Applications of Poincaré series on Jacobi groups [Elektronische Ressource] / vorgelegt von Kathrin Bringmann

Inaugural-DissertationzurErlangung der DoktorwurdederNaturwissenschaftlich-Mathematischen Gesamtfakult atderRuprecht-Karl-Universit atHeidelbergvorgelegt vonDiplom-MathematikerinKathrin Bringmannaus: MunsterTag der mundlichen Prufung: 23.07.04ThemaApplications of Poincare serieson Jacobi groupsGutachter: Prof. Dr. Winfried KohnenProf. Dr. Eberhard Freitag1Contents1 Introduction 42 Preliminaries 112.1 Basic facts about Siegel and Jacobi cusp forms . . . . . . . . . . . 112.2 Poincare series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Estimates for Fourier coecients of Siegel cusp forms 243.1 The full Siegel modular group . . . . . . . . . . . . . . . . . . . . 243.1.1 Poincare series of small weight . . . . . . . . . . . . . . . 243.1.2 The nal estimates . . . . . . . . . . . . . . . . . . . . . . 453.2 The subgroup (N) . . . . . . . . . . . . . . . . . . . . . . . . 46g,03.2.1 Poincare series for Jacobi cusp forms . . . . . . . . . . . . 463.2.2 EstimatesofthePeterssonnormofFourier-Jacobicoe cients 643.2.3 Final estimates . . . . . . . . . . . . . . . . . . . . . . . . 813.3 Subgroups of nites index and open questions . . . . . . . . . . . 834 Lifting maps 844.1 The generalized genus character and geodesic cycle integrals . . . 844.2 Construction of lifting maps . . . . . . . . . . . . . . . . . . . . .
Publié le : jeudi 1 janvier 2004
Lecture(s) : 34
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Source : ARCHIV.UB.UNI-HEIDELBERG.DE/VOLLTEXTSERVER/VOLLTEXTE/2004/4939/PDF/DOCUMENT.PDF
Nombre de pages : 124
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Inaugural-Dissertation
zur
Erlangung der Doktorwurde
der
Naturwissenschaftlich-Mathematischen Gesamtfakult at
der
Ruprecht-Karl-Universit at
Heidelberg
vorgelegt von
Diplom-Mathematikerin
Kathrin Bringmann
aus: Munster
Tag der mundlichen Prufung: 23.07.04Thema
Applications of Poincare series
on Jacobi groups
Gutachter: Prof. Dr. Winfried Kohnen
Prof. Dr. Eberhard Freitag
1Contents
1 Introduction 4
2 Preliminaries 11
2.1 Basic facts about Siegel and Jacobi cusp forms . . . . . . . . . . . 11
2.2 Poincare series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3 Estimates for Fourier coecients of Siegel cusp forms 24
3.1 The full Siegel modular group . . . . . . . . . . . . . . . . . . . . 24
3.1.1 Poincare series of small weight . . . . . . . . . . . . . . . 24
3.1.2 The nal estimates . . . . . . . . . . . . . . . . . . . . . . 45
3.2 The subgroup (N) . . . . . . . . . . . . . . . . . . . . . . . . 46g,0
3.2.1 Poincare series for Jacobi cusp forms . . . . . . . . . . . . 46
3.2.2 EstimatesofthePeterssonnormofFourier-Jacobicoe cients 64
3.2.3 Final estimates . . . . . . . . . . . . . . . . . . . . . . . . 81
3.3 Subgroups of nites index and open questions . . . . . . . . . . . 83
4 Lifting maps 84
4.1 The generalized genus character and geodesic cycle integrals . . . 84
4.2 Construction of lifting maps . . . . . . . . . . . . . . . . . . . . . 87
Bibliography 121
2Abstract
In meiner Arbeit betrachte ich Anwendungen von Poincarereihen zur verallge-
meinerten Jacobigruppe.
Im ersten Teil sch atze ich Fourierkoe zienten Siegelscher Spitzenformen ab.
Zun achst betrachte ich den Fall Siegelscher Spitzenformen kleiner Gewichte zur
vollenSiegelschenModulgruppe ;anschlie enduntersucheicheineUntergruppeg
(N) von .0,g g
Im zweiten Teil der Arbeit geht es um Liftungs-Abbildungen von einem Vek-
torraum verallgemeinerter Jacobi Spitzenformen in einen Teilraum elliptischer
Spitzenformen.
In our work we regard applications of Poincare series for generalized Jacobi
groups. The rst part deals with estimates of Fourier coe cients of Siegel cusp
forms. First we consider the case of Siegel modular forms for the full modular
group having small weight. Afterwards the case of a certain subgroup (N)g 0,g
of is regarded.g
In the second part we construct lifting maps from a vector space of generalized
Jacobi cusp forms to a subspace of elliptic modular forms.
3Chapter 1
Introduction
InthisthesiswediscussapplicationsofPoincareseriesoncertain(higher-dimen-
sional) Jacobi groups.
In the rst part (Chapter 3) we derive estimates of Fourier coe cients of Siegel
cusp forms. In the second part (Chapter 4) we construct certain lifting maps
from a vector space of Jacobi cusp forms to a certain subspace of elliptic modular
forms. Chapter 2 contains preliminary facts on Siegel and Jacobi modular forms.
Let F be a cusp form of integral weight k with respect to the Siegel modular
group = Sp (Z) GL (Z) with Fourier coecients a(T), where T is a posi-g g 2g
tive de nite symmetric half-integral gg matrix. Then a conjecture of Resniko
and Saldana˜ (cf. [RS]) says that
k/2 (g+1)/4+a(T) (detT) (> 0), ,F
where the constant implied in only depends on and F. For g = 1 this,F
conjecture is true (Ramanujan-Petersson conjecture, proved by Deligne for k 2
[DE], and by Deligne and Serre fork = 1 [DS]). This estimate is the best possible
one, because due to Rankin we have (for f = 0)

(1 k)/2 limsup A(T)T =∞.
T→∞
([Ra]).
For arbitary g there are known counter examples for the conjecture of Resniko
and Saldana˜ as the following theorem (cf. [K5]) shows
Theorem 1.1 Let g 1 (mod 4), gk (mod 2) and F∈S ( ) be a Heckek+g 2g
eigenform that is the Ikeda lift of a normalized Hecke eigenform f ∈ S ( ).2k 1
Then the conjecture of Resniko and Saldana˜ is not true.
For g 2 the estimate is at least known on average (cf. [K7]).
4
6From the classical Hecke argument the following bound
k/2a(T) (detT)F
follows readily, where the constant implied in only depends on F.F
For k >g+1, the at present best estimate is
k/2 c +ga(T) (detT) (> 0), (1.1) ,F
where  13 if g = 2 ([K1]), 36
1 if g = 3 ([Br]),c :=g 4
 1 1+(1 ) if g > 3 ([BK]),g2g g
where

g 1 21 := 4(g 1)+4 + . (1.2)g 2 g+2
Clearly
→ 0 for g→∞,g
i.e., one is far away from the conjecture of Resniko and Saldana.˜ To be more
precise, in [BK] it is proved that
1/2 +g (k 1)/2+a(T) (m (T)) (detT) (> 0), (1.3),F g 1
where
m :=min{T[U] |U∈GL (Z)},g 1 g 1 g
t twhere T[U] := U TU (U = transpose of U) and T[U] denotes the (g 1)g 1
(g 1) minor of T[U]. From (1.3) the estimate (1.1) follows directly if one uses
1 1/gtheboundm (T) (detT) , whichinturnfollowsreadilyfromreductiong 1 g
theory.
The method in [BK] is the following (for details cf. Chapter 2, Section 2 or

z (1,g 1)[BK]): If we write Z∈ H as Z = , where ∈ H, z∈ C , andg t 0z
0 ∈H , we see that the function F(Z)∈S ( ) has a so-called Fourier-Jacobig 1 k g
expansion of the kind
X
02itr (m ) 0F(Z) = (,z)e ( ∈H ),m g 1
m>0
wheremrunsthroughallpositivede nitesymmetrichalf-integral( g 1)(g 1)
matrices, andwherethecoe cients (,z)areJacobicuspforms(thede nitionm
of a Jacobi cusp form is given in Chapter 2). In [BK] the Fourier coe cients of
5

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