons une description concrète de l’espace(1, χ)-theta de type Fock-Bargmann

espace constitué de fonctions automorphes associées aux sous-groupe discret de rang 1 de l’espace (C2,+). Il ressemble à un produit tensoriel de l’espace de fonction (Γ, χ) au-tomorphes sur le plan complexe C et l’espace classique de Fock-Bargmann sur C. En outre, nous construisons une base orthonormée et nous donnons l’expression explicite de son noyau reproduisant en terme de la fonction theta de Riemann. Pour le cas général 1 ≤ r ≤ g, nous montrons que si le sous-groupe discret Γr est isotrope de rang r, par

rapport à la structure symplectique (Cg, E) alors l’epace des fonctions automorphes as-sociéFΓ,χ2,ν,H(Cg)est non triviale de plus c’est un espace de Hilbert à noyau reproduisant. Nous fournissons une base orthonormée explicite et l’expression explicite de son noyau reproduisant en termes de la fonction theta de Riemann de plusieurs variables avec des caractéristiques spéciales . Comme application , nous introduisons, et étudions la trans-formation de Segal-Bargmann pour les espacesF2

Γr(C

2)etF2,ν,H Γ,χ (Cg).

Mots-clées: Sous groupe discret de rang r, sous groupe isotropique, Space de Fock-Bargmann(Γ, χ), base orthonormée, noyau reproduisant, fonction theta de Riemann, trans-formation de Segal-Bargmann.

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5

Abstract

We give concrete description of the(1, χ)-theta Fock-Bargmann space consisting of

holomorphic automorphic functions associated to given discrete subgroup inC2 of rank one. It looks like a tensor product of a theta Fock-Bargmann space on the complex planeC and the classical Fock-Bargmann space onC. Moreover, we construct an orthonormal ba-sis and we give explicit expression of its reproducing kernel function in terms of Riemann theta function. For the general case 1 ≤ r ≤ g, we show that if the discrete subgroup Γ is isotropic of rank r with respect to the symplectic structure (Cg, E) then FΓ,χ2,ν,H(Cg) is nontrivial if and only if χ is a character. In this case,FΓ,χ2,ν,H(Cg)is an infinite reproducing kernel Hilbert space, and we provide an explicit orthonormal basis and explicit expression of its reproducing kernel function in terms of Riemann theta function of several variables with special characteristics. As application, we introduce and study the Segal-Bargmann transform for the spacesF2

Γr(C

2)andF2,ν,H Γ,χ (Cg).

Keywords: Rank r discrete subgroup; Holomorphic theta functions; Isotrope discret subgroup;(Γ, χ)-theta Fock-Bargmann space orthonormal basis; Reproducing kernel; Rie-mann theta function; Segal-BargRie-mann transformation.

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Résumé de la thèse

Dans ce travail, on se propose de considérer le cas où le sous groupe discret Γ est non cocompact (i.eCg/Γ n’est pas compact) de rang r, avec 1≤ r ≤g. Dans ce qui suit, nous dotonsCg = {z= (z1, z2,· · · , zg); zj∈ C}, l’espace complexe de dimension g ≥1, de son

produit Hermitien;

H(z, w) = zw¯ =z1w¯1+z2w¯2+ · · · +zgw¯g; (z, w) ∈CCg.

Soit Γr un sous group discret de rang r de (R2g,+), qu’on peut considérer comme un

Z-module de rang r = 0, 1,· · · , 2g, et on peut écrireΓr =1+ · · · +r où les vecteurs

ω1,· · · , ωr ∈ Cg sontR-linéairement indépendants. Un domaine fondamental Λ(Γr) de

Γr dans R2g peut s’identifier au groupe quotient R2g/Γr muni de la topologie quotient.

On considére aussi une application χ de Γr à valeurs dans le cercle unité U(1) = {z ∈

C,|z| = 1}, et on s’intéresse à l’équation fonctionnelle suivante : f(z+γ) =χ(γ)eH(z+

γ

2)f(z); z∈ Cg, γ∈ Γr. (0.1) Une telle fonction f sur Cg satisfaisant l’équation fonctionnelle (1.2) est dite fonction

(Γ, χ)-automorphe. Ainsi, on peut considérer l’espace fonctionnelOν,H

Γr(C

g)constitué des

fonctions holomorphes qui sont (Γ, χ)-automorphes. On munit cet espace de la norme suivante kfk2Γ r := Z Λ(Γr) |f(z)|2e−H(z,z)(z), (0.2) où dλ(z)désigne la mesure de Lebesgue surCg.

On peut aussi définir l’espace fonctionnelFΓ2,ν

r(C

g)des fonctions holomorphes(Γ, χ)

-automorphes surCgde normekfk2Γr finie . Ainsi

FΓ2,ν

r(C

g) : =nf holomorphe onCgtelle que f(z+

γ) =χ(γ)eH(z+ γ 2)f(z); z ∈Cg, γ∈ Γr et Z Λ(Γr) |f(z)|2e−H(z,z)(z) < +∞  . (0.3) 6

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7

Motivations et Objectifs

Les recherches dans le cas non-cocompact (i.e Γr de rang r; 0 ≤ r < 2g) sont donc

forte-ment motivés dans le cadre des variétés quasi-abéliennes voir par exemple ( [2, 1, 10] les réferences dedans ). Toutefois, la construction explicite des fonctions automorphes holo-morphes utilisant des séries de Fourier est encore loin d’être comprise dans le cas général de 0≤r≤2g, sauf pour r =0 et r =2g. Pour le casΓ0 ={0}, on aΛ(Γ0) =Cget l’espace FΓ2,ν

0(C

g) est trivial si χ(0) 6= 1. Sinon l’espace F2,ν Γ0(C

g) est isomorphe à l’espace de

Fock-Bargmann classique des fonctions holomorphes sur Cg de carrées intégrables par rapport à la gaussienne e−ν|z|2dλ. Pour le rang maximal r = 2g, le groupe Λ(Γ

2g) est

compact et l’espaceF2 Γ2g(C

g)est non trivial sous la condition

χ(γ+γ0) = χ(γ)χ(γ0)eiE(γ,γ

0)

(RDQ),

pour tout γ, γ0 ∈ Γ2g. Dans ce casFΓ,χ2 (Cg)est de dimension finie. Elle est donnée par une

formule explicite en terme de volume du tore complexeCg/Γ2g [13].

Les propriétés spectrales pertinentes de l’espace des fonctions(L2,Γ2g, χ) -thêta sont

examinées dans [7, 13]. Récemment, A.Ghanmi et A. Intissar ont étudié dans [14] le rang r = 1 avec g = 1. Ils ont montré que l’espace(Z, χ) - theta Fock-Bargmann FΓ,χ2,ν(C) est de dimension infinie pour χ étant un caractère. En outre, une base orthonormée de cet espace de Hilbert a été construite et l’expression explicite de son noyau reproduction a été donnée explicitement en terme de la fonction theta modifiée θα,β(z|τ).

Résultats obtenus

Dans ce travail, nous donnons une description complète de FΓ2,ν

r(C

g); g 2. On

com-mence par le cas où le sous groupe discretΓrest de rang un et nous généralisons les

résul-tats obtenus dans [14] pour g =1 à g=2 et ensuite à g quelconque. On montre que pour Γ = 1 telle que ω1 ∈ C2\ {(0, 0)}et ν = H(w1, w1), l’espaceFΓ,χ2 (C2) est non trivial

si et seulement si χ est un caractère de Γ. De plus, on construit une base orthonormée formée par le système de fonctions

ψm,n(z1, z2) =:  2ν π 1/4 νn+1 πn! 1/2 e−π2ν (m+α) 2 eν2z12+2iπ(α+m)z1zn 2,

pour m ∈ Z et nZ+. On montre aussi que l’espaceF2

Γ,χ(C2)est un espace de Hilbert à

noyau reproduisant donné par K((z1, z2);(w1, w2)) = √ 2ν π 3/2 eν2(z21+w12)θ α,0  z1−w1 2iπ ν  eνz2w2.

Pour g >2 et r =1, les résultats sont similaires. Voir Chapitre 1 pour l’ennoncé exacte et pour plus de détails. Pour introduire la transformée de Bargmann associée, on est

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8 amené à étudier l’espace L2

L,χ(R2), l’espace des fonctions mesurables sur R2satisfaisant

l’équation suivante φ(q1+ √ 2m1, q2) =χα(m1)φ(q1, q2) ∀∀(q1, q2) ∈R2 ∀m1 ∈Z, et telle que kφk2L := Z √ 2 0 Z +∞ −∞ | φ(q1, q2) | 2dq 1dq2 < +∞,

associée à la donnée d’un sous groupe discret L :=√2Z de R2et le caractére

χα(m):=e2iπαm; m∈ Z,

pour certain αR. On montre qu’une fonction mesurable φ : R2 −→ C appartient à L2L,χ(R2)si et seulement si elle s’écrit sous la forme

φ(q1, q2) = e √

2iπαq1g(q

1, q2),

où g est une fonction√2Z-périodique (i.e g(q1+ √

2m1, q2) = g(q1, q2)). De plus, l’espace

de HilbertL2

L,χ(R2)posséde la base orthonormale

φm,n(q1, q2) = ()−

1

4(2nn!)−12e

2iπ(α+m)q1e−21q22Hn(q2), où Hn désigne le polynôme de Hermite donnée par

Hn(ξ) = (−1)neξ

2n(e−ξ 2

)

nξ .

On définit alors la transformation de BargmannB par :

[Bφ](z1, z2) = 1 π 12 Z R2φ(x, y)e −1 2(z21+z22)+ √ 2(z1x+z2y)e−12(x2+y2)dxdy. Alors, on a B(φm,n) = ψm,n pour m∈ Z, et nZ+,

autrement dit transforme la base orthonormée de l’espace de Hilbert L2

L,χ(R2) à la base

orthonormée de l’espace de Hilbert F2

Γ,χ(C2). Cela signifie que B est une isométrie de L2

L,χ(R2)surFΓ,χ2 (C2).

Dans le cas où le sous groupe discretΓr est de rang r ≥ 2 , la description de l’espace F2

Γr(C

g)nécessite une hypothèse supplémentaire. On demande au sous groupe discret

Γr d’être isotropique c-à-d:

E(γ, γ0) := =mγ, γ0 =0 pour tout (γ, γ0) ∈ Γ2r.

Nous remarquons d’abord que l’équation fonctionnelle devient f(z+m, z⊥) = eνB(z+

m

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9 où z = (z1,· · · , zr) ∈ Cr, z⊥ = (zr+1,· · · , zg−r) ∈Cg−r , et m = (m1,· · · , mr) ∈Zr. Noter

que B est la matrice définie par

B = (H(ωj, ωk))1≤j,k≤r,

c’est une matrice réelle symétrique et non-dégénérée . Dans l’équation fonctionnelle (0.4), le z⊥ peut être considérer comme un paramètre . On montre que pour tout sous groupe discret isotropiqueΓr, l’ensemble des fonctions en,kα,ν(z, z⊥)données par :

eα,ν

n,k(z, z⊥) :=e

ν

2B(z,z)+2iπ(α+n)zzk

⊥, (0.5)

pour tout n ∈ Zr et k ∈ (Z+)g−r, constituent une base orthogonale de l’espaceF2 Γr(C g) avec e α,ν n,k 2 Γr =  1 √ det B  π  π ν g−2 k! ν|k|  e2π2ν (n+α)B −1(n+ α). (0.6)

De plus toute fonction f appartenant à l’espaceF2 Γr(C

g)peut s’écrire sous la forme :

f(z, z⊥):=

(n,k)∈Zr×(Z+)g−r an,ke ν 2B(z,z)+2iπ(α+n)zzk ⊥, (0.7)

(an,k)n;kest une suite de nombres complexes satisfaisant la condition suivante :  1 √ det B  π  π ν g−r

(n;k)∈Zr×(Z+)g−r  k! ν|k|  e2π2ν (n+α)B −1(n+α) |an,k|2 < +∞.

D’autre part l’espaceFΓ2

r(C

g)est un espace de Hilbert possédant un noyau reproduisant

K(u, v). Plus précisément, pour tout u = (z, z⊥)et v = (w, w⊥)dansCg, nous avons K(u, v) =√det B 2ν π  ν π g−r eν2(B(z,z)+B(w,w))Θ α,0  z−w 2πi ν B −1  eνhz⊥,w⊥iCg−r, oùΘα,0 est la fonction thêta Riemann définie par :

Θα,β(z

F) =

n∈Zr

e2iπ{12(α+n)F(α+n)+(α+n)(z+β)}, (0.8) pour α, βR, zCr, F ∈Cr×r telle que F est symétrique dont la partie imaginaire Im(F)

est définie positive , pour plus de détails voir [34, ?, 33, 22]. La condition définie positive de Im(F)garantie la convergence de la fonction (0.8), pour tout z∈ Cr.

Dans ce cas, on peut définir aussi une transformation de type Segal-Bargmann. Plus précisement, on considére [Bφ](z, z⊥) = e− 1 2B(z,z) Z Rgφ(u)e 2B(z,u)−B(u,u) (u), (0.9) qui définie un isomorphisme isométrique deL2

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10

Organisation

Dans le Chapitre 1, on introduit certains espaces fonctionellsOν

Γ,χ(C),CΓ,χ∞,ν(C)et L2,νΓ,χ(C)

associés à un sous groupe discret de rang maximal de l’espaceC, puis on cherche la con-dition nécessaire et suffisante de non trivialité de ces espaces. On montre que les epaces considérés sont des espaces de Hilbert. Nous donnons dans le Chapitre 2 une description concrète de l’espace(1, χ)-theta Fock-Bargmann constituée des fonctions automorphes

associées au sous-groupe discret de rang 1 de l’espace (C2,+). Nous construisons une base orthonormée et nous donnons l’expression explicite de son noyau reproduisant en terme de la fonction theta de Riemann.

Dans le Chapitre 3, on considére l’espaceV=VgC =Cg; g≥1 muni d’une forme her-mitienne H(u, v) définie positive, et on lui associe l’espace fonctionnelFΓ,χ2,ν,H(V). Nous montrons que si le sous-groupe discretΓ est isotrope de rang r par rapport à la structure symplectique (V, E)alorsFΓ,χ2,ν,H(V)est non triviale si et seulement si χ est un caractére. Comme application, nous introduisons dans le Chapitre 4 la transformation de Segal-Bargmann. On montre que cette transformation est une isométrie entre l’espaceL2

L,χ(R2)

et l’espace F2

Γ,χ(C2), et plus généralement entre l’espaceL2Γ(R2g) et l’espaceFΓ,χ2,ν,H(Cg).

On termine par des appendices, où on rappelle quelques notions sur les fonctions theta, les intégrals de type Gauss.

Conclusion et prespectives

Suite aux travaux menés dans cette thèse, nous proposons quelques pistes de recherche susceptibles d’en constituer un prolongement naturel. Dans les Chapitre 1 et 2 on peut faire l’étude spectrale du Laplacien magnétique agissant sur l’espace fonctionnelFΓ,χ2,ν,H(V)

à la lumiére de l’article [14]. D’autre part, on peut suivre le travail de Abdelkader In-tissar [16] pour étudier la chaoticité associée a l’espace F2

Γ,χ(C2). La transformation de

Bargmann est traitée dans le Chapitre 4, on peut aussi faire appel à la transformation de Fourier-Wigner. La construction d’une base de l’espace fonctionnel FΓ,χ2,ν,H(V)nous per-mettra de construire les états cohérents associés.

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Contents

1 Introduction 13

2 One dimension automorphic functions associated to maximal rank discrete

sub-groups of(C,+) 15

2.1 Notation and Definition . . . 15

2.2 On some functional vector spaces of automorphic function associated to the triplet(Γ, χ, ν) . . . 16

2.3 Hilbert structure, of the spacesOν Γ,χ(C)andL2,νΓ,χ(C) . . . 24

3 Holomorphic(Γ, χ)-theta function associated to rank one of discret subgroups in high dimension 28 3.1 Introduction and statement of main result . . . 28

3.2 Notation and preliminaries . . . 30

3.3 Background on the vector spaceF2 Γ,χ(C2) . . . 31

3.4 Main results . . . 34

3.5 Concluding remarks . . . 40

4 Holomorphic theta functions associated to rank r isotropic discrete subgroups of(Cg,+) 44 4.1 Introduction . . . 44

4.2 Preliminaries on isotropic discrete subgroups of(Cg,+) . . . 46

4.3 Statement of main results . . . 48

4.4 Proofs of main results . . . 51

5 The Segal Bargmann transform 57 5.1 Classical Segal-Bargmann transformB . . . 57

5.2 Connection of F2 Γ,χ(C2)toL2L,χ(R2)through the Segal-Bargmann transform 60 5.3 Connection of F2 Γr(C g)toL2 Γ(Rg)through the Segal-Bargmann transform 69 6 Appendix 72 6.1 theta function . . . 72

6.2 Gaussian integral . . . 75 11

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Contents 12 6.3 General Gaussian integral . . . 76

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Chapter

1

Introduction

Classical theta functions are holomorphic functions onC satisfying a quasi-periodic prop-erty

θ(z+γ) = eγ(z)θ(z), γ∈ Γ, z∈ C, (1.1)

with respect to given lattice Γ in the complex plan C, i.e., a discrete subgroup of (C,+)

which spans C = R2 as a real vector space and eγ is a holomorphic map on C. In the

contexte of algebraic geometry, they are interpreted as sections of certain line bundles on complex torusC/Γ.

The basic theta series

θα,β(z|τ) =

n∈Z

e(n+α)2τ+2iπ(n+α)(z+β)

α, βR

are basic example and satisfying the funtional equation.

θα,β(z|τ)e

iπα2

τ2iπl(z+β) =

θ(z+ατ+β|τ)

Here τ is confined to the upper half-plane H = {τC;=(τ) >0}, andΓ =+Z. The

function θα,β(z|τ) was introduced by Riemann [34] in connection with Riemann surfaces

and is called the Riemann (modified) theta function (or also first order theta function) with characteristic[α, β]. The case of integer characteristics leads to the four classical theta

series. An overview of the properties of θα,β(z|τ)can be found in Refs.[34, 22], and [20, 21].

The most general form of the Riemann theta function defined above was considered by Wirtinger[33].

Hereafter, we endow C with its standard hermitian structure defined through H(z, w) = zw¯ =z1w¯1+z2w¯2+ · · · +zgw¯g; (z, w) ∈CCg.

and denote by dλ(z) =dxdy; z =x+iy∈ C, the Lebesgue measure on C. We emphasize

the case of theta functions associated to an arbitrary discrete subgroupΓrof(C,+). More

precisely, we have to consider, among others functional spaces, the vector spaceOν,H

Γr(C

g)

of all holomorphic functions onC, f ∈ Oν,H

Γr(C

g), displaying the functional equation

f(z+γ) = χ(γ)eνH(z+

γ

2)f(z); zCg, γ∈ Γr. (1.2) 13

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14 for every z ∈ C and γ∈ Γr, where ν >0 and χ is a given map defined onΓrwith values in

the unit circle U(1) = {z∈ C,|z| =1}. LetΛ(Γr)be a fundamental domain, representing

inC=R2gthe orbital groupCg/Γr endowed with the quotient topology, and perform the

spaceFΓ2,ν r(C g)of functions f belonging toOν,H Γr(C g)such that kfk2Γ r := Z Λ(Γr) |f(z)|2e−νH(z,z)(z) < +∞. (1.3)

An other functional space to be considered is L2,ν(Cg)consisting of complex valued func-tions onCgdisplaying 1.2 and that are e−νH(z,z)dλ-square integrable onΛ(Γ

r). Three case

are envisaged:

(i) The trivial case ofΓr = {0}.

(ii) The case ofΓr = 1+ · · · +r for some spanned by R-linearly independent

vec-tors ω1,· · · , ωr ∈ Cg.

(iii) The maximal caseΓr is a latticeΓr =Γ2g = 1+ · · · +2g. This follows, sinceΓr

can be viewed as a Z-module of rank r =0, 1, 2,· · · , 2g.

For the trivial caseΓ0 = {0}, we haveΛ(Γ0) =Cgand it is obvious that 1.2 implies χ=1.

Therefore,FΓ2,ν

0(C

g)reduces to the usual Fock-Bargmann space consisting of eνH(z,z)

dλ-square integrable entire functions.

B2,νH (Cg):= {f holomorphic function onCg;

Z

Λ(Γr)

|f(z)|2e−νH(z,z)(z) < +}.

For the case r = 2g, the fundamental domainΛ(Γ2g) is compact and the functionel space FΓ2,ν

2g(C

g)is non trivial under de cocycle condition

χ(γ+γ0) =χ(γ)χ(γ0)eiνE(γ,γ

0)

(RDQ), for every γ, γ0 ∈ Γ2g. In this case F2

Γ,χ(Cg)is a finite dimensional vector space. It

dimen-sion is given by [13]

dimFΓ,χ2 (Cg) = ν

π

g

vol(Λ(Γ2g))

where vol(Λ(Γ2g))is the volume of the complex torusCg/Γ2g.

The general problem whenΓr is arbitrary discret subgroup of any rank 1 ≤r ≤2g−1 is

still for from being undestood. A first systematic study is done in [[14]] for g =r=1. In this thesis we consider the case of arbitrary isotropic discret subgroups in high dimen-sion.

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Chapter

2

One dimension automorphic functions

associated to maximal rank discrete

subgroups of

(

C,

+)

In this chapter, we introduce and discuss some basic properties of some functional of automorphic functions, associated to given discrete subgroup inC=R2of maximal rank. To this end, we begin by fixing notations and giving necessary and sufficient conditions to theses spaces to be non-zero.

2.1

Notation and Definition

LetC be the complex space endowed with its standard Hermitian form H(z, w) = zw and let E(z, w) = =mhz, wibe the associated symplectic form. We denote byM(C) = M(R2)

the space of Borel-Lebesgue measurable functions onC =R2 . LetΓ be a fixed lattice of space C = R2 (i.e Γ be a discrete subgroup in C = R2 of maximal rank). Let Λ(Γ) be a fundamental domain, representing in C = R2 the orbital group C/Γ endowed with the quotient topology. We are interested in the space of functions belonging to M(C) and displaying the functional equation

f(z+γ) = χ(γ)eνH(z+

γ

2)f(z); ∀∀zC,γ∈ Γ, (2.1) where the notation ”∀∀” is used, to mean "for almost all", ν > 0 and χ is a given map defined onΓ with values in the unit circle U(1) = {γC;|γ| =1}.

Definition 2.1. A function f on C satisfies the functional equation (2.1) is called automorphic function onR2associated to the triplet(Γ, χ, ν).

Denote by Df =Λ(Γ)\Nf the set where f is defined inΛ(Γ)with Nf is a negligible set

inΛ(Γ). Moreover

Df ,Γ =C\Nf ,Γ,

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2.2. On some functional vector spaces of automorphic function associated to the triplet

(Γ, χ, ν) 16

the set where f is well defined with

Nf ,Γ = [

γ∈Γ

(Nf +γ).

a set Nf ,Γis negligible set as countable union of negligible sets.

2.2

On some functional vector spaces of automorphic

func-tion associated to the triplet

(

Γ, χ, ν

)

We consider the functional spaceMν

Γ,χ(C)of all measurable functions onC satisfying the

functional equation (2.1)

Mν

Γ,χ(C) = {f ∈ M(C)/ f(z+γ) = χ(γ)eνH(z+

γ

2)f(z),∀∀z ∈C,γ∈ Γ}. The notation ∀∀z ∈ C designed ∀z ∈ Df ,Γ = C\Nf ,Γ . Added toMν

Γ,χ(C), we consider

the functional vector space Oν

Γ,χ(C) of all holomorphic functions f on C satisfying the

functional equation (2.1) for every z∈ C, and every γ∈ Γ, i.e, Oν

Γ,χ(C):= {f ∈ O(C)/ f(z+γ) =χ(γ)eνH(z+

γ

2)f(z) ∀z∈ Cγ ∈Γ},

as well as the vector space CΓ,χ∞,ν(C) of all functions f ∈ C∞(C) satisfying the functional equation (2.1) for every z ∈ C, and every γ∈ Γ;i.e,

CΓ,χ∞,ν(C) := {f ∈ C∞(C)/ f(z+γ) = χ(γ)eνH(z+

γ

2)f(z) ∀z ∈Cγ∈ Γ}. In parallel to the vector spaceMν

Γ,χ(C), we define the vector spaceGΓ,χν (C)of all

func-tions f ∈ Mν

Γ,χ(C), such that there exists a constant C > 0 verified |f(z)| ≤ Ce

ν 2|z|2, for every z ∈ Df ,Γ. That is Gν Γ,χ(C):= {f ∈ MνΓ,χ(C)/∃C >0 |f(z)| ≤ Ce ν 2H(z,z) ∀∀z ∈C} We consider also the vector space L2,νΓ,χ(C)of all functions f ∈ Mν

Γ,χ(C)satisfying Z

Λ(Γ) f

(z)e−νH(z,z)dm(z) < +∞,

where dm(z) = dxdy; z=x+iy∈C, is the Lebesgue measure on C,

L2,νΓ,χ(C) := {f ∈ Mν

Γ,χ/ Z

Λ(Γ) f

(z)e−νH(z,z)dm(z) < +}.

The relationship between the vector spacesCΓ,χ∞,νandGν

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2.2. On some functional vector spaces of automorphic function associated to the triplet

(Γ, χ, ν) 17

Lemma 2.2. Let f ∈ CΓ,χ∞,ν(C), then there exists a constant C >0 such that

|f(z)| ≤Ceν2H(z,z) for every z∈ C. That is CΓ,χ∞,ν(C) ⊂ Gν

Γ,χ(C).

Proof. Let f ∈ CΓ,χ∞,ν(C). We consider the function g defined by : g(z) = e−ν2 H(z,z)f(z) for every z ∈C. As f ∈ CΓ,χ∞,ν, then f satisfies the functional equation. It follows,

g(z+γ) =e −ν 2 |z+γ|2f(z+γ) =e−ν2 |z+γ|2χ(γ)eν2|γ|2+νzγf(z) =χ(γ)e −ν 2 |z|2f(z)e−ν2 (|γ|2++−|γ|2−2zγ) =χ(γ)g(z)e −νi 2 ()/2i

=χ(γ)g(z)e−=m() for every z∈ C, and every γ∈ Γ.

Using|χ(γ)| =1 and=m() ∈R, we have

|g(z+γ)| = |g(z)|, for every z∈ C, for every γ ∈Γ

,i.e Then the function z −→ |g(z)|isΓ−periodic. Since this function is continuous on C, we deduce that the function z −→ |g(z)|is bounded on C. Then, there exists a constant C >0, such that

|g(z)| ≤C , for every z∈ C and |f(z)| ≤Ceν2|z|2 for every zC. Then, we deduceCΓ,χ∞,ν(C) ⊂ Gν

Γ,χ(C).

Hence the following result is obtained

Corollary 2.3. We have the following algebraic inclusions:

Oν

Γ,χ(C) ⊂ CΓ,χ∞,ν(C) ⊂ GΓ,χν (C) ⊂ L2,νΓ,χ(C) ⊂ MνΓ,χ(C)

Definition 2.4. We say that the triplet(Γ, χ, ν)satisfies the(RDQ)condition if

χ(γ1+γ2) = χ(γ1)χ(γ2)eiνE(γ12) for every γ1, γ2 ∈Γ,

where E(γ1, γ2) = =m(γ1γ2)is the imaginary part of γ1γ2.

Lemma 2.5. If the triplet(Γ, χ, ν)satisfies the(RDQ)condition, then the map χ has the following properties:

χ(0) =1 and χ(−γ) = χ(γ)−1 =χ(γ) , for every γ ∈Γ.

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2.2. On some functional vector spaces of automorphic function associated to the triplet

(Γ, χ, ν) 18

Proof. We remark that

χ(0) = χ(0+0) = χ(0)χ(0)e=m(0) =χ(0)2.

Then χ(0)(1−χ(0)) =0 , and since

χ(0) ∈ U(1) = {λC/|λ| = 1},

we deduce that χ(0) = 1. On the other hand, we have

χ(γγ) = χ(γ)χ(−γ)e=m(γ(−γ)) =χ(γ)χ(−γ)

As γ(−γ) = −γγ= −|γ|2∈ R, and χ ∈ U(1), we have

1=χ(γ)χ(−γ) = |χ(γ)|2 =χ(γ)χ(γ).

Then

χ(γ) = χ(−γ) and χ(γ)−1 =χ(γ) , for all γ∈ Γ.

Let γ1, γ2 ∈ Γ, by interchanging the roles of γ1and γ2in the(RDQ)condition

χ(γ1+γ2) = χ(γ1)χ(γ2)e=m(γ1γ2). (*)

On the other hand, we have

χ(γ2+γ1) = χ(γ2)χ(γ1)e=m(γ2γ1). (**).

As=m(γ1γ2) = −=m(γ2γ1), we deduce from(∗)and(∗∗)that

e=m(γ1γ2) =e=m(γ2γ1) =e=m(γ1γ2). This yields

e2iν=m(γ1γ2) =1 and

ν=m(γ1γ2) ∈πZ.

The following proposition gives a sufficient and necessary condition on the triplet

(Γ, χ, ν)to the spaceMν

Γ,χ(C)be a nonzero space. Namely, we have

Proposition 2.6. Mν

Γ,χ(C) 6= {0}if only if the triplet(Γ, χ, ν)satisfies the(RDQ)condition.

Proof. For f ∈ Mν

Γ,χ(C) \ {0}, there exists z0 ∈ Df ,Γ ⊂ C such as f(z0) 6= 0. Then, we

have, f(z0+γ1+γ2) = f((z0+γ1) +γ2) =χ(γ2)e ν 2|γ2|2+ν(z0+γ1)γ2f(z 0+γ1) =χ(γ2)e ν 2|γ2|2+ν(z0+γ1)γ2 χ(γ1)e ν 2|γ1|2+ν(z0γ1)f(z 0) =χ(γ1)χ(γ2)e ν 2(|γ1|2+|γ2|2)+ν(z0+γ1)γ2+νz0γ1f(z 0). (?)

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2.2. On some functional vector spaces of automorphic function associated to the triplet

(Γ, χ, ν) 19

On the other hand, we have:

f(z0+γ1+γ2) = χ(γ1+γ2)e

ν

2|γ1+γ2|2+νz0(γ1+γ2)f(z

0). (??)

By combining(?)and(??). Keeping in mind that f(z0) 6=0, we conclude that

χ(γ1+γ2)e ν 2(γ1γ2+γ1γ2) = χ(γ1)χ(γ2)eνγ1γ2. Hence χ(γ1+γ2) = χ(γ1)χ(γ2)e− ν 2(γ1γ2+γ1γ2)+νγ1γ2

=χ(γ1)χ(γ2)eiνE(γ12), for every γ1, γ2 ∈Γ.

For the converse, can be proved using the following lemma.

Lemma 2.7. Suppose that the (RDQ) condition holds. Let ψ be a non-zero measurable function on C such as supp(ψ) ⊂ Dψ,Γ. We denote by PΓ,χν ψ the (Γ, χ) Poincaré periodization of the ψ

given by [Pν Γ,χψ](z) =

γ∈Γ χ(γ)e−νH(z+ γ 2)ψ(z+γ).

which is also equivalent to

[Pν Γ,χψ](z) =

γ∈Γ χ(γ)e−νH(z+ γ 2)ψ(z−γ). Then, we have i) The functionPν

Γ,χψ is a measurable function onC andPΓ,χν 6≡0.

ii) The functionPν

Γ,χψ belongs to the spaceMνΓ,χ(C).

Proof. i) The function z 7→ z−γ is continuous function on Λ(Γ). Then it is a

mea-surable function on Λ(Γ). Moreover ψ is a measurable function on C, so that the function

z7→ ψ(z−γ) is measurable function on Λ(Γ).

Note also that the function z 7→ e−ν2|γ|2+νzγis measurable on Λ(Γ). Thus the func-tion:

z7→ χ(γ)e−

ν

2|γ|2+νzγψ(z−γ) is measurable on Λ(Γ). Since Γ is a lattice on C, then Γ is countable , and hence Pν

Γ,χψ is a measurable on

Λ(Γ). As the translated cell Λ(Γ)forms a partition ofC, then

Pν

Γ,χψ is a measurable function on C.

On the other handΓ is discrete subgroup, and Dψ,Λ(Γ) ⊂Λ(Γ). Then for z ∈ Dψ,Λ(Γ),

we have

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2.2. On some functional vector spaces of automorphic function associated to the triplet (Γ, χ, ν) 20 Then ψ(z−γ) =0, so [Pν Γ,χψ](z) =χ(0)ψ(z) = ψ(z) (χ(0) = 1). It follows Pν Γ,χψ/Dψ(Γ) =ψ6≡0. ii) For every γ∈ Γ and z∈ C, we have

[Pν Γ,χψ](z+γ) =

h∈Γ χ(h)e− ν 2|h|2−ν(z+γ)hψ(z+γ−h) =

k∈Γ χ(γ+k)e− ν 2|γ+k|2−ν(z+γ)γ+kψ(z−k); [h=γ+k]. However, according to the (RQD)condition ,we have

χ(γ+k) = χ(γ)χ(k)eiνE(γ,k). Then [Pν Γ,χψ](z+γ) =χ(γ)e ν 2|γ|2+νzγ

k∈Γ χ(k)e− ν 2|k|2+νzkψ(z−k) =χ(γ)eνH(z+ γ 2)[PΓ,χν ψ](z). We conclude that Pν Γ,χψ ∈ MνΓ,χ(C).

This result proves that the vector spaceMν

Γ,χ(C)is nonzero.

Proposition 2.8. . The following assertions are equivalents i) Oν Γ,χ(C) 6= {0} ii) CΓ,χ∞,ν(C) 6= {0} iii) Gν Γ,χ(C) 6= {0} iv) LΓ,χ(C) 6= {0} v) Mν Γ,χ(C) 6= {0}

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2.2. On some functional vector spaces of automorphic function associated to the triplet

(Γ, χ, ν) 21

Proof. According to the Corollary 2.3 and the Proposition 2.6, we have i) =⇒ii) =⇒iii) =⇒iv) =⇒v) =⇒vi).

We need to prove vi) =⇒ i). To this end, one has to proceed differently since we do not dispose with holomorphic functions with compact support. For this the following theorem is used

Theorem 2.9. Suppose that the (RDQ) condition holds and let Kν

Γ,χ(z, w)the function defined on

C×C by: Kν Γ,χ(z, w) = ν π  eνH(z,w)

γ∈Γ χ(γ)e− ν 2|γ|2+ν(H(z,γ)−H(γ,w)). We have:

i) For every z, w∈ C, we have

Kν

Γ,χ(z, w) =KνΓ,χ(w, z).

ii) For every γ1, γ2 ∈ Γ and z, w∈ C, we have

Kν Γ,χ(z+γ1, w+γ2) = χ(γ1)eνH(z+ γ1 2 1)Kν Γ,χ(z, w)χ(γ2)eH(γ2,z+ γ2 2 ). In particular, the function z 7−→Kν

Γ,χ(z, w)belongs toOΓ,χν (C)for every w ∈C fixed.

iii) We denote by vol(Γ)the Lebesgue volume of the fundamental domain of the latticeΓ. Thus, we have : Z Λ(Γ)K ν Γ,χ(z, z)e−νH(z,z)dm(z) = ν π  vol(Γ). Proof. i) For given z, w ∈C, we have

Kν Γ,χ(z, w) = ν π  eνzw

γ∈Γ χ(γ)e− ν 2|γ|2+ν() =ν π  eνzw

γ∈Γ χ(−γ)e− ν 2|γ|2+ν() =ν π  eνwz

γ0∈Γ χ(γ0)e− ν 2|γ|2+ν(0−γ0z) =Kν Γ,χ(w, z).

ii) Starting from

Kν Γ,χ(z, w) = ν π  eνzw

γ∈Γ χ(γ)e− ν 2|γ|2+ν().

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2.2. On some functional vector spaces of automorphic function associated to the triplet (Γ, χ, ν) 22 We get Kν Γ,χ(z+γ, w) = ν π  eν(z+γ)w

γ0∈Γ χ(γ 0 )e−ν2|γ 0 |2+ν((z+γ)γ00) =ν π  eν(zw+γw)

γ0∈Γ χ(γ 0 )e−ν2|γ 0 |2+ ν(0+γγ0−0) =ν π  eνzw

γ0∈Γ χ(γ 0 )e−ν2|γ 0 |2+ ν(0+γγ0−w(γ0−γ)). By replacing γ” =γ 0 −γ, and then γ 0 =γ”+γ. By (RDQ) condition, we get χ(γ 0 ) = χ(γ”+γ) =χ(γ”)χ(γ)e=m(γγ). It follows then : Kν Γ,χ(z+γ, w) = ν π  eνzw

γ”∈Γ χ(γ”)χ(γ)e=m(γγ) e−ν2|γ”+γ|2+ν(”+γ+γγ”+γ”). As |γ”+γ|2 = |γ”|2+ |γ|2+γγ+γγ”, We obtain =m(γγ) = γγγγ 2i = γγγγ” 2i . A direct computation shows that

ν 2|γ+ γ|2+ν(”+γ+γγ”+γ”) is equal to −ν 2(|γ|2− | γ|2) −=m(γγ) +νzγ+ν(”−”). Therefore, we get Kν Γ,χ(z+γ, w) = ν π  eνzγ χ(γ)e ν 2|γ|2

γ”∈Γ χ(γ”)e− ν 2|γ”|2+ν(”−”) =χ(γ)e ν 2|γ|2+νzγKνΓ,χ(z, w). Thus Kν Γ,χ(z+γ1, w+γ2) = KΓ,χν (w+γ2, z+γ1) =χ(γ2)e ν 2|γ2|2+νwγ2Kν Γ,χ(w, z+γ1) =χ(γ2)e ν 2|γ2|2+νwγ2KΓ,χν (z+γ1, w)

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2.2. On some functional vector spaces of automorphic function associated to the triplet (Γ, χ, ν) 23 and hence Kν Γ,χ(z+γ1, w+γ2) = χ(γ1)e ν 2|γ1|2+νzwKν Γ,χ(z, w) =χ(γ1)e ν 2|γ1|2+νzwKν Γ,χ(z, w)χ(γ2)e ν 2|γ2|2+νwγ2. In particular Kν Γ,χ(z+γ1, w) = χ(γ1)e ν 2|γ1|2+νzwKν Γ,χ(z, w).

In addition the function z 7−→Kν

Γ,χ(z, w)belongs toOΓ,χν , for every fixed w ∈ C.

iii) We have Z Λ(Γ)K ν Γ,χ(z, z)e−ν|z| 2 dm(z) = Z Λ(Γ) ν π  eνzzeνzz

γ∈Γ χ(γ)e− ν 2+ν() =ν π 

γ∈Γ χ(γ)e− ν 2|γ|2 Z Λ(Γ)e 2iν=m(zw) dm(z)  . Using the following lemma

Lemma 2.10. If the (RDQ) condition holds, then for all γ ∈ Γ8{0}, we have

Z Λ(Γ)e 2iνE(w,γ) dm(w) = 0. Consequently Z Λ(Γ)K ν Γ,χ(z, z)e−νH(z,z)dm(z) = ν π  χ(0)vol(Γ) = ν π  vol(Γ). According to ii), we have the function

z7−→Kν

Γ,χ(z, w) belongs to OνΓ,χ for all fixed w∈ C.

Moreover, this function is nonzero. Indeed, if

∀z∈ C Kν

Γ,χ(z, w) =0 for every fixed w ∈C.

Then Kν

Γ,χ(w, w) =0 for every fixed w ∈C , thus KνΓ,χ(z, z) = 0 for every z∈C.

So Z

Λ(Γ)K

ν

Γ,χ(z, z)e−νH(z,z)dm(z) =0.

This is absurd with iii). We conclude that the spaceOν

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2.3. Hilbert structure, of the spacesOν

Γ,χ(C)andL2,νΓ,χ(C) 24

2.3

Hilbert structure, of the spaces

O

ν

Γ,χ

(

C

)

and

L

2,νΓ,χ

(

C

)

We assume in this part that the (RQD) condition holds, otherwise the spacesOν

Γ,χ(C),CΓ,χ∞,ν(C)

andL2,νΓ,χ(C)will be reduced to{0}. In the sequel, we try to build a norm on these spaces. To do this, we start with the following result:

Lemma 2.11. Let f1, f2∈ CΓ,χ∞,ν(C), then the function h defined onΛ(Γ)by

h(z) = f1(z)f2(z)e−νH(z,z)

isΓ-periodic ,i.e,

h(z+γ) = h(z) for every z∈ Λ(Γ) and every γ ∈ Γ.

Proof. For every z ∈ Λ(Γ)and every γ ∈Γ, we have

h(z+γ) = f1(z+γ)f2(z+γ)e−ν|z+γ| 2 =χ(γ)χ(γ)eν|γ| 2+ ν(+)eν|z+γ|2f 1(z)f2(z) = f1(z)f2(z)e−ν|z| 2 (|χ(γ)| =1) =h(z).

Lemma 2.11 show that the application

(f1, f2) 7−→ Z

Λ(Γ) f1(z)f2(z)e

νH(z,z)dm(z)

is a norm onCΓ,χ∞,ν(C). Moreover, we have the following result Lemma 2.12. kfk2

Γ := R

Λ(Γ)|f(z)|2e

νH(z,z)dm(z)is the norm on the spaceC∞,ν

Γ,χ(C).

Proposition 2.13. i) (CΓ,χ∞,ν(C),k.kΓ)is a préhilbertian vector space of infinite dimension. ii) (CΓ,χ∞,ν(C),k.kΓ)is dense in the vector spaceL2,νΓ,χ(C).

Proof. i) For simplification, we take a square of side 1 as the fundamental domain Λ(Γ). We divide this square into two equal rectangles, rect1 and rect2 Let ψ1 ∈ CΓ,χ∞,ν

that ψ1 6≡0 and supp(ψ1) ⊂ rect1, and let ψ2 ∈ CΓ,χ∞,νsuch as ψ1 6≡0 and supp(ψ2) ⊂

rect2.Define f1and f2by f1(z) = [PΓ,χν ψ1](z) =

γ∈Γ χ(γ)e− ν 2|γ|2+νzγψ1(z−γ), and f2(z) = [PΓ,χν ψ2](z) =

γ∈Γ χ(γ)e− ν 2|γ|2+νzγψ2(zγ).

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2.3. Hilbert structure, of the spacesOν

Γ,χ(C)andL2,νΓ,χ(C) 25

Then, one can check that f1 ∈ CΓ,χ∞,ν(C) and f2 ∈ CΓ,χ∞,ν(C). Moreover supp(ψ1) and

supp(ψ2)are disjoint, and the family(f1, f2)is linearly independent in theCΓ,χ∞,ν(C).

Similarly, if we divide the square Λ(Γ) by three equal rectangles, we can construct

ψ1, ψ2 and ψ3 the CΓ,χ∞,ν not identically zero, such as supp(ψ1) ⊂ rect1, supp(ψ2) ⊂

rect2and supp(ψ3) ⊂rect3. Then the family(f1, f2, f3)is linearly independent of the

spaceCΓ,χ∞,ν(C).

Doing so we can construct, for every n ∈ N∗, a family of n independent functions

(f1, f2, ..., fn)inCΓ,χ∞,ν(C). We conclude thatCΓ,χ∞,ν(C)has of infinite dimension.

ii) For the density of the(CΓ,χ∞,ν(C),k.kΓ)in the spaceL2,νΓ,χ(C), we will use the following theorem

Theorem 2.14. Let Ω be an open of R2. The space CC∞(Ω) of all the functions of class

C∞ on Ω with compact support in C, is dense in the vector space L2,ν(Ω, dλ(z)), where

(z) = e−ν|z|2dm(z)

Let f ∈ L2,νΓ,χ(C), then f ∈ L2,ν(Ω, dλ(z)), with Ω an open of the space C such as

Λ(Γ) ⊂ Ω. Using Theorem2.14, there exists a sequence(fn)n≥0 of the spaceCC∞(Ω)

such as

kfn(z) − f(z)kΓ −→ 0 when n −→ +∞.

We build the sequence(gn)n≥0such as

gn = [PΓ,χν fn] for every n∈ N.

Then gn is C∞ on the space C , for every nN. By Lemma 2.7, the sequence gn

satisfies the functional equation (4.14), so

gn ∈ CΓ,χ∞,ν for every n∈ N.

Moreover gn = fn on the compact support of fn, for every n∈ N.

So ,we have

kgn(z) −f(z)kΓ −→ 0 for every n−→ +∞.

We conclude that(CΓ,χ∞,ν,k.kΓ)is dense in the spaceL2,νΓ,χ(C). We endow the spacesL2,νΓ,χ(C)andOν

Γ,χ(C)with the normk.kΓ, and we show that they

are Hilbert space. Recall first that for given open setΩ⊂C, the space L2,ν(Ω, dλ(z)) = {f ∈ M(C)/

Z

Ω|f(z)|

2(z) < +}

is a Hilbert space, with dλ(z) = e−ν|z|2dm(z). So, we have the following result

Proposition 2.15. (L2,νΓ,χ(C),k.kΓ)is a Hilbert space. Proposition 2.16. (Oν

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2.3. Hilbert structure, of the spacesOν

Γ,χ(C)andL2,νΓ,χ(C) 26

For the proof of Proposition 2.15 and Proposition 2.16, we establish the following lemma

Lemma 2.17. LetΩ be an open bounded set of C =R2. Then, there exists an integer α(Ω) ≥ 1 and a real number β(Ω) >0 such as

β(Ω)Max{|f(z)|/z∈ Ω} ≤ Z Ω|f(z)| 2eν|z|2dm(z) ≤ α(Ω) Z Λ(Γ)|f(z)| 2eν|z|2dm(z) for every f ∈ L2,νΓ,χ(C).

Proof. Notice first thatΩ is a compact of C=R2forΩ being a bounded open set of C. On the other hand, we have

C = [

γ∈Γ

(Λ(Γ) +γ).

So there is a finite collection of open sets coveringΩ, and there exists α(Ω) ≥ 1 such as Ω⊂ [ 1≤i≤α(Ω) (Λi(Γ) +γi). Hence Z Ω|f(z)| 2 e−ν|z|2dm(z) ≤ α(Ω)

i=1 Z Λi(Γ)+γi |f(z)|2e−ν|z|2dm(z), and therefor Z Ω|f(z)| 2eν|z|2dm(z) ≤ α(Ω) Z Λ(Γ)|f(z)| 2eν|z|2dm(z).

Let(fn)n≥0be a Cauchy sequence ofL2,νΓ,χ(C). Then, Z

Λ(Γ)

|fn(z) − fm(z)|2e−ν|z|

2

dm(z) −→0 when n, m−→ +∞.

Making use of the previous Lemma, we deduce that

Z

Ω|fn(z) − fm(z)|

2eν|z|2dm(z) −→0 when n, m −→ +∞.

Then (fn)n≥0 is a Cauchy sequence in the complete space L2,ν(Ω, dµ). Then there exists

fΩ ∈ L2,ν(Ω, dµ) such as (fn)n≥0 converges to fΩ in the space L2,ν(Ω, dµ). For the

con-vergence in C, we take Ω0 a bounded open the C such as Ω ⊂ Ω0, and repeat the same process, so there is a function g0 ∈ L2,ν(Ω, dµ)such as

(fn)n≥0 converges to gΩ0 ∈ L2,ν(Ω, dµ) with g0/Ω= f.

So on, there is a function f such that (fn)n≥0 converges to fΩ in L2,ν, and consequently,

there is a sequence(fnk)k≥0the(fn)n≥0such as

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2.3. Hilbert structure, of the spacesOν

Γ,χ(C)andL2,νΓ,χ(C) 27

as k tends to infinity. Since fn(z+γ) =χ(γ)e

ν

2|γ|2+νzγfn(z) for almost all z ∈ C and every γ∈ Γ, we deduce that

fnk(z+γ) =χ(γ)e

ν

2|γ|2+νzγfn

k(z) for almost all z ∈C and every γ ∈ Γ. Thus by tending k to infinity, we obtain

f(z+γ) =χ(γ)e

ν

2|γ|2+νzγf(z) for almost all z∈ C and every γ ∈ Γ, so that f ∈ L2,νΓ,χ(C). We conclude that(LΓ,χ2,ν(C),k.kΓ)is a Hilbert space.

Let(fn)n≥0is a Cauchy sequence of(OΓ,χν (C),k.kΓ). We have

β(Ω)Max{|f(z)|/z ∈Ω} ≤ α(Ω)kf(z)kΓ.

Then(fn)n≥0is a Cauchy sequence, for the infinite norm inC. So (fn)n≥0 converges to a holomorphic function f .

On the other hand, it can be shown by the same method that f(z+γ) = χ(γ)e

ν

2|γ|2+νzγf(z) for every zC and every γ∈ Γ, so that f ∈ Oν

Γ,χ(C), so(OνΓ,χ(C),k.kΓ)is a Hilbert space.

Corollary 2.18. Oν

Γ,χ(C)is a closed space in the space(L2,νΓ,χ(C),k.kΓ).

Proof. We have Oν

Γ,χ(C) ⊂ L2,νΓ,χ(C). Moreover (OνΓ,χ(C),k.kΓ)is a complete space. Then Oν

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Chapter

3

Holomorphic

(

Γ, χ

)

-theta function

associated to rank one of discret

subgroups in high dimension

We give concrete description of the (1, χ)-theta Fock-Bargmann space consisting of

holomorphic automorphic functions associated to given discrete subgroup inC2 of rank one and given character χ. It looks like a tensor product of a theta Fock-Bargmann space on the complex planeC and the classical Fock-Bargmann space on C. Moreover, we con-struct an orthonormal basis and we give explicit expression of its reproducing kernel func-tion in terms of Riemann theta funcfunc-tion.

3.1

Introduction and statement of main result

Automorphic functions (or abelian functions [34, 20, 22, 4]) on the g-dimensional complex vector space V =Cg, related to given cocompact discret subgroupΓ in(R2g,+), given au-tomorphic factor j and given map χ onΓ such that|χ(γ)| =1, are holomorphic functions

satisfying a functional equation of type

f(u+γ) =χ(γ)j(γ, u)f(u); u∈ V, γ∈Γ. (3.1)

The existence problem for such functions requires a cocycle condition that must be satis-fied by χ (see for example [6, 35]). In this case they can be interpreted as sections of certain holomorphic line bundle on the g-dimensional complex torusCg/Γ. They play important roles in many fields of mathematics and technology, especially in algorithmic number the-ory, cryptography and coding thethe-ory, as well as in quantum field thethe-ory, including string theory.

However, whenΓ is non cocompact (i.e., Cg/Γ is not compact) less things are known. The case ofΓ = {0}leads to the usual standaidre structure onCg(see Remark (3.4) below). Precisely, associated to the standard Hermitian scalar product H(z, w) on Cg and given discrete subgroup Γr of rank r (generated by 2g R-linearly independent vectors in Cg =

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3.1. Introduction and statement of main result 29 R2g), we consider the space F2,ν

Γr(C

g) of holomorphic functions on Cg displaying the

functional equation

f(z+γ) =χ(γ)eH(z+

γ

2)f(z); z∈ Cg, γ∈ Γr, and such that

kfk2Γ r := Z Λ(Γr)|f(z)| 2e−H(z,z) (z) < +∞,

where Λ(Γr) =R2n/Γr is a fundamental cell and dλ(z) denotes the usual Lebesgue

mea-sure onCg(see Section 2 for more precisions).

Recently, A. Ghanmi and Ahmed Intissar have investigated in [14] the case of rank one with g = 1 andΓ = Z. It is shown there that the(Z, χ)-theta Fock-Bargmann space

FΓ,χ2,ν(C)is of infinite dimension for χ being a character. Moreover, an orthonormal basis of such Hilbert space is constructed and the explicit expression of its reproducing kernel is given in terms of the modified theta function θα,β(z|τ)defined for any complex number

z and any τ in the upper half-plane by ([34, 22]):

θα,β(z|τ):=

n∈Z

e(n+α)2τ+2iπ(n+α)(z+β)

; α, βR. (3.2)

In [16], Abdelkader Intissar has studied chaoticity of a shift operator on the obtained basis and gave some applications in the context of Gribov operator.

In this paper, we generalize the results in [14] to any g ≥ 2 and give a complete de-scription of FΓ,χ2,ν(Cg). To not cumbersome with additional notations, we state here our main results for g=2.

Theorem 3.1. Let Γ = 1 for given nonzero vector ω1 ∈ C2 and set ν = H(w1, w1). Let

χ: Γ−→U(1) = {λC; |λ| =1}. Then, we have

i) F2

Γ,χ(C2)is a nonzero vector space if only if χ is a character ofΓ.

ii) An othonormal basis ofF2

Γ,χ(C2)is given by ψm,n(z1, z2) =:  2ν π 1/4 νn+1 πn! 1/2 e−π2ν (m+α) 2 eν2z12+2iπ(α+m)z1zn 2

for varying m∈ Z and nZ+. iii) F2

Γ,χ(C2)is a Hilbert space.

iv) The Hilbert spaceF2

Γ,χ(C2)possesses a reproducing kernel given by

K((z1, z2);(w1, w2)) = √ 2ν π 3/2 eν2(z21+w12) θα,0  z1−w1 2iπ ν  eνz2w2, where θα,0is as in (3.2).

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3.2. Notation and preliminaries 30

Remark 3.2. The exact statement of this result for g >2 will be given in the last section. Its proof is similar to the one that we will provide when g =2.

The organisation of the paper is as follows. In Section 2, we fix notations and collect some elementary remarks on the vector space F2

Γ,χ(Cg). In Section 3, we restrict ourself

to g = 2. Mainly, we show that the vector space F2

Γ,χ(C2) is nontrivial if and only if χ is

a character. In Section 4, we prove our main results summarized in Theorem (3.1) above. We end the paper by stating the result for any g ≥2 and some concluding remarks.

3.2

Notation and preliminaries

By (Cg, H); g ≥ 1, we denote the g-complex vector space equipped with its standard Hermitian scalar product

H(z, w) =zw¯ =z1w¯1+z2w¯2+ · · · +zgw¯g,

whose the associated symplectic form is E(z, w) = =mhz, wi. Let Γr be a discrete

sub-group of rank r in (R2g,+). It can be viewed as aZ-module of rank r =0, 1,· · · , 2g, that is Γr = 1+ · · · +r for some R-linearly independent vectors ω1,· · · , ωr ∈ Cg. A

fundamental cell Λ(Γr) ofΓr inR2gis defined to be the orbital abelian groupR2g/Γr

en-dowed with the quotient topology. More precisely, a fundamental cellΛ(Γr)ofΓrsatisfies

i) Cg =R2g = ∪γ∈Γr(Λ(Γr) +γ).

ii) (Λ(Γr) +γ)γ∈Γr are pairwise disjoints; that is (Λ(Γr) +γ) ∩ (Λ(Γr) +γ 0

) = ∅ if

γ, γ0 ∈ Γr, γ6=γ0.

Associated to suchΓr and given mapping χ onΓr with|χ| = 1, we consider the

func-tional equation

f(z+γ) =χ(γ)eH(z+

γ

2)f(z); z∈ Cg, γ∈ Γr. (3.3) Whence, we assert the following

Lemma 3.3. For given f, g satisfying (3.3), the function hf ,gdefined onCgby z 7−→hf ,g(z) = f(z)g(z)e−H(z,z),

isΓr-invariant in the sense that

hf ,g(z+γ) = hf ,g(z); z∈ Cg, γ∈ Γr.

Proof. Let f , g satisfying (3.3) ,then

hf ,g(z+γ) = f(z+γ)g(z+γ)e−H(z+γ,z+γ) =χ(γ)eH(z+ γ 2)f(z)χ(γ)eH(z+2γ,γ)g(z)e−H(z+γ,z+γ) =χ(γ)χ(γ)f(z)g(z)eH(z+ γ 2)eH(z+γ2)e−H(z+γ,z+γ) = f(z)g(z)e−H(z,z) =hf ,g(z)

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3.3. Background on the vector spaceF2

Γ,χ(C2) 31

Therefore, the quantity

hf , giΓ

r = Z

Λ(Γr)

f(z)g(z)e−H(z,z)(z) (3.4) makes sense under some additional regularity conditions. Moreover, it is independent of the choice of the fundamental cell Λ(Γr) = R2g/Γr and defines a hermitian inner scalar

product. Above dλ(z) denotes the usual Lebesgue measure onCg. The associated norm is given by kfk2Γ r := Z Λ(Γr) |f(z)|2e−H(z,z)(z). (3.5) By using the functional equation 3.3 and the definition of fundamental cell it is not difficult to see thatkfk2Γr is a norm associated to the inner scalar product 3.4 . Therefore, one can perform the functional space FΓ2,ν

r(C

g) of all holomorphic complex valued functions on

Cgsatisfying (3.3) and such thatkfk2

Γr is finite . That is

FΓ2,ν

r(C

g) : =nf holomorphic onCgsuch that f(z+

γ) =χ(γ)eH(z+ γ 2)f(z); z∈ Cg, γ∈ Γr and Z Λ(Γr) |f(z)|2e−H(z,z)(z) < +∞  . (3.6)

Remark 3.4. For the trivial caseΓ0 = {0}, we haveΛ(Γ0) = Cg and the spaceFΓ2,ν

0(C

g) will

be trivial if χ(0) 6= 1. Where χ(0) = 1, the space FΓ2,ν

0(C

g) becomes isomorphic to the usual

Fock-Bargmann space consisting of e−ν|z|2dλ-square integrable holomorphic functions onCg.

Remark 3.5. For the cocompact case (i.e. Γ is of maximal rank r = 2g), the Λ(Γ2g) is compact

and the spaceF2 Γ2g(C

g)is nontrivial under the cocycle condition

χ(γ+γ0) = χ(γ)χ(γ0)eiE(γ,γ

0)

(RDQ), for every γ, γ0 ∈ Γ2g. In this case F2

Γ,χ(Cg) is of finite dimension that is given by explicit

for-mula involving the volume of the complex torus Cg/Γ2g. The spectral properties relevant to the (L2,Γ2g, χ)-theta functions are investigated in [7, 13]. This was worked out by rather different

methods by Folland in [11] for χ≡1.

3.3

Background on the vector space

F

2

Γ,χ

(

C

2

)

Let g =2 andΓ =1be a discrete subgroup of rank one in(C2,+), where ω1is nonzero

vector inC2. Below, we give necessary and sufficient conditions to the spaceFΓ,χ2,ν(C2)be non trivial.

Proposition 3.6. If the vector spaceF2

Références

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