Quantization and arithmetic

150 Pages·2008·0.86 MB·English

by  Andre Unterberger

Pseudo-Differential Operators
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André Unterberger
Quantization and Arithmetic
Birkhäuser
Basel · Boston · Berlin
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André Unterberger
Département Mathématiques et Informatique
Université de Reims
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Contents
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1 Weyl Calculus and Arithmetic
2 A non-arithmetic prologue . . . . . . . . . . . . . . . . . . . . . . . 7
3 A family of arithmetic coherent states for the metaplectic
representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4 Wigner functions of pairs of arithmetic coherent states . . . . . . . 24
5 Matrix elements of operators from the Weyl calculus
against arithmetic coherent states. . . . . . . . . . . . . . . . . . . 35
2 Quantization
6 Discrete series of SL(2,R) and the hyperbolic half-plane
as a phase space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
7 Reinstalling R2 as a phase space: the horocyclic calculus . . . . . 58
3 Quantization and Modular Forms
8 An extension of the Rankin-Selberg unfolding method . . . . . . . 72
9 Discrete series and arithmetic coherent states . . . . . . . . . . . . 81
10 Radial horocyclic calculus and arithmetic . . . . . . . . . . . . . . 91
11 Beyond the radial case: automorphic distributions . . . . . . . . . 102
4 Back to the Weyl Calculus
12 Letting N go to infinity . . . . . . . . . . . . . . . . . . . . . . . . 117
13 Spaces of combs and Dirichlet series . . . . . . . . . . . . . . . . . 130
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
1 Introduction
Let χ(12) be the unique even non-trivial Dirichlet character mod 12, and let χ(4)
be the unique (odd) non-trivial Dirichlet character mod 4. Consider on the line
the distributions
(cid:3) (cid:4)
(cid:2)
m
d (x)= χ(12)(m)δ x− √ ,
even
12
m(cid:2)∈Z (cid:5) (cid:6)
m
d (x)= χ(4)(m)δ x− . (1.1)
odd 2
m∈Z
UnderaFouriertransformation,orundermultiplicationbythefunctionx(cid:3)→eiπx2,
thefirst(resp. second)ofthesedistributionsonlyundergoesmultiplicationbysome
24th(resp. 8th) rootofunity. Then, considerthe metaplectic representationMet,
a unitary representation in L2(R) of the metaplectic group G(cid:7), the twofold cover
of the group G = SL(2,R), the definition of which will be recalled in Section 2:
it extends as a representationin the space S(cid:3)(R) of tempered distributions. From
what has just been said, if g˜ is a point of G(cid:7) lying above g ∈ G, and if d = d
even
or d , the distribution dg˜ = Met(g˜−1)d only depends on the class of g in the
odd
homogeneousspaceΓ\G=SL(2,Z)\G,uptomultiplicationbysomephasefactor,
by which we mean any complex number of absolute value 1 depending only on g˜.
On the other hand, a function u ∈ S(R) is perfectly characterized by its scalar
productsagainstthedistributionsdg˜,sinceonehasforsomeappropriateconstants
C , C the identities
0 1
(cid:8)
|(cid:6)dg˜ , u(cid:7)|2 dg =C (cid:8)u(cid:8)2 if u is even,
even 0 L2(R)
(cid:8)Γ\G
|(cid:6)dg˜ , u(cid:7)|2 dg =C (cid:8)u(cid:8)2 if u is odd. (1.2)
odd 1 L2(R)
Γ\G
Each of the two formulas is called a resolution of the identity since its polarized
version makes it possible to write a general function u with a given parity as
an integral superposition of the corresponding distributions dg˜. In view of these
equations, together with the fact that the distributions dg˜ of a given parity are
essentially permuted under the metaplectic representation, the family (dg˜), g˜ de-
noting now a point in an appropriate homogeneous space ofG(cid:7) coveringΓ\G, will
becalledafamilyofcoherentstatesforthemetaplecticrepresentation:admittedly,
this terminology will be frowned upon by some readers,since the distributions dg˜
do not lie in L2(R).
Inspiteofthis,onecanstillcharacterizealinearoperatorA: S(cid:3)(R)→S(R),
preserving the parity of functions, by means of its matrix elements against the
given family. By this, we mean the set of scalar products (dg˜2|Adg˜1): one can
consider only the even (resp. odd) coherent states dg˜ in the case when, moreover,
2 Introduction
A kills all odd (resp. all even) functions, in which case we shall say that A is of
even-even (resp. odd-odd) type. A very different question, central to the present
work, is to what extent an operator A can be recovered from its diagonal matrix
elements, i.e., those for which g =g .
1 2
Readers familiar with the so-called Wick symbol of an operator (a notion
originating from Physics) or with the Berezin theory of quantization, will have
a feeling of d´eja` vu. Indeed, instead of starting from the arithmetic distribution
deven, let us start from the standard Gaussian function x(cid:3)→241 e−πx2: it is invari-
ant, up to phase factors, under the operators of the metaplectic representation
lyingabovethe maximalcompactsubgroupK =SO(2)ofG,whichmakesitpos-
sible to build a family (u ) of coherent states (belonging this time to the space
gK
L2(R)) parametrizedby points in G/K. The associatedfamily of diagonalmatrix
elements (u |Au ) is just a symbol of some Berezin kind of the operator A:
gK gK
similar considerations bring to light the Wick symbol, but using the Heisenberg
representationin placeof the metaplectic representation.We identify in the usual
way the homogeneous space G/K with the hyperbolic upper half-plane Π: by the
way, the space Γ\G, which is up to finite covering the parametrizing space of the
family of arithmetic coherent states introduced above, can be identified, just as
classically, with a set of lattices in the plane. It is one of the purposes of the
presentworkto push the analogybetween Γ and K as far as possible,staying en-
tirely within classical analysis, i.e., away from adeles: there is considerable room
in R2 as soon as distributions enter the picture, and even though there is no fun-
damental domain for the linear action of Γ in R2, there is a well-defined concept
of automorphic distribution in the plane [30, 31]. This notion is advantageous on
several accounts: in particular, after it has been transferred in the right way (to
wit,undersomeassociateoftheRadontransformation)fromthehalf-planetothe
plane,theoperatorΔ−1 factorsasπ2E2,where2iπE =x ∂ +ξ ∂ +1istheEuler
4 ∂x ∂ξ
operator in the plane. As a consequence, spectral decompositions of automorphic
functions give way, in the plane, to decompositions of automorphic distributions
into homogeneous components: this makes explicit calculations easier in general.
Coming back to the family of matrix elements (uz|Auz)L2(R), where z ∈ Π
and A is assumed to be of even-even type, understanding the amount of infor-
mation carried by this function demands that we should first characterize A by a
symbolinsomegood symboliccalculus:inthiscase,itisofcoursetheWeylcalcu-
lus, for many reasons,one of which is that it establishes an isometry Op fromthe
space L2(R2) to the space of Hilbert-Schmidt operators in L2(R). The operator
A = Op(h) is of even-even type if and only if its symbol h is an even function
invariantunderarescaledversionG (byafactor2)ofthesymplecticFouriertrans-
formationintheplane.Ifsuchisthecase,andifonesets(Ch)(z)=(uz|Auz)L2(R),
one has the spectral-theoretic equation
(cid:8)Ch(cid:8)L2(Π) =2 (cid:8)Γ(iπE)h(cid:8)L2(R2), (1.3)
which captures exactly the loss of information incurred in considering Ch instead
of h, or A. Since the Gamma function is rapidly decreasing at infinity on vertical
1. Introduction 3
lines in the complex plane, the operator C is far from invertible: on the other
hand,the Gammafunction doesnotvanishatanyfinite point,whichimplies that
the map C is one-to-one, even when given the space of all even-even tempered
distributionsasadomain.Thislastfactcouldalsobe obtainedbyanargumentof
analyticcontinuationsince,aswillbe recalled,u isthe productof(Im (−z−1))14
z
by an antiholomorphic function of z: then, the set of matrix elements (u |Au )
z2 z1
is characterized by its subset of diagonal ones.
Let us now switch to the arithmetic situation, assuming that h ∈ S(R2) so
that the associated Weyl operator should act from S(cid:3)(R) to S(R), that A is of
even-even type and that h(0) = 0: we also assume that h is a radial function –
most of the analysis remains without this condition, but must be formulated in
termsofautomorphicdistributions,notautomorphicfunctions–sothatthescalar
product (dg˜ |Adg˜ ) depends only on z = g.i ∈ Π. The function (Ah)(z) so
even even
definedisautomorphic,andonehastheidentity,againofaspectral-theoretictype,
(cid:3) (cid:4)
1
8 2
(cid:8)Ah(cid:8)L2(Γ\Π) = π (cid:8)(1−22iπE)(1−32iπE)ζ(1−iπE)h(cid:8)L2(R2). (1.4)
Since the zeta function has no zero on the line Re s = 1, the inverse operator
A−1 becomes continuous if followed by the spectral projection, relative to the
self-adjointoperator2πE, correspondingto anyclosedintervalnotcontainingany
point 2πn or 2πn with n∈Z. Of course, we cannot say that a function h, homo-
log2 log3
geneous of degree −1−iλ for such an exceptional value of λ, lies in the nullspace
of A since we have to work with symbols in S(R2). Needless to say, something
analogous works with odd functions too. Finally, the reader may wish to know
what will happen if one uses the Dirac comb, the sum of unit masses at points
of Z, in place of d or d . Then, one cannot use the full modular group Γ,
even odd
but only a certain subgroup, isomorphic to the so-called Hecke congruence group
Γ (2): the fundamental domain, in this case, has two cusps, which complicates a
0
little bit the discussion, but not much. The matters discussed so far make up the
first chapter of the present work.
Itisinterestingtocompare(1.3)and(1.4),especiallyinviewofthefactthat,
aswillbediscussedpresently,thereisaverynaturalgeneralizationofthefirstiden-
tityinwhichtheoperatorΓ(iπE)hastobereplacedbyΓ(τ+1+iπE)forsomereal
2
number τ >−1 but otherwise arbitrary. One might expect that it should be pos-
sible, generalizing the second identity in a similar way, to manage so as to let the
restrictionofthe function zeta to some possibly arbitrary,or atleastnot overspe-
cialized,lineenterthepicture:then,thenon-existenceofzerosofzetaonsuchaline
would be given an interesting interpretation. For some reasons which seem to us
ratherdeep,andwhichleaveroomfor furtherinvestigation,weweredisappointed
inthis hope, but not completely. Beforewecometo this point, let us describethe
generalizationof (1.3), depending on the parameter τ, which we have in mind.
The idea,certainly not a novelone – but only up to some point – consists in
regardingthe evenandodd partsof the metaplectic representation,up to unitary
4 Introduction
equivalence,asspecialcasesoftheholomorphicdiscreteseriesofSL(2,R)or,more
precisely,ofaprolongationoftheholomorphicdiscreteseriesoftheuniversalcover
ofthatgroup:theparameterτ labellingtherepresentationsinthisseriesliesinthe
interval ] −1, ∞[, the two special cases already considered corresponding to the
values τ = ∓1. The representation obtained for each value of τ has at least two
2
useful realizations: one, denoted as D , in a weighted L2-space H of func-
τ+1 τ+1
tions onthe half-line (0, ∞), andone,denoted asπ , ina space ofholomorphic
τ+1
functions in Π, easily described only in the case when τ ≥ 0. In the first realiza-
tion, there is a family (ψzτ+1)z∈Π of functions on the half-line substituting for the
formerfamily(u ):the familyofdiagonalmatrixelements ofanoperatorinH
z τ+1
against the set of coherent states just referred to is called the Berezin-covariant
symbol of A.
Chapter2isdevotedtothe constructionofagood symboliccalculusofoper-
ators in H . Let us hasten to say that this calculus is none of the calculi, using
τ+1
Π as a phase space (this is the space where symbols live), some readers may be
familiar with, such as the Berezin calculus [2, 3], or the active-passive calculus
[26, 29]. It will be necessary, in Section 6, to recall a few facts concerning these
calculi: but this will only serve as a preparation to the introduction, in Section
7, of the horocyclic calculus. The construction of this second-generation calculus
is more involved than that of the preceding ones, as it does not admit any ob-
vious definition. The phase space is in this case R2, and symbols have to satisfy
some specific symmetry property, expressed with the help of the rescaled Fourier
transformation G. This is the good calculus we had been aiming for, and (1.3)
generalizes in the way already alluded to.
The thirdchapterrevisitsthenotionofarithmeticcoherentstatesinconnec-
tion with the family of representations (D ), taking advantage of the symbolic
τ+1
calculi studied in Chapter 2. We consider the diagonal matrix elements of oper-
ators in H , characterized by their horocyclic symbols, against a τ-dependent
τ+1
family (sgτ˜)g∈G of arithmetic coherent states: these are built from a discrete mea-
sure s on the half-line, invariant, up to phase factors, under all transformations
τ
D (g˜),g˜beinganelementoftheuniversalcoverofSL(2,R)lyingaboveamatrix
τ+1
g ∈SL(2,Z).Eventhoughsuchadiscretemeasureisnotunique,thereisalwaysat
leastone possiblechoice, obtainedwiththe help ofsomepowerofthe Ramanujan
Δ-function. The resolution of the identity expressed in (1.2) easily generalizes in
theobviousway.Itismuchmoredifficult,andthisistheobjectofChapter3,toex-
tend(1.4)totheτ-dependentcase.Thehorocycliccalculuscouldnotbedispensed
withatthis point,but there are otherdifficulties as well,whichdemandreconsid-
ering in particular the Rankin-Selberg unfolding method of modular form theory.
The version needed here starts from the consideration of series of the Poincar´e
style built not from the function z = x+iy (cid:3)→ ys, in the way Eisenstein’s is,
butfromtheWhittakerfunctionz (cid:3)→y12 Ks−1(2πky)e2iπkx.Yes,theserieswould
2
diverge,butonecanbypassthisdifficulty inacertaincanonicalway,whichmakes
it possible to complete the proof of the the τ-dependent generalization of (1.4).
1. Introduction 5
This is the spectral decomposition of the automorphic function taking the
place of the function Ah in (1.4): instead of the zeta function, it is a certain
convolution L-function, to wit L(f ⊗ f, s), built with the help of the modular
form f of real weight τ +1 used in the construction of s , that appears. This
τ
function has to be considered on the “spectral line” Re s = 1. When τ = −1
2 2
(the same goes when τ = 1), it is essentially (up to one or two elementary extra
2
factors)therestrictionofzetatotheboundaryofthecriticalstrip.Onthecontrary,
when τ +1 is an even integer and f is a Hecke cusp-form, it follows from results
of Shimura [23] (which we learned from [13, 14]) that the function L(f ⊗ f, s)
is “divisible” by ζ(s), thus letting the critical zeros of zeta participate in the
non-invertibilityofthe mapA.Itwouldcertainlybe anice thingif aτ-dependent
theorywereavailable:itisratherunlikely–aslongastheRiemannhypothesishas
not been proved – that a spectral-theoretic interpretation of all non-trivial zeros
of zeta (the “Hilbert-Polya dream” [17, p. 7]) could exist; having such a theory
for those (hopefully non-existent) lying on any given line Re s = a, 1 < a <1,
2
is another matter. However, the move – in the non-arithmetic situation – from
Γ(iπE) to Γ(τ + 1 +iπE) does not generalize in the way one might have hoped
2
fortothe arithmetic case.Onthe otherhand,since muchdepends onsomebetter
understandingof holomorphicmodular forms of weightτ+1 and their associated
convolutionL-functions,resultsinthedesireddirectionarenotyettobeexcluded,
at least for some very special values of τ: but nothing easy can be expected.
Going beyond the case of radial (horocyclic) symbols, as treated in Section
10,isnotcompletelyobvious,asitistantamounttosubstitutingthehomogeneous
space SL(2,Z)\SL(2,R) for the double quotient SL(2,Z)\SL(2,R)/SO(2), and
it is the subject of Section 11. Avoiding the use of the three-dimensional first
spaceis neverthelesspossible,relyingonthe conceptofautomorphicdistributions
(SL(2,Z)-invariantdistributionsintheplane)alreadyreferredtointhisintroduc-
tionandusedforadifferent(insomesensedual)purposeinautomorphicpseudod-
ifferential analysis [31]: there, non-holomorphic modular distributions were used
assymbolswhile,inthepresentwork,itisthefunctions–or,rather,distributions
– the operators are applied to that carry the arithmetic.
In the fourth, and last, chapter,we come back to the Weyl calculus, andob-
tain in a naturalway a generalizationof (1.4)in which the product of two factors
on the right-hand side is replaced by an arbitrary partial product of the Eule-
rianexpansionofthe operator(ζ(2iπE))−1. Itrequiresthatone substitute forthe
distribution d in (1.1) a finite collection of distributions, depending on some
even
integer N, the product of 4 by a squarefree odd integer but otherwise arbitrary.
Again (Theorem 12.4), a formula of “resolution of the identity” exists: it is espe-
ciallyinterestingtoanalyzewhathappenswhenN →∞,morepreciselywhenthe
setofprimedivisorsofN tendstothesetofallprimenumbers.Inconnectionwith
astudyofthedistributionsintheplaneobtainedintheprocess,wemakethefirst
few steps, in the last section, towards the development of an analysis specifically
adapted to the study of combs or of their associated Dirichlet series. Note that it