Arithmetic properties of the Shimura–Shintani– Waldspurger correspondence

80 Pages·2009·1.01 MB·English

Invent.math.176,521–600(2009)
DOI:10.1007/s00222-008-0169-z
Arithmetic properties of the Shimura–Shintani–
Waldspurger correspondence
Withanappendix byBrian Conrad(cid:1),(cid:1)(cid:1)
KartikPrasanna
DepartmentofMathematics,UniversityofMaryland,CollegePark,MD20742,USA
(e-mail:[email protected])
Oblatum12-VIII-2006&3-XI-2008
Publishedonline:19December2008–© Springer-Verlag2008
Tomyparents.
Abstract. We prove that the theta correspondence for the dual pair ((cid:1)SL ,
2
PB×), for B an indefinite quaternion algebra over Q, acting on modular
formsofoddsquare-freelevel,preservesrationalityand p-integralityinboth
directions. As a consequence, we deduce the rationality of certain period
ratiosofmodularformsandeven p-integrality oftheseratiosundertheas-
sumptionthat pdoesnotdivideacertain L-value.Therationalityisapplied
togiveadirectconstructionofisogeniesbetweennewquotientsofJacobians
ofShimuracurves, completelyindependent ofFaltings’isogenytheorem.
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521
2 Modularformsofintegralandhalf-integralweight. . . . . . . . . . . . . . . . . 528
3 Explicitthetacorrespondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540
4 ArithmeticpropertiesoftheShintanilift . . . . . . . . . . . . . . . . . . . . . . 549
5 ArithmeticpropertiesoftheShimuralift . . . . . . . . . . . . . . . . . . . . . . 558
6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587
AppendixA.AnintegralitypropertyfortheAtkin–Lehneroperator. . . . . . . . . . 591
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599
1. Introduction
In his seminal paper [27], Shimura initiated the systematic study of holo-
morphic modular forms ofhalf-integral weight and showed that one could
(cid:1) DepartmentofMathematics,StanfordUniversity,SloanHall,Stanford,CA94305,USA
(e-mail:[email protected])
(cid:1)(cid:1) PartiallysupportedbyNSFgrantDMS-0600919.
522 K.Prasanna
associate to a Hecke eigenform h of half-integral weight k + 1 a Hecke
2
eigenform f ofintegralweight2k suchthatthe p2-thFouriercoefficientof
hiscloselyrelatedtothe p-thFouriercoefficientof f.Thecorrespondence
whichassociates f toh isoften described astheShimuracorrespondence,
and f iscalledtheShimuraliftofh.Later,Shintani[33]describedamethod
togointheotherdirection,namelyconstructmodularformsofhalf-integral
weight beginning with forms of integral weight using the theta correspon-
dence. At around the same time, Niwa [21] also explained the original
Shimura lift in terms of theta lifts. (In the case of Maass forms, there is
a much earlier construction due to Maass [20] of the lift to forms of half-
integralweight;see[8]foranexposition.)
The relation between f and the square-free Fourier coefficients aν(h)
ofh remainedhighlymysterious, butforasuggestion ofShimura[29]that
theseshouldsomehowberelatedtospecialvaluesofL-functionsassociated
to f.Intworemarkable articles [36,37] Waldspurger settled this question,
showing (roughly) that aν(h)2 is proportional (as ν varies) to the value
L(k√, f ⊗χν) where χν is the quadratic character associated to the field
Q( ν).AcentraltoolthatWaldspurgeremploysisthethetacorrespondence
(cid:1)
between the groups SL and PGL as in the work of Shintani and Niwa.
2 2
In a later article [38], Waldspurger also studied the theta correspondence
for the pair ((cid:1)SL , PB×) for B a quaternion algebra, and its relation to the
2
Jacquet–Langlands correspondence betweenPGL andPB×.
2
Waldspurger’s results are representation-theoretic in nature. In particu-
lar, he does not study the arithmetic properties of the theta-lifts in either
direction.ThisissuewashoweverconsideredbyShimura[31],whoshowed
that (for suitable choices of theta function) the theta lift from (cid:1)SL to PB×
2
isalgebraic andfurther, intheopposite direction, thereisacanonical tran-
scendentalperiodmodulowhichthethetaliftisalgebraic.Inthisarticle,we
willproveanalogsofShimura’sresultsforrationalityoverspecifiednumber
fields and also p-adic integrality. As a consequence we deduce several re-
sultsrelating periodsofmodularformsondifferent Shimuracurves. These
results,infact,constitutedthemainmotivationforthisarticleandwebegin
bydescribing theminmoredetail.
Let N = N+N− beanoddsquare-free integer with N− aproduct ofan
even number of primes. Let f be a holomorphic newform of even weight
2k on Γ (N), g a holomorphic newform with respect to the unit group
0
of an Eichler order O(cid:3) of level N+ in the indefinite quaternion algebra B
ramified at the primes dividing N−, and with the same Hecke eigenvalues
as f.Let(F ,Φ)beapairconsisting ofaGaloisextension ofQthatsplits
0
B along with a suitable splitting Φ : B⊗F (cid:4) M (F ) (see Sect. 2.2.1).
0 2 0
Set F˜ = Q if 2k = 2 and F˜ = F otherwise. Let F be any number
0 0 0
˜
field containing F and all the Hecke eigenvalues of f, let p be a prime
0
not dividing N and λ a prime in F lying over p. As shown in [22] and
as will be recalled below, f and g may be normalized canonically up
to λ-adic units in F. One has attached to f and g, canonical fundamental
periodsu±(f,F,λ)andu±(g,F,λ),welldefineduptoλ-adicunitsinF.For
ArithmeticpropertiesoftheShimura–Shintani–Waldspurgercorrespondence 523
σ ∈Aut(C/F˜0),letu±(fσ,Fσ,λσ)andu±(gσ,Fσ,λσ)bethefundamental
periods attached to fσ and gσ. These periods are chosen such that the
period pair (u±(f,F,λ),u±(fσ,Fσ,λσ)) gives a well defined element in
(C× ×C×)/(1,σ)F× (and likewise with f replaced by g). To begin with,
wehavethefollowingtheoremonrationality ofperiodratios.
Theorem1.1.
(cid:2) (cid:3)
u±(f,F,λ) σ = u±(fσ,Fσ,λσ).
u±(g,F,λ) u±(gσ,Fσ,λσ)
In the special case k = 2, the above theorem can be used to construct
directly isogenies definedoverQbetweenquotients ofJacobians ofdiffer-
ent Shimura curves, without the crutch of Faltings’ isogeny theorem. This
application is treated in the last section of the article. (The idea that one
should be able to construct such isogenies by proving the rationality of
periodratioswassuggestedbyShimura[31].)Inthecaseofhigherweight,
one might be able to use Theorem 1.1 to derive relations between the mo-
tives associated to the forms f and g, but we have not pursued this theme
furtherinthisarticle.
Indeed, our main interest is in integrality results for the ratios appear-
ing above. With this in mind, let us define u±(f) (resp. u±(g)) to be
u±(f,F,λ) (resp. u±(g,F,λ) for any choice of F, so that both periods
are well defined up to λ-adi(cid:4)c u(cid:5)nits. Let ν be a quadratic discriminant and
χν the quadratic character ν· . It is known under rather general condi-
tions (see [34]) that A(f,ν) := |ν|k−1g(χν)(2πi)−kL(k, f,χν)/u±(f) =
|ν|k−1g(χν)(2πi)−kL(12,πf ⊗χν)/u±(f)isaλ-adicinteger,whereg(χν)is
the Gauss sum attached to χν and the ± sign holds according as χν(−1)·
(−1)k = ±1. Hereπ denotes the automorphic representation ofPGL at-
f 2
tachedto f andtheL-functionisbeingevaluatedatthecenterofthecritical
strip, thisbeing thepoint s = k inthe classical normalization and s = 1/2
intheautomorphic normalization.
The integrality result we have in mind is motivated by the following
observation. If f hasweight2,andλisnotEisensteinfor f (i.e.the modλ
Galois representation associated to f is irreducible), one may show, again
using Faltings’ isogeny theorem that u±(f)/u±(g) is a λ-adic unit. So it
is reasonable to ask if such a result holds for arbitrary even weights. The
followingtheorem providesaconditional resultinthatdirection.
(cid:6)
Theorem1.2. Suppose p >2k+1and p(cid:1)N˜ := q(q+1)(q−1).Let
q|N
χν bethequadratic character associated toanoddfundamental quadratic
discriminant ν and set (cid:7) = sign((−1)kν). Suppose A(f,ν) (cid:6)≡ 0mod λ.
Then
(cid:2) (cid:3)
vλ uu(cid:7)(cid:7)((gf)) ≥ 0.
524 K.Prasanna
It is naturally of interest then to ask if there always exists a quadratic
discriminantνwithprescribedsignandparitysuchthat A(f,ν) (cid:6)≡0mod λ.
Thisquestioningeneralseemstobeextremelyhard.However,asmentioned
above, in the case of weight 2 (for instance for elliptic curves) and non-
Eisensteinprimesλ,weknowapriorifromFaltings’isogenytheoremthat
u(cid:7)(f)/u(cid:7)(g)isaλ-adicunit.Feedingthisinformationintothemethodsand
resultsofthisarticle,oneobtainsinterestingapplicationstoquestionsabout
the p-divisibility of the central values of quadratic twists of f. Assuming
theexactformoftheBirch–Swinnerton–Dyer conjectureforellipticcurves
of rank 0, one further gets applications to questions about p-torsion of
Tate–Shafarevich groups. These applications are treated in a subsequent
article ([24]), in which we also explain an intriguing relation between the
(cid:1)
Waldspurger packet on SL and congruences of modular forms of integral
2
andhalf-integral weight.
ThereaderwillnotethatthestatementsofTheorems1.1and1.2donot
involveformsofhalf-integralweight.Nevertheless,theirproofdependscru-
ciallyonarithmeticpropertiesoftheShimuracorrespondence andofforms
ofhalf-integral weight. Wenowgiveanintroduction toourmaintheorems
regardingtheShimuracorrespondence andthemethodsofthisarticle.
Suppose χ isacharacter ofconductor N(cid:3) dividing 4N withχ(−1) = 1
(cid:4) (cid:5) (cid:4) (cid:5)
and set M = lcm(4,NN(cid:3)).Setχ = χ · −1 k,let χ = χ · −1 k+τ (where
0 · ·
τ = 0 or 1) be such that χ is unramified at the prime 2, and use the same
symbolsχ0andχtodenotetheassociatedadeliccharacters.Alsosuppose fχ
andgχarenewformsinπ⊗χandπ(cid:3)⊗χrespectivelywhereπandπ(cid:3)arethe
automorphic representations ofGL (A)and B×(A)associated to f and g.
2
ItfollowsthenfromworkofWaldspurgerthatthespace Sk+1(M,χ, fχ)
consisting of holomorphic forms of weight k + 1 on Γ (M)2 with char-
2 0
acter χ, and whose Shimura lift is fχ, is two dimensional. Further this
spacehasaunique onedimensional subspace, calledtheKohnensubspace
S+ (M,χ, fχ),consisting offormswhoseonlynonvanishingFourierco-
k+1
effic2ientsaξ are(possibly)thosesuchthat(−1)τξiscongruentto0,1 mod4.
Letusdenotebyhχ anonzerovectorinthissubspacewithalgebraicFourier
coefficients. We may normalize hχ to have all its Fourier coefficients be
λ-adic integers in Q(f,χ), and further so that at least one is a λ-adic unit.
Here Q(f,χ) is the field generated over Q by the Hecke eigenvalues of f
andthevaluesofthecharacterχ.
Theformhχ mayinfactbeobtainedasathetaliftfromPB×asfollows.
For q|N, denote by w and w(cid:3) the signs of the Atkin–Lehner involutions
q q
acting on f and g respectively, so that w = ±w(cid:3), the + (resp. −) sign
q q
holding exactly when B is unramified (resp. ramified) at q. Fix ν, an odd
quadratic fundamental discriminant such that (−1)τ = sign(ν) and such
thatthefollowinglocalconditions aresatisfiedattheprimesdividing N:
(a) Ifq|N butq(cid:1)ν,χ0,q(−1) =w(cid:3)q ·χν,q(q).
(b) If q|N and q|ν, χ0,q is ramified exactly when q|N− and for such q,
χ0,q(−1) =−1.
ArithmeticpropertiesoftheShimura–Shintani–Waldspurgercorrespondence 525
Letusdenotebyg(cid:3)theformgχ⊗(χχν◦Nm)−1 ∈π(cid:3)⊗χν.Onenowcon-
sidersthethetacorrespondence forthedualpair((cid:1)SL ,PB×).Itisshownin
2
Sect.3belowthattheconditions(a)and(b)aboveimply(againfromworkof
Waldspurger)thattheformhχ occursinthethetaliftΘ(π(cid:3)⊗χν,ψ(cid:3))where
ψ(cid:3) = ψ1/|ν| and ψ is the usual additive character on Q\AQ. Let V be the
subspaceof Bconsistingofthetrace0elements.Foranappropriateexplicit
choiceofSchwartzfunctionϕ ∈ V(A)(seeSect.3),onehasθϕ(g(cid:3))=α0hχ
and θϕt(hχ) = βg(cid:3) for scalars α0 and β. The arithmetic properties of the
complex numbers α and β are then of crucial importance. It will turn out
0
that β is algebraic, while α0 is an algebraic multiple of the period u(cid:7)(g)
where(cid:7) =sign((−1)kν).Infactitisnaturaltowriteα0 =αg(χ)u(cid:7)(g),and
ik+τβ =g(χ)−1β,whereg(χ)istheGausssumattachedtoχ.Thefollowing
isone ofourmaintheorems regarding theShimura–Shintani–Waldspurger
correspondence.
Theorem1.3. The complex numbers α,β are algebraic, and α ∈ F(χ),
β ∈ Q(f,χ).Further,assuming p > 2k+1and p(cid:1)N˜,wehave
(a) vλ(α)≥ 0,
(b) vλ(β) ≥ 0.
The algebraicity of αand β is due to Shimura [31]; our contribution is
therationality oftheseover F(χ),Q(f,χ)respectively andtheλ-adicinte-
grality. It turns out that the theorem for α is quite easy and with adequate
preparation, is almost tautological (see Sect. 4). Onthe other hand, the ra-
tionality and integrality ofβ is much harder and requires the very detailed
analysis of Sect. 5. Here is a brief description of the ideas involved. To
checkforrationality orintegrality ofβ,itsufficestoevaluateθϕ(hχ)=βg(cid:3)
at specific CM points j : K (cid:15)→ B associated to an imaginary quadratic
field K andcheckthattheresultingvaluesarerationalorintegralmultiples
ofappropriate CMperiods.Fromacomputational pointofview,itiseasier
to compute a sum of values at all Galois conjugates of a Heegner point,
twistedbyaHeckecharacterη(cid:3);theresultingsumisinterpretedasaperiod
integral Lη(cid:3) on a torus. Now one applies see-saw duality. It turns out that
this israther subtle, involving the choice of twocharacters κ,µdepending
onη(cid:3).HereκisaHeckecharacterofK ofweight(k,0)atinfinity,whileµis
afiniteordercharacterofQ×.Furtherthepair(µ,η)isonlywelldefinedup
A
toreplacementby(κ·(ω◦NmK/Q),µ·ω2)foranyfiniteordercharacterω
ofQ×A.Letπµdenotetheautomorphicthetarepresentation of(cid:1)SL2(A)asso-
ciatedtoµandπκ theautomorphicrepresentation ofGL2(A)associated to
the Hecke character κ. Then by an application of see-saw duality one gets
roughly anexpression for Lη(cid:3) asatripleintegral
(cid:7)
(cid:4) (cid:5)
(1.1) Lη(cid:3) θϕt(hχ) = hχ(σ)θµ(σ)θκ(σ)dσ,
SL2(Q)\SL2(A)
for some vectors θµ and θκ lying in πµ and πκ respectively. Let K0 be the
trace 0 elements of K, and K⊥ the orthogonal complement to K for the
526 K.Prasanna
norm form on B. With respect to the decompositi(cid:8)on V = K0 + K⊥, the
Schwartz function ϕ ∈ V(A) splits up as a sum i∈I ϕ1,i⊗ϕ2,i over an
indexingset I.Moreprecisely,whatonegetsthenisnotasingleintegralof
the form (1.1) but in fact a sum of such integrals indexed by the set I and
depending on the splitting of the pure tensor ϕ as a sum of pure tensors.
The data of such splitting is in general highly ramified, as are the local
representations involved, and so one needs an elaborate argument to show
that the sum of integrals so obtained may indeed be replaced by a single
integral with convenient choices of vectors in πµ and πκ. This argument
occupies allofSect.5.2.Weshould remarkherethattheweightsofhχ,θµ
and θκ are k + 1, 1 and k +1 respectively. As for the possibilities for the
2 2
local representations at non-archimedean primes, many different types of
ramification could occur, including forinstance thepossibility that πµ and
πκ areboth supercuspidal, eventhough wehaverestricted theramification
ofπ tobeatworstSteinberg.1
f
Theupshot oftheargument isthatonehasanexpression fortheperiod
integral as c·(cid:12)H,θκ(cid:13) for some constant c (that depends on f,χ,κ,µ) and
a modular form H of weight k + 1 with coefficients that are λ-integral
and lie in Q(f,χ), (cid:12)·,·(cid:13) being the usual Petersson inner product. (It is
at this point we make use of the appendix due to Brian Conrad; indeed
the form H is naturally presented as w H for a form H with λ-integral
Q 0 0
Fourier coefficients and an Atkin–Lehner operator w with Q|N2. The
Q
maintheoremoftheappendixguaranteesthenthat H hasλ-integralFourier
coefficients as well.) Now one applies an argument similar to that of the
authors’previousarticle[22]toshowthatc·(cid:12)H,θκ(cid:13)/Ωisaλ-adicintegerfor
asuitableCMperiodΩ.Oneneedstouseherearefinedstudyofcongruences
betweenθκandotherformsaswellasthemainconjectureofIwasawatheory
fortheimaginaryquadraticfield K,whichisadeeptheoremofRubin[25].
The constant c above arises from the delicate computations with the local
integrals mentioned above, and is a p-integer but not necessarily a p-unit.
Miraculously, its p-adicvaluation turnsouttobeexactlywhatisneededto
maketheargumentusingIwasawatheoryandcongruencesgothrough.One
needstobeparticularly carefulheresincethechoiceofauxiliary quadratic
discriminantνintroducesextralevelstructureintotheproblem,andwithan
eyeonapplications, onedoesnotwanttomakeanyassumptionsonνother
than those inTheorem 1.2. Therationality proceeds somewhat differently:
theCMperiodmustbechosenmorecarefully(todependonκ),andonethen
needstoapplytherationality resultsofBlasius[2]forthespecialvaluesof
L-functions ofGrossencharacters of K.
To use the integrality of α and β we need several formulas. In what
follows we will use the symbol ∼ to denote equality up to less important
factors, andreferthereadertothemaintextofthearticle formoreexplicit
1 Ifµisthetrivialcharacter,θµ isanEisensteinseries.Inthiscase,theintegral(1.1)is
identifiedwiththevaluesats = k (intheclassicalnormalization) oftheRankin–Selberg
DirichletseriesD(s,hχ,θκ)associatedbyShimuratothecuspformshχ andθκ ofweights
k+ 1 andk+1respectively.
2
ArithmeticpropertiesoftheShimura–Shintani–Waldspurgercorrespondence 527
equations.CrucialtousisaformulafortheFouriercoefficientsofthetheta
liftθϕ(g(cid:3))thatisprovedin[23].Thisformulastatesroughly that
(cid:2) (cid:3) (cid:2) (cid:3)
(1.2) (cid:9)(cid:9)aξ(cid:4)θϕ(g(cid:3))(cid:5)(cid:9)(cid:9)2 ∼ L 21,π ⊗χν L 12,π ⊗χξ0 (cid:12)(cid:12)gf,,gf(cid:13)(cid:13)
for ξ = (−1)τξ satisfying a particular set of congruence conditions. This
0
formula is used in two ways. Firstly it shows that the theta lift θϕ(g(cid:3))
is nonvanishing for the particular choice of Schwartz function ϕ since
L(1,π⊗χν) (cid:6)= 0 and we can find a ξ such that L(1,π⊗χξ ) (cid:6)= 0. Sec-
2 2 0
ondly, comparing it with the following formula of Baruch–Mao [1] which
isprovedusingtherelativetraceformulaofJacquet,
(cid:4) (cid:5)
(1.3) |aξ(hχ)|2 ∼ L 12,π ⊗ξ0 ,
(cid:12)h,h(cid:13) (cid:12)f, f(cid:13)
andapplying see-sawduality
(cid:10) (cid:11) (cid:10) (cid:11)
(1.4) θϕ(g(cid:3)),hχ = g(cid:3),θϕt(hχ),
oneobtainsthefollowingimportantformula
(cid:2) (cid:3)
1
(1.5) L ,π ⊗χν ∼αβu(cid:7)(g).
2
The integrality of u(cid:7)(f)/u(cid:7)(g) follows immediately from (1.5) using the
integrality of α and β and the assumption on A(f,ν) being a p-unit. As
abonus, ifonecombines(1.5)with(1.4),onegets
(cid:2) (cid:3)
(cid:10) (cid:11)
1
(1.6) θϕt(hχ),θϕt(hχ) ∼ L ,π ⊗χν (cid:12)hχ,hχ(cid:13)
2
whichisnothingbuttheexplicitversionoftheRallisinnerproductformula
in this situation, obtained in a completely different way than the original
methodofRallis!
It would be very interesting to generalize the results of this article to
totallyrealfieldsotherthanQ,butthisseemstobemuchharder.Forinstance,
for a real quadratic field, one would like integral period relations between
the periods usually denoted u++,u+−,u−+ and u−−. Another interesting
questionistostudytheintegralitypropertiesofthetaliftsfrom(cid:1)SL toPB×
2
for B a definite quaternion algebra over Q. Very surprisingly, this seems
harderthantheindefinitecase:thereadermayfindadiscussionoftheissues
involved inthearticle[24].
The article is organized as follows. Sect. 2 contains preliminaries on
modular forms of integral and half-integral weight and some results ex-
tracted from Waldspurger’s article [37]. In Sect. 3, wework out, using the
resultsofWaldspurger’s article[38], somefactsregarding thethetacorres-
pondence for ((cid:1)SL ,PB×) and study the same for a certain explicit choice
2
528 K.Prasanna
of theta function. Sections 4 and 5 are devoted to proving the rationality
and integrality of the Shintani and Shimura lifts respectively. Finally, in
Sect. 6 we explain in more detail the various formulas mentioned above,
and discuss the applications to arithmeticity of period ratios and isogenies
betweennew-quotients ofJacobians ofShimuracurves.
Acknowledgements. TheauthorwouldliketothankDonBlasius,HaruzoHida,SteveKudla,
JonRogawski,ChrisSkinnerand AkshayVenkatesh foruseful discussions, PeterSarnak
forpointingouttheworkofMaassreferredtoabove,BrianConradforverykindlyagreeing
toprovidetheAppendixandMichaelHarrisforhiscommentsandacorrectiontoanearlier
version of thisarticle. Inaddition, thanks aredue totheanonymous refereefor acareful
reading of the article and numerous comments towards improving it. Finally, it will be
cleartothereaderthattheauthorowesatremendousintellectualdebttoShimura,Shintani
andespeciallyWaldspurger,whoseverypowerfultechniquesandresultsprovideastepping
stoneonwhichthisarticlebuilds.
2. Modularformsofintegralandhalf-integralweight
2.1. Preliminaries
2.1.1. Metaplectic groups. Here we follow the exposition and notations
of [37, II §4]. If v is a place of Q, let(cid:12)Sv denote the metaplectic (degree 2)
ocofvSeLr2o(fAS)L.2W(Qev)m.aLyikiedweinsteif,yle(cid:12)tS(cid:12)SvA(rdeesnpo.t(cid:12)SeAth)ewmitehtaSpLle2c(tQicv()d×egr{e±e12})(croevsepr.
SL (A)×{±1}),theproductoftwoelements(σ,(cid:7)),(σ(cid:3),(cid:7)(cid:3))beinggivenby
2
(σ,(cid:7))(σ(cid:3),(cid:7)(cid:3))=(σσ(cid:3),(cid:7)(cid:7)(cid:3)β(σ,σ(cid:3))),
where βv is defined as follows. For σ = (ac db) ∈ SL2(Qv), let x(σ) = c if
c(cid:6)=0,x(σ)=d,ifc =0.Forvreal,letsv(σ)=1.Forv =qafiniteplace,
let sv(σ) = (c,d)v if cd (cid:6)= 0 and vq(c) is odd, sv(σ) = 1 otherwise. Here
(·,·)v denotestheHilbertsymbol.Then
βv(σ,σ(cid:3))=(x(σ),x(σ(cid:3)))v(−x(σ)x(σ(cid:3)),x(σσ(cid:3)))vsv(σ)sv(σ(cid:3))sv(σσ(cid:3)).
If σ ∈ SL2(Qv), we denote(cid:6)also by the same symbol σ the element
(σ,1) ∈(cid:12)Sv. The map σ (cid:15)→ (σ, v(sv(σ))),σ ∈ SL2(Q)is a homomorph-
ismofSL2(Q)into(cid:12)SA,theimageofwhichwedenotebythesymbolSQ.
Forx ∈Qv,α ∈Q×v,definen(x),n(x)andd(α)tobetheelementsof(cid:12)Sv
givenby
(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)
1 x 1 0 α 0
n(x)= , n(x)= , d(α)= .
0 1 x 1 0 α−1
(cid:4) (cid:5)
Letw= −01 10 ∈(cid:12)Sv andnoticethat
(2.1) n(x) =d(−1)·w·n(−x)·w,
(cid:12)
inSv,arelationthatwewilluserepeatedly.
ArithmeticpropertiesoftheShimura–Shintani–Waldspurgercorrespondence 529
2.1.2. For t ∈ Q×v and ψ an additive character of Qv, let γψ(t) be the
constant associated by Weilto the character ψ and the quadratic form tx2.
Recallthatforv =q,afiniteprime,γψ(t)maybecomputedtobe
(cid:7)
(2.2) γψ(t)= lim ψ(tx2)dtx,
n→∞ q−nZq
whered x isHaarmeasurechosentobeautodualwithrespecttothepairing
t
(x,y)(cid:15)→ψ(txy).Wedenoteγψ(1)simplybythesymbolγψ.Define
(2.3) µψ(t) =(t,t)vγψ(t)γψ(1)−1 =γψ(1)γψ(t)−1.
Thenonehastheequalities:
µψ(tt(cid:3))=(t,t(cid:3))vµψ(t)µψ(t(cid:3)).
µψ(t2)=1.
Thus µψ defines a genuine character of Q×v, the extension of Q×v by
{±1}givenbytheHilbertsymbol.Forα ∈Q×,letψα denotethecharacter
v
definedbyψα(x) =ψ(αx).Onecheckseasilythat
µψα(t)=(α,t)vµψ(t).
2.1.3. Let(V,(cid:12),(cid:13))beaquadraticspac(cid:8)eoverQvandψanadditivecharacter
of Qv. Suppose Q(x) := 12(cid:12)x,x(cid:13) = di=1aixi2 in terms of an orthogonal
basisfor V,whered = dim(V).Set
(cid:13)d
γ := γ .
ψ,Q ψai
⎧ i=1
⎪⎪ (cid:13)d
⎪
⎪⎪(−1)(d−1)/2 a ifd isodd,
⎨ i
D := i=1
Q ⎪⎪ (cid:13)d
⎪
⎪⎪⎩(−1)d/2−1 ai ifd iseven.
i=1
Thenthereexistsarepresentationrψ of(cid:12)SvonSψ(V),theSchwartzspace
of V,calledtheWeilrepresentation, whichischaracterized by
(2.4) rψ(n)ϕ(x)= ψ(nQ(x))ϕ(x),
(2.5) rψ(d(α))ϕ(x)= µψ(α)d(α,DQ)v|α|d/2ϕ(αx),
(2.6) rψ(w)ϕ(x)= γψ,QFψ(ϕ)(x),
(2.7) rψ(1,(cid:7))ϕ(x)= (cid:7)dϕ(x),
where Fψ denotes the Fourier transform with respect to the pairing
(x ,x ) (cid:15)→ ψ((cid:12)x ,x (cid:13)), the Haar measure on V being chosen such that
1 2 1 2
Fψ(Fψ(ϕ))(x)=ϕ(−x)forallϕ ∈Sψ(V).
530 K.Prasanna
2.1.4. Suppose q is an odd prime. Let ψˆ be the character on Z/qZ given
byψˆ(1) = e−2πi/q andχˆ anycharacter on(Z/qZ)×,extended toZ/qZby
settingχˆ(0)= 0.DefinetheGausssum
(cid:18)
G(χˆ,ψˆa)= χˆ(δ)e−2πiaδ/q,
δ∈(Z/qZ)×
sothat G(χˆ,ψˆa)=χˆ−1(a)G(η,ψˆ).If(cid:21) isthe unique nontrivial quadratic
character of(Z/qZ)×,
(cid:19)√
G((cid:21),ψˆ) = √q, ifq ≡1 mod 4.
i q, ifq ≡3 mod 4.
Hence G((cid:21),ψˆ)2 =(cid:21)(−1)q.
2.1.5. LetqbeafixedfiniteprimeandψthecharacteronQ withkernelZ
q q
such that ψ( 1 ) = e−2πi/qn.Ifq (cid:6)= 2and t ∈ Z×,one easily computes that
qn q
γψ(t)=1andµψ(t) =1.Thus
(2.8) µψα(t) =(α,t)v,
foranyα ∈Q×q.Ifq = 2,µψ(t) = 12[1−i+(1+i)χ−1,2(t)]fort ∈Z×2.In
particular,µψ(−1)= −i.Notethatµψα(−1)=(−1,α)2·i andµψ(α)3 =
(α,α)2µψ(α)=(−1,α)2µψ(α).
Suppose now that q is odd, and ψ(cid:3) = ψα with v (α) = −1, qα ≡
q
a mod q,a ∈ (Z/qZ)×. Then set G(χˆ,ψ(cid:3)) := G(χˆ,ψˆa). One computes
from(2.2)that
(2.9) γψ(cid:3) =q−1/2G((cid:21),ψˆa)=q−1/2G((cid:21),ψ(cid:3))= (cid:21)(a)q−1/2G((cid:21),ψˆ).
Ifq = ∞,andψ(x)= e2πix wehaveµψ(−1)=i.
2.1.6. Let χ be a Dirichlet character of conductor M. We denote by χ
theassociated Grossencharacter ofQ×,satisfying χ (q)= χ(q)foralmost
A q
all q. If χ is a character of Q× of conductor q, we denote (in Sect. 3.2
q q
alone)byχˆ theinducedcharacter onZ×/(1+qZ )(cid:4)(Z/qZ)×.
q q q
2.1.7. Measures. Weusethesameconventionshereasin[22].Intheinter-
estofbrevity,thereaderisreferredtoSect.1ofthatarticleforthemeasure
normalizations used on the different local and adelic groups, the only dif-
ference being that the indefinite quaternion algebra is called D in [22] as
opposed to B inthepresent article.
2.2. Modularformsofintegralweightonanindefinitequaternionalge-
bra. Thereisnothingoriginalinthissection,theonlypurposeofwhichis
tosetupnotation.