THE THETA CORRESPONDENCE, PERIODS OF AUTOMORPHIC FORMS AND SPECIAL VALUES OF STANDARD AUTOMORPHIC L-FUNCTIONS 5 1 PATRICKWALLS 0 2 n a Contents J 7 Introduction 1 1. The Theta Correspondence 2 ] 2. Periods of Automorphic Forms 4 T 3. Special Values of Standard Automorphic L-functions 14 R References 17 . h t a m Introduction [ The zerosand poles ofstandardautomorphic L-functions attachedto representationsofclassical 1 v groups are linked to the nonvanishing of lifts in the theory of the theta correspondence. This deep 5 and subtle connection is formulated in terms of the Rallis inner product formula, the Siegel-Weil 6 formula and its extensions, and was developed over several years by many authors (for symplectic- 5 orthogonaldualpairssee[Ral84a],[Ral84b],[Ral87],[KR88a],[KR88b],[KR94],forotherdualpairs 1 see [Ike96], [Ich01], [Ich04], [Ich07], [Yam11] and [Yam13], and Weil’s original work [Wei65]). We 0 . referto the recentwork[GQT]foranexcellentaccountofthis theoryaswellasthe proofofthe last 1 remaining piece of the regularized Siegel-Weil formula in the second term range. 0 In this paper, we take the following as our point of departure. Let σ be a cuspidal automorphic 5 1 representationofthesymplecticgroupSpndefinedoveranumberfieldF. Thereisanentireoceanof : orthogonalgroups to which σ may lift by the theta correspondence. We may imagine the collection v oforthogonalgroupsasacollectionofWitttowers(cf.Section 1.4)withorthogonalgroupsattached i X to anisotropic quadratic spaces of all different dimensions and characters at their bases. Rallis’s r tower property shows that σ must lift to a nonzero cuspidal representation at some index in each a Witt towerand Rallis introduced his inner product formula to derive nonvanishing criteriain terms of automorphic L-functions. Theresultsofthispapershowthatwhenacuspidalrepresentationσ ofSp does lifttoacuspidal n representation π = θ (σ) of an orthogonal group O(V) (where1 dim V = m is even), the Fourier ψ F coefficients of automorphic forms in σ are linked to the orthogonalperiods of automorphic forms in π (cf. Theorem 2.4.1 and Theorem 2.5.1). Consequently, when our results are combined with the Rallisinner productformulainthe convergentrangeorthe secondtermrange(ie.whenm>2n+1 or V is anisotropic,cf. (3.1.1) and (3.3.1)), we prove a special value formula (cf. Theorem 3.5.1) for the standardautomorphicL-function L(s,σ,χ ) attachedto σ (and twisted by the characterχ of V V Date:January8,2015. 1Wehave chosen to restrictto symplectic-orthogonal duals pairs(Spn,O(V))wheredimFV =miseven forthe sakeofconvenience andclarity. Ourresultsapplywithminormodifications tothecase whenmisoddandGisthe metaplecticcover ofSpn,aswellastothecaseforunitarydualpairs. 1 2 PATRICKWALLS V) at the point s =m/2−(2n+1)/2 in terms of the Fourier coefficients of automorphic forms m,2n in σ and orthogonal periods of forms in π. Notation. Weusethefollowingnotationthroughoutthepaper. LetF beanumberfieldwithringof integersO andring ofadeles A=A . We fix anadditive unitary class characterψ :A/F −→C×. F F Let A = F be the product of the completions of F at the archimedean places and let ∞ v|∞ v A = ′ F be the ring of finite adeles (ie. the restricted product of the completions of F at f v<∞Qv nonarchimedean places with respect to the rings of integers O ). For an algebraic group G over Q Fv F, let [G] = G(F)\G(A) be the adelic quotient, dg is any fixed Haar measure on G(A) and we write vol[G] for the measure of [G] with respect to this measure. The results of this paper are independent of the choice of measures. The inner product of square-integrable functions on [G] is written hf1,f2i = [G]f1(g)f2(g)dg. Finally, tA is the transpose of a matrix A and |·|A = v|·|v is the adelic norm. R Q 1. The Theta Correspondence LetF beanumberfieldwithA=A itsringofadelesandfixonceandforallanadditiveunitary F classcharacterψ :A/F −→C×. Inthissection,weintroducethethetacorrespondenceandtheWeil representation for symplectic-orthogonal dual pairs, the Rallis tower property and the cuspidality of theta lifts. This material is standard and so our treatment will be brief. We refer to [MVW87], [Kud] and [RR93] for an introduction to the Weil representation and the theta correspondence (see also Weil’s original paper [Wei64]) and Rallis’s original work [Ral84b] for his tower property. 1.1. SymplecticandOrthogonalGroups. LetV beavectorspaceoverF suchthatdim V =m F is even and let ( , ) be a nondegenerate symmetric bilinear form on V. Let H = O(V) be the orthogonalgroup over F attached to the pair V,( , ). Let W be the standard vector space over F of dimension 2n equipped with the nondegenerate skew-symmetric bilinear form h , i defined by the matrix 0 1 J = n −1 0 n (cid:18) (cid:19) where1 istheidentitymatrixofsizen. LetG=Sp bethesymplecticgroupoverF attachedtothe n n pairW,h, i. Inparticular,weviewW asthespaceofrowvectorsofsize2nandtheautomorphisms of W as matrices acting by right multiplication therefore G=Sp = g ∈GL :gJtg =J . (1.1.1) n 2n TheSiegelparabolicsubgroupP ⊂Sp h(cid:8)asadecompositionP(cid:9)=MN wheretheLevicomponent n M is a 0 M = m(a)= :a∈GL (1.1.2) 0 ta−1 n (cid:26) (cid:18) (cid:19) (cid:27) and the unipotent radical N is 1 b N = n(b)= :b∈Sym (1.1.3) 0 1 n (cid:26) (cid:18) (cid:19) (cid:27) where Sym denotes symmetric matrices of size n. n We restrict to the case that m is even for convenience and clarity. The experienced reader will see that the results contained in this paper extend with minor modifications to the case where m is odd and G is the metaplectic cover of Sp . n THE THETA CORRESPONDENCE, PERIODS AND SPECIAL VALUES OF L-FUNCTIONS 3 1.2. The WeilRepresentation. Attachedtothedualpair(G,H)isaWeilrepresentationω =ω ψ (relative to the fixed character ψ) which is an action of G(A)×H(A) on the space S(V(A)n) of Schwartz functions on V(A)n ={v =(v ,...,v ):v ,...,v ∈V ⊗ A} . (1.2.1) 1 n 1 n F This model of the Weil representation satisfies the familiar formulas for ϕ∈S(V(A)n) ω(h)ϕ(v)=ϕ(h−1v) h∈H(A) ω(m(a))ϕ(v)=χ (deta)|deta|m/2ϕ(va) m(a)∈M(A) (1.2.2) V A ω(n(b))ϕ(v)=ψ(trbQ[v])ϕ(v) n(b)∈N(A) where m(m−1) χV(x)= x,(−1) 2 detV (1.2.3) A (cid:16) (cid:17) is the character of the quadratic space V and ( , )A is the global Hilbert symbol, and 1 Q[v]= ((v ,v )) ∈Sym (A) (1.2.4) 2 i j i,j n where ((v ,v )) is the symmetric matrix of inner products of the components of v =(v ,...,v ). i j i,j 1 n 1.3. The Theta Correspondence. Each ϕ∈S(V(A)n) defines a theta function θ (g,h)= ω(g)ϕ(h−1v) , g ∈G(A) , h∈H(A) (1.3.1) ϕ v∈Vn X which is an automorphic form on the product G(A)×H(A). In particular, θ is left invariant by ϕ G(F)×H(F)andisofmoderategrowth. Notethattheformationofthetafunctionsdefinesa(G(A)× H(A))-equivariantmapfromtheWeilrepresentationω tothespaceA(G×H)ofautomorphicforms on G(A)×H(A) ω −→A(G×H):ϕ7→θ . (1.3.2) ϕ Theta functions serve as kernel functions for the theta correspondence which map automorphic forms on one group to automorphic forms on the other. In particular, let fH be a cuspidal auto- morphic form on H, let ϕ∈S(V(A)n) and define the theta lift of fH to G by θ fH(g)= θ (g,h)fH(h)dh . (1.3.3) ϕ ϕ Z[H] This integral is convergent since fH is cuspidal and θ is of moderate growth. For a cuspidal ϕ automorphicrepresentationπ ofH,the theta lift ofπ to Gis the spaceθ (π) ofalltheta lifts θ fH ψ ϕ for ϕ∈S(V(A)n) and fH ∈π. Similarly, let fG be a cuspidal automorphic form on G, let ϕ ∈ S(V(A)n) and define the theta lift of fG to H by θϕ∨fG(h)= θϕ(g,h)fG(g)dg . (1.3.4) Z[G] Again, this integralis convergentsince fG is cuspidal and θ is of moderate growth. For a cuspidal ϕ automorphicrepresentationσ ofG,thethetaliftofσ toH isthespaceθψ(σ)ofallthetaliftsθϕ∨fG for ϕ∈S(V(A)n) and fG ∈σ. Our definitions of theta lifts imply the following adjoint property: if fG and fH are cuspidal automorphic forms on G and H respectively and ϕ ∈ S(V(A)n) such that both theta lifts θϕ∨fG and θ fH are cuspidal (ensuring that every integral below is absolutely convergent),then ϕ fG,θϕfH G = θϕ∨fG,fH H . (1.3.5) (cid:10) (cid:11) (cid:10) (cid:11) 4 PATRICKWALLS Furthermore,weknowbytheworkofMoeglin[Mœg97](generalizing[Ral84b])thatifσisacuspidal representation of G such that its theta lift consists of cusp forms, then π = θ (σ) is an irreducible ψ cuspidal automorphic representation of H and θ (θ (σ))=σ . (1.3.6) ψ ψ Finally, the analogous statement for representations of H also holds. 1.4. Cuspidality and Rallis’s Tower Property. Natural questions arise immediately: When is a theta lift nonzero? When is a theta lift cuspidal? These questions were first addressed by Rallis and the tower property. Let(V ,Q )beananisotropicquadraticspaceoverF andletH bethecorrespondingorthogonal 0 0 0 group. LetHbethehyperbolicspaceofdimension2equippedwiththequadraticformQ(x,y)=xy. For each r ≥1, define the quadratic space V =V ⊕H⊕···⊕H (1.4.1) r 0 rcopies and let H be the corresponding orthogonal group. The increasing sequence of groups r | {z } H ⊂H ⊂H ⊂···⊂H ⊂··· (1.4.2) 0 1 2 r is called the Witt tower attached to H . There is a Weil representation for each dual pair (G,H ) 0 r and we let θ (σ) denote the theta lift of a cuspidal representation σ of G(A) to H (A). Rallis’s ψ,r r tower property is the following. Theorem 1.4.1 ([Ral84b]). Let σ be a cuspidal automorphic representation of G(A) and let i be the smallest integer such that θ (σ) is nonzero. Then: ψ,i (1) i≤2n, (2) θ (σ) is cuspidal, ψ,i (3) θ (σ) is nonzero for all r ≥i. ψ,r Furthermore, the space of cusp forms on G(A) decomposes into the orthogonal sum L2 (G(F)\G(A))=I(Q )⊕I(Q )⊕···⊕I(Q ) cusp 0 1 2n where I(Qr) is the space of σ’s such that θψ,r(σ)6=0 and θψ,r′(σ)=0 for r′ <r. The integer i in the theorem is called the first occurrence index of σ in the Witt tower of H . Note 0 that Rallis’s tower property shows that a cuspidal representation σ of G must lift to a cuspidal representation θ (σ) of H for some 0 ≤ i ≤ 2n but the theorem does not give any information ψ,i i about the index itself. The goalofthis paper isto showthatthere aregeneralrelationsbetweenperiodsofautomorphic forms of σ and π in the case that σ is a cuspidal representation of G such that π = θ (σ) is a ψ cuspidal representation of H. 2. Periods of Automorphic Forms The main result of this paper is the period identity in Theorem 2.4.1 which shows a general relation between Fourier coefficients and orthogonal periods of cuspidal automorphic forms which correspondbythethetacorrespondence. Asaconsequence,weproveinthelastsectionofthispaper a special value formula for standard automorphic L-functions by combining our identity with the Rallis inner product formula (cf. Theorem 3.5.1). We begin by introducing Fourier coefficients of automorphic forms on symplectic groups and periods of cuspidal automorphic forms on orthogonal groups. The connection between Fourier coefficients and orthogonal periods in the theta correspondence is shown in Proposition 2.1.1 from which we derive Theorem 2.4.1. THE THETA CORRESPONDENCE, PERIODS AND SPECIAL VALUES OF L-FUNCTIONS 5 2.1. Fourier Coefficients and Orthogonal Periods. Let G = Sp and H = O(V) as in n Section 1.1. The symplectic group G contains the Siegel parabolic subgroup P = MN with unipo- tent radical 1 b N = n(b)= :b∈Sym (2.1.1) 0 1 n (cid:26) (cid:18) (cid:19) (cid:27) where Sym denotes the space of symmetric matrices of size n. For T ∈ Sym (F), the ψ -Fourier n n T coefficient of an automorphic form fG on G is W (fG)= fG(n)ψ (n)dn (2.1.2) T T Z[N] where ψ (n)=ψ(trTb) for n=n(b) as in (2.1.1) above. T Let x= (x1,...,xn)∈Vn be an n-tuple of rational vectors in V and let Hx be the stabilizer of x in H where H acts on Vn componentwise. The Hx-period ofa cuspidalautomorphic form fH on H is P (fH)= fH(h)dh . (2.1.3) Hx Z[Hx] Fourier coefficients and orthogonalperiods are related by the following (well-known) calculation. GivenacuspidalautomorphicformfH onH andaSchwartzfunctionϕ∈S(V(A)n),theψ -Fourier T coefficient of the theta lift θ fH is ϕ W (θ fH)= ω(n)ϕ(h−1v)fH(h)dh ψ (n)dn T ϕ T Z[N] Z[H]v∈Vn ! X = ϕ(h−1v)ψ(tr(bQ[v]−Tb))db fH(h)dh Z[H] Z[Symn]v∈Vn ! X = ϕ(h−1v)fH(h)dh Z[H] vX∈Vn Q[v]=T If we assume that T is nondegenerate, then the group H(F) acts transitively on the set in the sum above and so we chose some x∈Vn such that Q[x]=T and we let Hx be its stabilizer. If there is no such x (in other words,if Q does not representT), then W (θ fH)=0 for all fH. We continue T ϕ W (θ fH)= ϕ(h−1γ−1x)fH(h)dh T ϕ Z[H]γ∈Hx(XF)\H(F) = ϕ(h−1x)fH(h)dh ZHx(F)\H(A) = ϕ(h−1x) fH(h′h)dh′dh . ZHx(A)\H(A) ZHx(F)\Hx(A) Thus the ψT-Fourier coefficient of the theta lift of fH is written in terms of a Hx-period where Q[x]=T. To push this integral one step further, we introduce the following lemma. Lemma 2.1.1. Let x ∈Vn such that Q[x] = T is nondegenerate and let Hx be the stabilizer of x in H. For each ϕ∈S(V(A)n), there is a smooth function ξ on H(A) which is rapidly decreasing on H(A ) and compactly supported on H(A ) satisfying ∞ f ϕ(h−1x)= ξ(h h)dh . (2.1.4) 0 0 ZHx(A) In this case, we say (ϕ,ξ;x) is a matching datum (or that ξ matches ϕ relative to x). 6 PATRICKWALLS We will defer the proof of the lemma until the next section. Continuing with the integral above with the function ξ matching ϕ relative to x as in the previous lemma, we have W (θ fH)= ϕ(h−1x) fH(h′h)dh′dh T ϕ ZHx(A)\H(A) ZHx(F)\Hx(A) = ξ(h h)dh fH(h′h)dh′dh 0 0 ZHx(A)\H(A)ZHx(A) ZHx(F)\Hx(A) = ξ(h) fH(h′h)dh′dh ZH(A) ZHx(F)\Hx(A) = ξ(h)fH(h′h)dhdh′ . ZHx(F)\Hx(A)ZH(A) Therefore we have proved the following formal identity which is the main input into our period identities Theorem 2.4.1 and Theorem 2.5.1. In the next section, we prove Lemma 2.1.1 and give an explicit formula for the local functions ξ at finite places. v Proposition 2.1.1. Let x ∈ Vn such that Q[x] = T is nondegenerate and let Hx be the stabilizer of x in H. Let ϕ ∈ S(V(A)n) and let ξ be a smooth function which matches ϕ relative to x as in Lemma 2.1.1. For any cusp form fH ∈L2 (H(F)\H(A)), we have cusp W (θ fH)=P (R fH) (2.1.5) T ϕ Hx ξ where R fH(h)= ξ(h′)fH(hh′)dh′. ξ H(A) 2.2. Matching FuRnctions. In this section, we will prove Lemma 2.1.1 along with an explicit for- mula for the local functions ξ at finite places. v Lemma. Let x ∈ Vn such that Q[x] = T is nondegenerate and let Hx be the stabilizer of x in H. For each ϕ ∈ S(V(A)n), there is a smooth function ξ on H(A) which is rapidly decreasing on H(A ) and compactly supported on H(A ) satisfying ∞ f ϕ(h−1x)= ξ(h h)dh . (2.2.1) 0 0 ZHx(A) In this case, we say (ϕ,ξ;x) is a matching datum (or that ξ matches ϕ relative to x). Furthermore, if ϕ = ⊗ ϕ is factorizable, then ξ = ⊗ ξ is factorizable where, for any finite place v, the local v v v v function ξ is given by the finite sum v ϕ (γ−1x) v ξ (h)= C (h) (2.2.2) v γX∈C vol(Hx(Fv)∩γUvγ−1) γUv γ−1x∈suppϕv where (1) U ⊂H(F ) is any compact open subgroup which acts trivially on ϕ v v v (2) C is a set of representatives of the double coset space Hx(Fv)\H(Fv)/Uv (3) C is the characteristic function of γU . γUv v Let us make few remarks before we continue with the proof of this lemma. Notice that if ξ(h) is any function which satisfies (2.2.1), then ξ(h′h) also satisfies (2.2.1) for any h′ ∈ Hx(A). Also, the localfunctionξ describedin(2.2.2)depends on the choice C of representatives forthedouble coset v space in the sum. Finally, although the set C may be infinite, the sum is always finite. THE THETA CORRESPONDENCE, PERIODS AND SPECIAL VALUES OF L-FUNCTIONS 7 Proof. Wemayassumethatϕ=⊗ ϕ isfactorizablethereforewewillprovetheanalogousequality v v ϕ (h−1x)= ξ (h h)dh , h∈H(F ) (2.2.3) v v 0 0 v ZHx(Fv) for each place v. The group H(A) can be written as the restricted product ′ H(F ) with respect v v to the compact open subgroups K = Aut(L ) ⊂ H(F ) where L = L⊗ O for a fixed global v v v v OFQ Fv O -lattice L⊂ V. We will show that ξ is the characteristic function of K for almost every place F v v v. Suppose v is a finite place. If V is anisotropic, the group H(F ) is compact therefore v v 1 ξ (h)= ϕ (h−1x) , h∈H(F ) (2.2.4) v v v volHx(Fv) is a smooth compactly supported function which matches ϕ relative to x. v Suppose V is isotropic. LetU ⊂H(F ) be a compactopensubgroupsuchthat ϕ (kv)=ϕ (v) v v v v v for all k ∈U and v ∈Vn. Use the notation C to denote the characteristicfunction of a set S and v v S write ϕ (h−1x)= ϕ (γ−1x)C (h) v v γUv γ∈HX(Fv)/Uv γ−1x∈suppϕv = ϕ (γ−1x) C (h h) . v γUv 0 γ∈Hx(FXv)\H(Fv)/Uv h0∈(Hx(Fv)∩XγUvγ−1)\Hx(Fv) γ−1x∈suppϕv We claim that the outer sum is finite. Since detT 6= 0, the group H(F ) acts transitively on the v set Ω = {v ∈ Vn : Q[v] = T} and therefore the map γ 7→ γ−1x is a homeomorphism between T v Hx(Fv)\H(Fv) and ΩT. Since ϕv has compact support, its restriction to the closed subset ΩT has compactsupportandthereforethe supportofϕv(h−1x) inHx(Fv)\H(Fv)is alsocompact. Finally, since Hx(Fv)\H(Fv)/Uv is discrete, the set {γ ∈Hx(Fv)\H(Fv)/Uv :γ−1x∈suppϕv} is finite. We make the observation CγUv(h0h)dh0 =vol(Hx(Fv)∩γUvγ−1) CγUv(h0h) (2.2.5) ZHx(Fv) h0∈(Hx(Fv)∩XγUvγ−1)\Hx(Fv) for all γ,h ∈ H(F ). Now we must make a choice C for a set of representatives γ of the double v coset space Hx(Fv)\H(Fv)/Uv and then define a function given by the finite sum ϕ (γ−1x) v ξ (h)= C (h) (2.2.6) v γX∈C vol(Hx(Fv)∩γUvγ−1) γUv γ−1x∈suppϕv Then ξ is a locally constant compactly supported function which matches ϕ relative to x. v v We claim that, for almost all v, the sum (2.2.6) has a single term and ξ (h)=C (h) where K v Kv v is the maximal compact subgroup K = Aut(L ) ⊂ H(F ) for a fixed global lattice L ⊂ V. For v v v almost all v, we are in the following situation: (1) ϕ =ϕ ⊗···⊗ϕ is the n-fold tensor product of the characteristic function of L v Lv Lv v (2) x=(x ,...,x )∈Ln 1 n v (3) detQ[x]∈O× Fv (4) Q is O -valued on L Fv v 8 PATRICKWALLS ThesymmetricmatrixQ[x]representsthequadraticformQrestrictedtoΛx =spanOFv{x1,...,xn} relative to the basis x ,...,x consisting of the components of x. Since detQ[x]∈O× , the lattice 1 n Fv Λx is regular therefore Lv =Λx⊕Λ′ is an orthogonaldirect sum of OFv-lattices where Λ′ ={v ∈Lv :(v,w)=0 for all w∈Λx} . (2.2.7) Suppose γ ∈H(Fv) such that xi ∈γLv for each i=1,...,n. Then Λx is a regular subspace of γLv therefore we have the orthogonal direct sum γLv =Λx⊕Λ′′ of OFv-lattices where Λ′′ ={γv∈γLv :(γv,w)=0 for all w∈Λx} . (2.2.8) Since Λx⊕Λ′ and Λx⊕Λ′′ are isometric, we must have that Λ′ and Λ′′ are isometric by the Witt cancellation property for local rings. In other words, there is some δ ∈ Hx(Fv) (note that Hx(Fv) is the orthogonal group of span {x ,...,x }⊥) such that δΛ′′ = Λ′. Finally, δγL = L and so Fv 1 n v v δγ ∈K . Therefore, in this most unramified case, v #{γ ∈Hx(Fv)\H(Fv)/Uv :γ−1x∈suppϕv}=1 (2.2.9) therefore the sum (2.2.6) has a single term and ξ (h) = C (h). Here we have used the fact that v Kv vol(Hx(Fv)∩Kv)=1 for almost all v. Suppose v is an infinite place. Then V is anisotropic and the group H(F ) is compact therefore v v 1 ξ (h)= ϕ (h−1x) (2.2.10) v v volHx(Fv) is a smooth compactly supported function which matches ϕ relative to x. v Finally, suppose v is an infinite place such that V is isotropic. This is the only case where we v cannot give an exact definition of the local function ξv. The group H(Fv) is a Lie group, Hx(Fv) is a Lie subgroup and the natural map p : H(Fv) −→ ΩT : h 7→ h−1x is a princial Hx(Fv)- bundle where Ω = {v ∈ Vn : Q[v] = T}. Let {U ,Φ } be an open cover of Ω with local T v i i i∈I T trivializations Φi : p−1(Ui) ∼= Hx(Fv)×Ui. On each of the open sets p−1(Ui), we can arbitrarily chose a smooth rapidly decreasing function fi on Hx(Fv) and define a smooth rapidly decreasing function on p−1(Ui) by fi(γ)ϕv(v) using the isomorphism Φi(h) = (γ,v) ∈ Hx(Fv)×Ui. We can use a partition of unity to glue the functions f (γ)ϕ (v) on each p−1(U ) together to get a smooth i v i rapidly decreasing function ξ on the whole group H(F ). Since the functions f were arbitrary we v v i can scale them so that ξ (h h)dh =ϕ (h−1x). Hx(Fv) v 0 0 v (cid:3) R 2.3. Example: Local Matching Functions for PGL (Q ). In this section, we will compute the 2 p local functions ξ by the recipe (2.2.2) in the following particular case. Let p be a prime not equal v to 2 and consider the quadratic space (V,Q) over Q consisting of traceless 2 by 2 matrices p a b V = :a,b,c∈Q (2.3.1) c −a p (cid:26)(cid:18) (cid:19) (cid:27) equipped with the quadratic form Q(x)=−det(x) (equivalently, x2 =Q(x)·id). (2.3.2) Note that this corresponds to the inner product (x,y)=tr(xy) , x,y ∈V . (2.3.3) The special orthogonalgroup is given by SO(V)=PGL (Q ) (2.3.4) 2 p THE THETA CORRESPONDENCE, PERIODS AND SPECIAL VALUES OF L-FUNCTIONS 9 via the natural action x 7→ gxg−1 for g ∈ GL (Q ). (Again, we note that even though this paper 2 p restricts to the case that dim V is even, all our results apply with trivial modifications to the case F when dim V is odd and G is the metaplectic cover of Sp .) F n LetH =PGL (Q )andletK ⊂H bethemaximalcompactsubgroupwhichstabilizesthelattice 2 p of integral elements a b L= :a,b,c∈Z . (2.3.5) c −a p (cid:26)(cid:18) (cid:19) (cid:27) In other words, K is equal to the image of GL (Z ) projected to H. 2 p Let ϕ ∈ S(V) be the characteristic function of the lattice L ⊂ V, let x ∈ V such that Q(x) 6= 0 and let H be the stabilizer of x in H. The goal of this section is to find a smooth compactly x supported function ξ on H such that ξ matches ϕ relative to x as in (2.2.1). In other words, we want to find ξ such that ϕ(h−1x)= ξ(h′h)dh′ (2.3.6) ZHx and we will use the recipe (2.2.2). Note that since ϕ is an even function, we can extend ξ to a function on the whole orthogonal group O(V) by the projection map O(V) → SO(V). We begin with a few reduction steps. Firstly, if Q(x)6∈Z then ϕ(h−1x)=0 for all h∈H since Q(L)⊂Z . Thereforewe may assume p p Q(x)∈Z . p Secondly, if Q(x) = pαε for α ∈ Z and ε ∈ Z×, then x generates a quadratic extension of Q . ≥0 p p Therefore we need only consider three cases (recall, p>2): 0 ε (inert) : if α is even and ε is a nonsquare, then we may assume x=pα/2 1 0 (cid:18) (cid:19) 0 pε (ramified) : if α is odd, then we may assume x=p(α−1)/2 1 0 (cid:18) (cid:19) 1 0 (split) : if α is even and ε is a square, then we may assume x=pα/2 0 −1 (cid:18) (cid:19) Inparticular,ify ∈V isanyelementsuchthatQ(y)isintegralandnonzerotheny is inthe H-orbit of one of the elements in the three cases above. Finally, note that the compact open subgroup K fixes ϕ and therefore it plays the role of U v appearing in (2.2.2). Therefore, to compute ξ we need to: (1) determineasetC ofrepresentativesforH \H/Kanddeterminethesubset{γ ∈C :x∈γL} x (2) compute vol(H ∩γKγ−1) for γ ∈{γ ∈C :x∈γL} x Lemma 2.3.1. . (1) The subgroup H is equal to the image in H of: x a bε (inert): :a,b∈Q , a2+εb2 6=0 b a p (cid:26)(cid:18) (cid:19) (cid:27) a bεp (ramified): :a,b∈Q , a2+pεb2 6=0 b a p (cid:26)(cid:18) (cid:19) (cid:27) a 0 (split): :a,b∈Q , ab6=0 0 b p (cid:26)(cid:18) (cid:19) (cid:27) 10 PATRICKWALLS (2) A set C of representatives for the double coset space H \H/K is given by: x pd 0 (inert): C = γ = :d≥0 d 0 1 (cid:26) (cid:18) (cid:19) (cid:27) pd 0 (ramified): C = γ = :d≥1 d 0 1 (cid:26) (cid:18) (cid:19) (cid:27) pd 1 (split): C = δ = :d≥0 d 0 1 (cid:26) (cid:18) (cid:19) (cid:27) (3) The set {γ ∈C :x∈γL} is given by: (inert): γ = pd 0 :0≤d≤ α , and vol(H ∩γ Kγ−1)= volHx d 0 1 2 x d d pd+pd−1 (cid:26) (cid:18) (cid:19) (cid:27) (ramified): γ = pd 0 :1≤d≤ α+1 , and vol(H ∩γ Kγ−1)= volHx d 0 1 2 x d d 2pd−1 (cid:26) (cid:18) (cid:19) (cid:27) pd 1 α (split): δ = :0≤d≤ , and vol(H ∩δ Kδ−1)=vol(H ∩K) d 0 1 2 x d d x (cid:26) (cid:18) (cid:19) (cid:27) Proof. Wewillbebriefsinceallthecomputationsrequiredtoprovethislemmaarequiteelementary. It is a straight forwardcomputation to determine H in each case. A set of coset representativesof x H/K is given by pd u 1 0 :d≥0 and 0≤u<pd ∪ :d>0, 0≤u<pd and p|u 0 1 u pd (cid:26)(cid:18) (cid:19) (cid:27) (cid:26)(cid:18) (cid:19) (cid:27) and it is easy to determine C and {γ ∈ C : x ∈ γL} in each case. Finally, we will discuss how to derive the quantity vol(H ∩γ Kγ−1) in the inert case via the action of H on the tree H/K. The x d d other cases are proved in a similar way. We may visualize the set H/K as a connected graph with a vertex for each element of H/K which is connected to exactly p+1 neighbouring vertices. We measurethedistanceofacosethK fromthebasepointK bythenumberofedgesalongtheshortest path from K to hK. It is easy to show that H stabilizes K and acts transitively on the pd+pd−1 x vertices ata distance d from K. The subgroupγ Kγ−1 is the stabilizer of the vertex γ K therefore d d d [H :H ∩γ Kγ−1]=pd+pd−1 and so x x d d volH vol(H ∩γ Kγ−1)= x . x d d pd+pd−1 (cid:3) Finally,inthisspecificcase,wehavealltheingredientstodeterminethelocalmatchingfunctions ξ according to the recipe (2.2.2). Proposition 2.3.1. If ϕ is the characteristic function of L, then, in each of the three cases above, a smooth function ξ on H that matches ϕ is given by: α/2 1 pd 0 (inert): ξ(h)= C (h)+ (pd+pd−1)C (h) , γ = (2.3.7) volH  K γdK  d 0 1 x d=1 (cid:18) (cid:19) X (α+1)/2  1 pd 0 (ramified): ξ(h)= 2pd−1C (h) , γ = (2.3.8) volH γdK d 0 1 x d=1 (cid:18) (cid:19) X α/2 1 pd 1 (split): ξ(h)= C (h)+ C (h) , δ = (2.3.9) vol(H ∩K) K δdK  d 0 1 x d=1 (cid:18) (cid:19) X  