Quantum variance on quaternion algebras, I

0.53 MB·

by  Paul D. Nelson

QUANTUM VARIANCE ON QUATERNION ALGEBRAS, I
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1 PAULD.NELSON
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2

v
o
Abstract. We determine the quantum variance of a sequence of families of
N
automorphicformsonacompactquotientarisingfromanon-splitquaternion
3 algebra. OurresultscomparetothoseobtainedonSL2(Z)\SL2(R)inworkof
Luo, Sarnak andZhao, whose method requiredacusp. Ourmethod uses the
1
theta correspondence to reduce the problem to the estimation of metaplectic

Rankin–Selbergconvolutions. Weapplyitheretothefirstnon-splitcase.
]
T
N
Contents
.
h
1. Introduction 1
t
a 2. Preliminaries 9
m
3. Definitions of some theta functions 11
[ 4. Comparison with arithmetic variance 12

5. Reduction of the proof of the main result 12
2
v 6. Further remarks 14
6 7. Proof of the algebraic input 15
2 8. Computation of partial orbital integrals 17
5
9. Fluctuations of fixed lines 19
2
10. Proof of the analytic input 23
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. 11. Classical correlations 29
1
12. Rallis inner product formula 32
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6 References 37
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:
v
i
X
1. Introduction
r
a 1.1. Overview. LetM beanegativelycurvedcompactRiemannianmanifoldwith
d:=dim(M)>2. It is known in various senses that a random Laplace eigenfunc-
tion on M of large eigenvalue is uniformly distributed, but very rarely known just
how uniformly. To explain, denote by µ the probability measure on M that is
a multiple of the volume measure and let T be a positive parameter. For
→ ∞
each t [T,2T], choose an orthonormal basis for the t2-eigenspace of the Lapla-
∈
cian acting on L2(M,µ). Let denote the union of these bases; it is finite set of
T
F
cardinality Td. To each ϕ , attach the L2-mass
T T
|F |≍ ∈F
µ := ϕ2µ, µ (Ψ):= Ψ(x)ϕ2(x)dµ(x).
ϕ ϕ
| | | |
Zx∈M
Date:November15,2016.
2010 Mathematics Subject Classification. Primary11F27;Secondary 11F37,58J51.
1
QUANTUM VARIANCE ON QUATERNION ALGEBRAS, I 2
ItisaprobabilitymeasureonM. Aconsequenceofthequantumergodicitytheorem
is that the mean −1 µ converges weakly to µ. The quantum variance
|FT| ϕ∈FT ϕ
problem concerns the asymptotic determination of the quadratic form
P
1
(Ψ):= µ (Ψ) µ(Ψ)2 (1)
T ϕ
V | − |
T
|F |ϕX∈FT
1
= Ψ(x)Ψ(y)(ϕ(x)2 1)(ϕ(y)2 1)dµ(x)dµ(y) (2)
| | − | | −
|FT|ϕX∈FTZx,y∈M
and its bilinearization (Ψ ,Ψ ), which quantify the correlations of the fluctua-
T 1 2
V
tions of the µ . They extend naturally to the phase space X=S∗M by replacing
ϕ
µ withitsmicrolocalliftω (Ψ):= Op(Ψ)ϕ,ϕ andµwiththeLiouvillemeasure.
ϕ ϕ
h i
Zelditch introduced these sums and showed using semiclassical techniques that
(Ψ) = O(1/logT) (see [71, 72, 62]). A prediction of Feingold–Peres (see [18],
T
V
[70, 15.6], [52, 4.1.3]) suggests that for “generic” M, the renormalized quantum
§ §
variance Td−1 is asymptotic to the classical variance of the geodesic flow. The
T
V
quantum variance problem is thus extremely delicate: a solution in the expected
form
Td−1 (Ψ)= (Ψ)+o(1) (3)
T ∞
V V
demands not only the enormous improvement (Ψ) = O(T1−d) upon the semi-
T
V
classical estimate, but the further extraction of a main term.
Luo–Sarnak [43] insightfully observed that on the modular curve SL (Z) H,
2
\
thequantumvarianceproblemforHecke–Laplaceeigenfunctions maybe profitably
attackedusingarithmetictechniquesstemmingfromthetheoryofHeckeoperators.
The solution,obtained insharpestformby Sarnak–Zhao[61]after the increasingly
sophisticatedworks [41, 32, 42, 43, 73], is arithmetically interesting: central values
of L-functions quantify the deviation of the generic prediction from the truth.
This article introduces a method for determining the quantum variance of fam-
ilies of automorphic forms on compact arithmetic quotients attached to non-split
quaternion algebras B. The non-compact quotient SL (Z) H considered by Luo–
2
\
Sarnak–Zhao arises algebraically from the split quaternion algebra B = M (Q).
2
Theproblemhadbeenopenineverynon-splitcasepriortothiswork. Weaimhere
to introduce our method by application to the simplest non-trivial non-split case.
The first step in the method of Luo–Sarnak–Zhao reduces the expression (2),
which is quadrilinear in ϕ, to a complicated bilinear expression to which trace
formulasapplyandfromwhichacolossallimitingbilinearformeventuallyemerges.
This is achievedby combining the close relationbetween Fourier coefficients atthe
cusp of SL (Z) H and Hecke eigenvalues λ(n) with the Hecke multiplicativity
2
∞ \
λ(m)λ(n) = λ(mn/d2). Unfortunately, such an approach is unavailable on
d|m,n
compactarithmetic quotients, which have no cusps and hence no notion of Fourier
P
expansion that interacts suitably with the Hecke theory.
In our method, the theta correspondence is decisive. The well-known seesaw
diagram
O(B) Mp Mp (4)
P♥P♥P♥P♥P♥P♥P♥P♥P♥P♥P♥P♥PP2× 2
O(1) O(B0) SL
2
×
QUANTUM VARIANCE ON QUATERNION ALGEBRAS, I 3
substitutes for the linearizing role played by Fourier coefficients and Hecke multi-
plicativity in the method of Luo–Sarnak–Zhao. The basic input powering conver-
gence to the limit is decay of matrix coefficients for the dihedral1 spectrum of SL .
2
The Rallis inner product formula and the Maass–Shintani–Waldspurger theta lift
are shown to give natural interpretations to the main terms obtained here and in
earlier works. We discover “secondary main terms” to the quantum variance sums
at the square-root cancellation threshold. The primary novelty is that we prove
asymptotic formulas for quantum variance sums on a non-split quaternion algebra
for the first time.2
AgeneraltoolavailableonanyquaternioniccongruencequotientΓ Histhetriple
\
product formula (see [25, 67, 29]). When both φ and Ψ are Hecke eigenfunctions,
it relates µ (Ψ)2 to the L-value L(ϕ ϕ Ψ,1/2) and (Ψ) to a weighted
ϕ T
| | × × V
average whose analysis succumbs to well-developed techniques (see [43, 73, 61])
giving Lindelo¨f-consistent asymptotics of the form (3). Unfortunately, the triple
product formula is non-linear; it does not apply when, for instance, Ψ is the sum
of two inequivalent mean zero Hecke eigenfunctions Ψ ,Ψ . For the correlations
1 2
1
(Ψ ,Ψ )= µ (Ψ )µ (Ψ )
T 1 2 ϕ 1 ϕ 2
V
T
|F |ϕX∈FT
1
= Ψ (x)Ψ (y)ϕ(x)2 ϕ(y)2
1 2
| | | |
|FT|ϕX∈FTZx,y∈M
one then expects (following [43]) the further cancellation (Ψ ,Ψ ) = o(1/T)
T 1 2
V
comingfromtheindependenceofvariation insign ofthequantitiesµ (Ψ ),µ (Ψ )
ϕ 1 ϕ 2
beyond the prediction µ (Ψ )=O(T−1/2+o(1)) of the Lindelo¨f hypothesis for their
ϕ i
magnitude. Detecting such cancellation is the fundamental difficulty overcome in
the works of Luo–Sarnak–Zhao. It is achieved by lengthy analysis of symmetry
propertiesoftheiremergentbilinearform,whichisfound(remarkably)tobeHecke
self-adjoint. The analogous difficulty in our approach is ultimately addressed by
the unramified case of the local theta correspondence for (Mp ,O(B0)).
2
Ourmainresultconcernsautomorphicformswithramificationvaryinginanon-
archimedean aspect. Thesetupdifferssuperficiallyfromwhatwasdiscussedabove,
but retains the fundamental difficulty while allowingus to introduce the core ideas
of the method as accessibly as we can.
1.2. Statement of main result. Terminology to follow is standard, and will be
reviewed more thoroughly in 2. Set G := GL (Q ), K := GL (Z ). We fix a dis-
2 2 2 2
§
crete cocompact subgroup Γ < G arising from a maximal order in the quaternion
algebra ramified at ,23 (see 2.1). (The significance of the pair of numbers
{∞ } §
(2,23) is that it is the lexicographically smallest for which the problem to be dis-
cussed has all essential features; we have focused our discussion in this way to
simplify notation and exposition.) Let N be a positive integral parameter tending
off to . We consider the family of L2-normalized newforms (see 2.5) on a
N
∞ F §
sequence of congruence covers
Y :=Γ G/K of Y :=Γ G/K,
N −N..N
\ \
1Inthisarticle,only“splitdihedral”forms(i.e.,unitaryEisensteinseries)contribute.
2Wereferto[36,35]forsomeanalogousresultsinasimplerarithmeticsetting.
QUANTUM VARIANCE ON QUATERNION ALGEBRAS, I 4
where more generally
K :=K Z2 2N1Z2 . (5)
−N1..N2 ∩ 2N2Z2 Z2
(cid:18) (cid:19)
The sets Γ G,Y ,Y come equipped with compatibly-defined Hecke correspon-
N
\
dences T for odd integers n>1 (see 2.2); on Y, T is defined for all n>1.
n n
§
ThesetsYandΓ Gareroughly2-adicanaloguesofacompactarithmetichyper-
\
bolicsurfaceandits cotangentbundle,respectively. ThebasespaceY turnsoutto
have cardinality three. One may regardit as a (2+1)-regular directed multigraph
on three vertices and its cover Y as (with suitable interpretation, taking into ac-
N
count torsion) the set of non-backtracking paths y = (y y ) of length
−N N
→···→
2N +1 on Y, with the covering map Y Y given by the projection y y .
N 0
→ 7→
One mayalso identify Y withthe set ofisomorphismclassesof supersingularellip-
tic curve E in characteristic 23 (see [23, 2]) and Y with the cover obtained by
N
§
consideringlevelstructureconsistingofpairsC ,C ofcyclicsubgroupsoforder2N
1 2
withC C = 0 ;infact,itwillbeconvenienttoadoptthisalgebraicperspective
1 2
∩ { }
for some calculations ( 9).
§
ThespaceΓ Ghasanaturalinvariantmeasure(see 2.4)whichinducesanatural
\ §
measure on Y. The space L2(Y) is three-dimensional, and decomposes as
L2(Y)=C CΨ CΨ ,
1 2
⊕ ⊕
where Ψ ,Ψ are orthonormal mean zero real-valued Hecke eigenfunctions, well-
1 2
defined up to permutation and sign. For orientation, we record that under the
Eichler/Jacquet–Langlandscorrespondence,Ψ ,Ψ correspondtotheweight2new-
1 2
forms ΨJL,ΨJL on Γ (23) and to the set of weight2 newforms onΓ (22N 23).
1 2 0 FN 0 ·
To each ϕ we attach the “harmonic weight” ι := L(2)(adϕ,1) of size
N ϕ
∈ F
2o(N) (see 2.8); these mild weights play a role (non-obviously) similar to that in
§
[61] and earlier works, and should be ignored on a first reading.
Thefundamentalobjectofstudyinthisarticleistherealsymmetric2 2matrix
×
V :=(Vkl) with entries
N N k,l=1,2
1
Vkl := ι Ψ (x)Ψ (y)ϕ2(x)ϕ2(y).
N 22N ϕ k l | | | |
ϕX∈FN Zx,y∈Γ\G
Equivalently, let µ denote the measure on Y given by µ (Ψ):= Ψϕ2; then
ϕ ϕ Γ\G | |
1 R
Vkl = ι µ (Ψ )µ (Ψ ).
N 22N ϕ ϕ k ϕ l
ϕX∈FN
Forthediscussionof 1.1specializedtothecaseofsurfaces,thenumericalanalogue
§
hereofthe quantity T is 2N; forinstance,the family cardinalityis 22N (see
N
|F |≍
9.5 for an exact formula). To justify interpreting Vkl as a variance, we verify in
§ N
9.5 that
§ 1
µ =µ (6)
ϕ
N
|F |ϕX∈FN
withµthe probabilitymeasureonY thatisamultiple ofthe pushforwardof .
Γ\G
The entries of V quantify the correlationsof the fluctuations of the L2-masses µ
N R ϕ
of a random newform ϕ on the congruence cover Y when tested against
N N
∈ F
the meanzeroHecke eigenfunctions Ψ ,Ψ onthe basespace Y. The setupis thus
1 2
the natural 2-adic analogue of that in 1.1.
§
QUANTUM VARIANCE ON QUATERNION ALGEBRAS, I 5
We normalize the Hecke eigenvalues of Ψ ,Ψ by T Ψ = √nλ (n)Ψ and
1 2 n i Ψi i
the standard L-functions L(Ψ ,s) by analytic continuation of the Dirichlet series
k
λ (n)/ns from the half-plane Re(s) > 1, so that s = 1/2 is the central
n>1 Ψk
point.
P
Theorem 1 (Main result).
(i) Existence of the limiting quantum variance.
The 2 2 matrix limit lim 2NV exists.
N→∞ N
×
(ii) Diagonalization and determination of the limit.
That limit is given by3
L˜(Ψ ,1) 0
V := 1 2
∞ 0 L˜(Ψ ,1)
(cid:18) 2 2 (cid:19)
with L˜(Ψ ,1):=P(λ (2))L(Ψ ,1), P(x):=π2(15 4√2x)/69.
k 2 Ψk k 2 −
(iii) Effective rate of convergence to the limit.
2NVkl Vkl 6CN2−N for some absolute effective C >0.
| N − ∞|
EachassertioninTheorem1 is highly non-trivial– forinstance, the existence of
the limitalreadyimplies somewhatmorethanthe strongimprovementVkl 2−N
N ≪
over the trivial bound Vkl 1 – but the most novel assertion is that for k=l,
N ≪ 6
lim 2NVkl =0 (7)
N
N→∞
(in the strong quantitative form 2NVkl =O(N2−N))
N
which is the first of its kind in any non-split/cocompact setting and lies genuinely
beyond existing methods such as Fourier expansions at cusps, trace formulas, and
triple product formulas, which fail even to reduce its proof to a technical problem.
One can interpret (7) as reflecting a non-obvious Hecke symmetry enjoyed by the
fluctuations of the measures µ .
ϕ
1.3. Application to moments of (square roots of) L-functions. The values
µ (Ψ ) are basic examples of triple product periods, which have been extensively
ϕ k
studied.4 The triple product formula relates the squares (but not the signs) of
these periods to the central values of triple product L-functions. Unlike in the
workofLuo–Sarnak–Zhao,wedonotusethetripleproductformulainthe proofof
Theorem1;nevertheless,itisinstructivetoindicatebrieflyhowourresultstranslate
thereunder. The formula in question (see [29, 51, 27]) has the shape
L˜(ϕ ϕ Ψ ,1)
ι µ (Ψ )2 =2−N × × k 2
ϕ ϕ k
| | L(adϕ,1)
whereL˜(ϕ ϕ Ψ ,1):=c (Ψ ,ϕ)L(ϕ ϕ Ψ ,1)forsomeexplicitnonnegative
× × k 2 0 k × × k 2
quantity c (Ψ ,ϕ) that we expect5 is uniformly bounded as ϕ varies. (The reader
0 k
will lose little in what follows by ignoring the factor c (Ψ ,ϕ), whose explication
0 k
3Onehas{L(Ψ1,12),L(Ψ2,12)}≈{0.552,0.450}accordingto[38,23.2.1a.0],[38,23.2.1a.1]
4See forinstance the articles [24, 5, 17, 7], whichfocus on definite quaternion algebras as we
dohere,or[51],whoseconsiderationsapplytotheanalogousperiodsobtainedfromnewformson
thetowerofcoversΓ0(22N)\H→SL2(Z)\H ofthemodularcurve.
5By[49,Thm34],thisexpectationholdswhenϕisprincipalseries. Itshouldfollowingeneral
fromthemethodsof[27]. Onecouldrewritethisarticlewiththeroleofp=2replacedbythatof
someoddprime. Weexpectthenthatc0(Ψk,ϕ)wouldbeindependentofΨk,ϕ. Thisexpectation
holdswhenϕisprincipalseriesby[49,Thm34].
QUANTUM VARIANCE ON QUATERNION ALGEBRAS, I 6
is beyond the scope of this paper.) The Lindelo¨f hypothesis thus predicts that the
magnitude ofthe realnumberµ (Ψ )istypically 2−N/2,butsaysnothing about
ϕ k
≈
itssign. Theorem1tellsusthatthesignsofµ (Ψ )andµ (Ψ )arehighly unbiased
ϕ k ϕ l
for k =l; its diagonal and off-diagonal k =l cases translate to
6 6
1 L˜(ϕ ϕ Ψ ,1) log
× × k 2 =L˜(Ψ ,1)+O |FN| , (8)
22N L(adϕ,1) k 2
ϕX∈FN |FN| !
p
1 L˜(ϕ×ϕ×Ψk,21) L˜(ϕ×ϕ×Ψl,12) =O log|FN| (9)
q L(adqϕ,1)
|FN|ϕX∈FN |FN| !
with the choice of square-rootin (9) given by the sign of the realpnumber µϕ(Ψk).
The diagonal estimate (8) is consistent with the Lindelo¨f hypothesis on average.6
Theoff-diagonalestimate(9)representssquare-rootcancellationbeyondtheLindel¨of
hypothesis,anddoesnotfollowfromstandardconjecturessuchasGRH.Wediscuss
the error terms further in 6.5. For orientation, we record that 22N and
N
§ |F | ≍
typically C(ϕ ϕ Ψ ) 24N.7
k
× × ≍
1.4. Application to quantum unique ergodicity. We touch briefly on the re-
lationship of our results to the study of worst-case behavior of the measures µ .
ϕ
It is expected that µ µ for any sequence of ϕ with N , i.e.,
ϕN → N ∈ FN → ∞
that µ (Ψ ) = o(1) := o (1) for k = 1,2. This expectation is much weaker
ϕN k N→∞
than the prediction µ (Ψ ) 6 2−(1+o(1))N/2 of the Lindelo¨f hypothesis, but re-
| ϕN k |
mains an open variant of the arithmetic quantum unique ergodicity conjecture as
in [46, 51, 27] which can be shown to follow from subconvexity. The special case
in which ϕ generates a principal series representation of GL (Q ) was recently
N 2 2
confirmed unconditionally in [49]. Theorem 1 and Chebyshev’s inequality imply
that the Lindelo¨f predictionis essentiallysharp, that it holds for a density 1 o(1)
−
subset of , and that all but a density O( −1/2+o(1)) subset of ϕ
N N N N
F |F | ∈ F
satisfy µ µ as N (compare with [41, 42, 43]).
ϕN → →∞
1.5. Method. We now indicate in high-level terms why is it natural to study the
quantum variance problem using theta functions and the diagram (4); a concrete
implementation of this discussion may be found in 5.
§
The arithmetic quotients X of interest to us are parametrized by quaternion
Q-algebras B; one can take for X the adelic quotient B×(Q) B×(A) or a further
\
quotient thereof. Given a pair of cusp forms Ψ ,Ψ on X with trivial central
1 2
character,we would like to understand quantum variance sums of the shape
ϕ2,Ψ Ψ , ϕ2 (10)
1 2
h| | ih | | i
ϕ∈F
X
for some “nice enough family of automorphic forms” .
F
6A direct proof of (9) using the approximate functional equation, Petersson formula and
Voronoi summation is heuristically straightforward; technical complications arise because of
“weight2.” TheproofofTheorem1does notusethetripleproductformula.
7We remarkalsothat by replacing the ϕ,Ψk withsome modifications belonging to the auto-
morphic representation that they generate, it should be possible to derive explicit analogues of
(8),(9)withoutthefactorsc0(Ψk,ϕ). Forinstance,itshouldbepossibletoachievethisbyleaving
Ψk as it is and replacing ϕ by the 2-adic microlocal lift introduced in [49] when ϕ is principal
seriesandbyananalogous constructioninvolvingnon-splittoriwhenϕissupercuspidal.
QUANTUM VARIANCE ON QUATERNION ALGEBRAS, I 7
Our first observationis that sums of this generalshape are relatedvia the theta
correspondence to four-fold integrals of theta functions: Suppose given a pair of
elementary theta functions θ ,θ and ternary theta lifts h ,h of the cusp forms
1 2 1 2
Ψ ,Ψ . More precisely, θ ,θ ,h ,h are functions on SL (Q) Mp (A) attached
1 2 1 2 1 2 2 \ 2
to Schwartz–Bruhat functions φ′,φ′ on A and φ′′,φ′′ on B0(A), where B0 is the
1 2 2 2
trace zero subspace of B. The Parseval formula and seesaw duality (4) then give
identities roughly of the shape
generalized quantum variance sums
θ h ,θ h = . (11)
h 1 1 2 2i tested against Ψ1,Ψ2
(cid:18) (cid:19)
We explain this shortly ( 1.6) in a “toy example” and in the body of this article
§
very concretely.
Theidentity(11)(togetherwithitsextensiontonon-puretensorsφ = φ′
i α i,α⊗
φ′′ )suggestsa naturalstrategyforattackingthe quantumvarianceproblem. It is
i,α P
notapriori clearthatthisstrategyshouldsucceed,anditsimplementationrequires
further novelties:
(1) The precise sums appearing on the RHS of (11) depend heavily upon the
localdataφ′,φ′,φ′′,φ′′. One mustthus confrontthe inversion problem for
1 2 1 2
a family of automorphic forms which consists of exhibiting local data φ
i
F
sothattheRHSof (11)resemblesthe quantumvariancesums(10)over .
F
(2) There is then the unprecedented analytic problem of asymptotically eval-
uating the integral of four-fold products of theta functions on the LHS of
(11)asthelocaldatavary;weaddressthisbyprovingasymptoticformulas
of the shape
θ h ,θ h θ ,θ h ,h . (12)
1 1 2 2 1 2 1 2
h i≈h ih i
The two problems are intertwined in that one does not expect the nice asymptotic
behavior (12) for completely general variation of the local data.
1.6. Quantum variance sumswith Eisensteinobservablesvia L-functions.
We include this subsection to convey the flavorof the first step (11) of our method
toreadershavingsomefamiliaritywithintegralrepresentationsofL-functions,but
not necessarily with the theta correspondence and seesaw duality.
We aim to illustrate (11) for a “toy problem” (unrelated to the main result of
this article) involving the Eisenstein series E on SL (Z) H, defined for a complex
s 2
\
parameter s by meromorphic continuation of the sums
1 ys
E (z):= =ys+ .
s 2 cz+d2s ···
(c,d)∈ZX2−{(0,0)}:| |
gcd(c,d)=1
ForRe(s)largeenoughandupto(important)normalizingfactors,Rankin–Selberg
theory and Shimura’s symmetric square integral representationgive
∞ λ (n)2
ϕ
L(ϕ ϕ,s):=ζ(2s) | | µ (E ),
× ns ≈ ϕ s
n=1
X
∞ λ (n2)
L(sym2ϕ,s):=ζ(2s) ϕ θE˜ ,ϕ ,
ns ≈h s i
n=1
X
QUANTUM VARIANCE ON QUATERNION ALGEBRAS, I 8
forsomeJacobithetafunctionθandhalf-integralweightEisensteinseriesE˜ . Com-
s
bining these identities with the factorization8 L(ϕ ϕ,s)=ζ(s)L(sym2ϕ,s) gives
×
with E˜′ :=ζ(s)E˜ that
s s
µ (E ) θE˜′,ϕ . (13)
ϕ s ≈h s i
Notethatϕ θE˜′,ϕ islinear, unlike ϕ µ (E ), andthatthe non-lineariden-
7→h s i 7→ ϕ s
tity (13), like the triple product formula,applies only when ϕ is an eigenfunction.9
Consider now the pair of unitary Eisenstein series Ψ ,Ψ := E ,E
1 2 1/2+it1 1/2+it2
attached to t ,t R. Let be an orthonormal basis of eigenfunctions for the
1 2
∈ F
discretespectrumofL2(SL (Z) H); write forthecorrespondingcontributionof
2
\ ···
thecontinuousspectrum,andignore(forthepurposesofthisformaldiscussion)that
most of the sums/integrals written below are divergent. From (13) and Parseval,
one obtains with h ,h :=E˜′ ,E˜′ that
1 2 1/2+it1 1/2+it2
µ (Ψ )µ (Ψ )+ θh ,ϕ ϕ,θh + = θh ,θh . (14)
ϕ 1 ϕ 2 1 2 1 2
···≈ h ih i ··· h i
ϕ∈F ϕ∈F
X X
Formally, quantum variance sums equal integrals of four-fold products of half-
integral weight theta functions.
The theta correspondence and seesaw duality give non-formal analogues (11) of
(14); the point of this article is thus to determine quantum variance asymptotics
byderivingsomesuchanaloguesandthenprovingasymptoticformulas(12)forthe
inner products that arise.
1.7. Further perspectives. We conclude this introduction by noting some rea-
sons to have anticipated an approach to the quantum variance problem using the
theta correspondence and half-integral weight forms:
(1) The quantumvariancetheoremsofLuo–Sarnak–Zhaogive new proofsthat
L(π,1) > 0 (see [43, p773, (2)]); earlier proofs as in [34, 33] used half-
2
integral weight forms.
(2) TheRallisinnerproductformulaimpliesthatthecompositionoftheMaass–
Shintani–Waldspurger lift (for fixed local data) with the Petersson inner
product has similar properties to those of the limiting bilinear formidenti-
fied in the works of Luo–Sarnak–Zhao;the presentwork confirms that this
similarity is no coincidence.
(3) Any progressonthe quantumvarianceproblempresupposessome progress
towardsthemultiplicityonetheorem;theknownconstructiveproofofthat,
due to Eichler and Shimizu, used the theta correspondence much as we do
here.
Acknowledgements. We are grateful to many people for their contributions to
this project. P.Sarnakintroducedus to his workwithW. Luoandemphasizedthe
problemofobtaininganalogousresultsoncompactquotients;wethankhimalsofor
much encouragement, inspiration and helpful feedback concerning earlier versions
of this work. Ph. Michel patiently entertained and offered helpful feedback and
encouragementonsomepresentationsofearlierformsofthemethod. Conversations
with K. Prasanna in connection with the work [47] were ultimately helpful also in
8This is the special case |λ(n)|2 = P λ(n2/d2) of the Hecke multiplicatively exploited by
d|n
Luo–Sarnak–Zhao.
9The analysis oftripleproduct averages mentioned above exploits the relatedidentity L(ϕ×
ϕ×Ψ,1/2)=L(sym2ϕ×Ψ,1/2)L(Ψ,1/2).
QUANTUM VARIANCE ON QUATERNION ALGEBRAS, I 9
connection with the present work. A. Venkatesh offered helpful encouragement,
feedback and comments which have improved our exposition. We thank also O¨.
Imamoglu, H. Iwaniec, E. Kowalski, D. Ramakrishnan, A. Saha, J. Tsimerman
for their encouragement and interest and W.T. Gan, B. Gross, R. Holowinsky, N.
Templier, Z. Rudnick, C. Skinner, K. Soundararajan for their feedback, interest
and/or questions on various aspects of this work.
We gratefully acknowledge the support of NSF grant OISE-1064866 and SNF
grant SNF-137488 during the work leading to this paper.
2. Preliminaries
We record definitions, some postponed from 1.
§
2.1. Generalities. Recall that G := GL (Q ), K := GL (Z ). Let B be the
2 2 2 2
quaternion algebra ramified at ,23 . Let R 6 B be a maximal order. Choose
{∞ }
an embedding of B into the matrix algebra M (Q ) under which R embeds in
2 2
M (Z ). Take for Γ the image of R[1/2]× under this embedding. Then Γ<G is a
2 2
discrete cocompact subgroup. The quotient X:=Γ G is compact.
\
The reduced norm and trace on B and its extensions are denoted nr,tr. A
superscripted 0 denotes “trace zero subgroup.” A subscripted 2 denotes “2-adic
completion.”
2.2. Hecke operators. For anodd positive integer n, the Hecke operatorT acts
n
on functions ϕ : X C by T ϕ(x) := ϕ(α−1x) where M := R[1/2]
→ n α∈Mn/Γ n ∩
nr−1(nZ[1/2]×). The group G acts on such functions by right translation, and
P
commutes with the Hecke operators.
2.3. Automorphic forms. Denote by (X) the space of smooth functions ϕ :
A
X C, i.e., those that are right invariant by some open subgroup of G. For a
→
subgroup S 6G, denote by (X)S the subspace of right S-invariant functions, or
A
equivalently, those that factor through the quotient X/S = Γ G/S. We may and
\
shall identify (X)S with a space of functions on X/S. The Hecke operators act
A
on it.
2.4. Measures. Equip G with the Haar measure assigning volume 2 to K, and X
with the quotient measure, denoted simply ; define L2(X) with respect to that
X
measure.
R
Recall that Y := Γ G/K = X/K. By the convention of 2.3, functions on Y
\ §
are identified with right K-invariant functions on X. For each E Y, choose a
representative g G. Set R := R[1/2] g R g−1; it is a max∈imal order in
E ∈ E ∩ E 2 E
B. Set w := #R×/Z× = (1/2)#R×. For Ψ : Y C, one then has Ψ =
E E E → X
Ψ(E)/w .
E∈Y E R
2P.5. Families of balanced newvectors. Let N be a positive integer. Recall the
definition (5) of K . By the multiplicity one theorem and local newvector
−N1..N2
theory,there is a unique (up to signs) maximalsubset N (X)K−N..N with the
F ⊂A
properties:
Each ϕ is an eigenfunction for the Hecke operator T for all odd
N n
• ∈ F
natural numbers n. We accordingly write T ϕ=√nλ (n)ϕ.
n ϕ
Each ϕ generates an irreducible representation of G under right
N
• ∈ F
translation.
QUANTUM VARIANCE ON QUATERNION ALGEBRAS, I 10
Each ϕ is real-valued and orthonormal, and any two ϕ,ϕ′ are
N N
• ∈F ∈F
orthogonal to one another.
Eachϕ isorthogonaltoanyfunctiononXthatisK -invariant
• ∈FN −N1..N2
fromsomeorderedpair(N ,N )withN 6N andN 6N and(N ,N )=
1 2 1 2 1 2
6
(N,N).
The multiplicity one theorem implies moreover that each ϕ is determined
N
∈ F
by its system of Hecke eigenvalues λ (n). The family is analogous to (and in
ϕ N
F
Hecke-equivariantbijectionwith;see 2.7)thesetofnormalizednewformsofweight
§
2 on Γ (22N 23).
0
·
2.5.1. Remark. We considerhere the families arisingfromthe “balanced”sub-
N
F
groupsK . Ourmethodappliestotheothersubgroups,suchastheunbalanced
−N..N
ones K more commonly denoted“K (2N)”, but the results obtained are nicer
0..N 0
for those considered here.
2.6. Conventionsonmodularforms. Wedenotebyz :=x+iyatypicalelement
of the upper half-plane and write q := e2πiz. “Modular” always means “modular
with respectto some congruence subgroupΓ′ of Γ (4).” We analytically normalize
0
holomorphic modular forms Φ of weight k 1Z by the factor yk/2, so that
∈ 2 >0
q-expansions read Φ(z) = yk/2 a qn and dilations Φ(z) Φ(az) for a Q×
n 7→ ∈ +
are unitary for the Petersson inner product, which we normalize by Φ ,Φ :=
1 2
P h i
z∈Γ′\HΦ1(z)Φ2(z)dν(z) for Γ′ small enough in terms of Φ1,Φ2 and ν = νΓ′ the
probability measure that is a multiple of y−2dxdy. Thus, for instance, 1,1 = 1,
R h i
regardless of the congruence quotient Γ′ H on which the constant function 1 is
\
regardedas living. Write Ψ := Ψ,Ψ 1/2.
k k h i
2.7. Eichler/Jacquet–Langlands lifts. For N >1 and ϕ , set
N
∈F
Φ (z):=y √nλ (n)qn
ϕ ϕ
n>1:
X
gcd(n,2)=1
It is known10 that Φ defines a weight 2 newform on Γ (22N 23) with Hecke
ϕ 0
·
eigenvalues √nλ (n) for odd natural numbers n.
ϕ
2.8. Harmonic weights. Recall from 1 that ι :=L(2)(adϕ,1). It is known[26]
ϕ
§
that ι = 2o(N) as N . By the theory of Eisenstein series,11 Φ 2 is the
ϕ ϕ
→ ∞ k k
residue as s 1+ for some sufficiently divisible M Z of the integral
>1
→ ∈
( Im(γz)s)Φ 2(z)y−2dxdy
ϕ
| |
ZΓ1(M)\H γ∈Γ∞X\Γ1(M)
which unfolds to ∞ ys nλ (n)2y2e−4πnyy−2dy and then simplifies to
y=0 n | ϕ |
∞ R dy P λ (n)2 Γ(s+1) L(S)(adϕ,s)ζ(S)(s)
ys+1e−4πy | ϕ | = ζ (s+1)
y ns (4π)s+1 23 ζ(S)(2s)
Zy=0 n∈XZ>1:
(n,2)=1
10ThiscanbededucedfromresultsofEichler(see[14],[53,§2])andAtkin–Lehner [1,Thm3
(iii)].
11seeforinstance[30,p138],[40,§5.1]