Averages and nonvanishing of central values of triple product $L$-functions
aa r X i v : . [ m a t h . N T ] J a n AVERAGES AND NONVANISHING OF CENTRAL VALUES OFTRIPLE PRODUCT L -FUNCTIONS BIN GUAN
Abstract.
Let f, g, h be three normalized cusp newforms of weight 2 k onΓ ( N ) which are eigenforms of Hecke operators. We use Ichino’s periodformula combined with a relative trace formula to show exact averages of L (3 k − , f × g × h ). We also present some applications of the average for-mulas to the nonvanishing problem, giving a lower bound on the number ofnonvanishing central L -values when one of the forms is fixed. Contents
1. Introduction 21.1. Main results 21.2. Ichino’s period formula 41.3. Jacquet’s relative trace formula 51.4. A sketch of the proof and the structure of the paper 62. Notation and Assumptions 72.1. Quaternion Algebras 72.2. Normalization of measures 92.3. Jacquet–Langlands correspondence 102.4. Representation theory of SU(2) 122.5. L -functions and adelic version of Main Theorem 153. Spectral Side of the RTF 183.1. Test function 193.2. Cusp ⊗ Cusp 203.3. Res ⊗ Res 213.4. Res ⊗ Cusp 233.5. Application of Ichino’s formula 244. Geometric Side of the RTF 264.1. Orbital decomposition 284.2. Nontrivial orbits and Waldspurger’s formula 334.3. Local calculation on ramified quaternion algebras 374.4. Local calculation on split quaternion algebras 394.5. Compatibility of two nontrivial orbits 464.6. Local calculation: Archimedean 485. Examples of small weights 535.1. Weight 2 or 4 535.2. Weight 6 55
Date : February 1, 2021.
Key words and phrases. triple product L -function, central L -value, nonvanishing, relative traceformula, quaternion algebra, period integral.
6. Applications 566.1. Sum over three forms 566.2. The nonvanishing problems 60Appendix A. Hilbert Symbol 63A.1. An application to quaternion algebras 63A.2. An application to quadratic forms 64Acknowledgements 65References 651.
Introduction
Main results.
The aim of this paper is to establish an exact average formulaof central values of triple product L -functions associated to three normalized cuspnewforms, while one of the three forms is fixed. We also give some applications ofthe average formulas to the nonvanishing problems.Let N, k ≥ N be square-free. Let F k ( N ) be the set ofnormalized cusp newforms of weight 2 k on Γ ( N ) which are eigenforms of Heckeoperators. Normalizing f ( z ) = P n ≥ a n ( f ) e πinz , g, h ∈ F k ( N ) such that a = 1,we can define the triple product L -function as the Euler product L fin ( s, f × g × h ) := Y p L p ( s, f × g × h )(see Section 2.5 for the data of local L -factors). B¨ocherer and Schulze–Pillot[BSP96] have proved thatΛ( s, f × g × h ) := (2 π ) k − − s Γ( s )Γ( s + 1 − k ) L fin ( s, f × g × h )has an analytic continuation to the entire s -plane and satisfies the functional equa-tionΛ( s, f × g × h ) = εN k − − s ) Λ(6 k − − s, f × g × h ) , where ε = − Y p | N ε p = ± . One can observe that the central value is at s = 3 k −
1. But after a translation s s + 3 k − , the functional equation may be written in the formΛ( s, f × g × h ) = εN − s ) Λ(1 − s, f × g × h )so that the central value is Λ( , f × g × h ). See (7).In this paper we will develop a relative trace formula (RTF) for automorphicforms on a specific quaternion algebra. We use this RTF, along with the Jacquet–Langlands correspondence and Ichino’s triple product formula, to translate thetriple product average value into a sum of orbital integrals. Finally we evaluate thissum explicitly and apply Waldspurger’s formula to rewrite certain orbital integralsas central values of Rankin–Selberg L -functions.Before we state the main theorem, we recall the CM modular forms arise fromHecke characters. For an imaginary quadratic extension E/ Q with discriminant − d ,consider a character Ω on E × \ A × E , whose restriction on A × = A × Q is trivial. Assumethat it is unramified everywhere at finite places. At infinity we have Ω ∞ = sgn m for some m ∈ Z where sgn( z ) := z/ | z | . Recall that, when Ω does not factorthrough the norm map N E : A × E → A × there is a modular form Θ Ω of level d , VERAGES AND NONVANISHING OF TRIPLE PRODUCT CENTRAL L -VALUES 3 weight 2 | m | + 1 and nebentypus χ − d such that L ( s, Θ Ω ) = L ( s, Ω). For h ∈ F k ( N )let L ( s, h × Θ Ω ) denote the completed Rankin–Selberg L -function which satisfies afunctional equation relating the value at s to 2 k + 2 | m | + 1 − s . Theorem 1.1 (Main Theorem) . Let N be a square-free integer with an odd numberof prime factors. For any h ∈ F k ( N ) , (1) N k − π k − X f,g ∈F k ( N ) ε p = − , ∀ p | N L fin (3 k − , f × g × h )( f, f )( g, g )( h, h ) = 1 − δ ( k ) /ϕ ( N )2 ω ( N ) Γ(2 k )Γ(2 k − + Γ(2 k − k ) Γ(3 k −
1) 4 L · ord ( N ) Q p | N − χ − ( p )2 + 6 √ L · ord ( N ) Q p | N − χ − ( p )2 (4 π ) k ( h, h ) , where L = I ( ) · L fin ( k, h ) L fin ( k, h ⊗ χ − ) + X =Ω ∈ d [ E × ] I (Ω) · L fin ( k + | m | + 12 , h × Θ Ω ) ,L = I ( ) · L fin ( k, h ) L fin ( k, h ⊗ χ − ) + X =Ω ∈ d [ E × ] I (Ω) · L fin ( k + | m | + 12 , h × Θ Ω ) . Here ( · , · ) is the Petersson inner product on F k ( N ) defined by (2) ( f , f ) := Z Γ ( N ) \H f ( z ) f ( z ) y k dx dyy ; I ( ) , I ( ) , I (Ω) , I (Ω) (defined in Theorem 4.3) are constants depending onlyon k and Ω (and on a ( h ) , a ( h ) when or | N respectively); ϕ ( N ) is the Euler’stotient function; ω ( N ) := P p | N is the number of distinct prime factors of N ; and δ ( k ) := ( , if k = 1 , , otherwise. Remark 1.2.
1) Since ε ∞ = − and N has an odd number of prime factors, theglobal ε -factor (the global root number) for the triple product L -function is in thecase of the Main Theorem.2) Here χ d is the Dirichlet character defined by the Kronecker symbol ( d · ) , where d ≡ , is a fundamental discriminant. The product over p | N can beseen as a congruency condition. For example, when N is square-free, ord ( N ) Y p | N − χ − ( p )2 = ( , if N has a prime factor ≡ ; , otherwise.If, in addition, N has a prime factor ≡ and one ≡ (or if N has a prime factor ≡ ), for any h ∈ F k ( N ) , the Main Theoremsimplifies to (3) N k − π k − X f,g ∈F k ( N ) ε p = − , ∀ p | N L fin (3 k − , f × g × h )( f, f )( g, g )( h, h ) = 1 − δ ( k ) /ϕ ( N )2 ω ( N ) Γ(2 k − Γ(2 k ) .
3) Feigon and Whitehouse [FW10] have shown an exact average formula of tripleproduct central L -values associated to three newforms of weight and of the same BIN GUAN prime level p . Their approach does not use the relative trace formula. Rather, usingthe classical period formula of Gross–Kudla [GK92] , they write L fin (2 , f × g × h ) asa finite sum of functions defined on a finite set.Our Main Theorem is a generalization of [FW10] ’s result to the case of generalweight and level. In Section 5 we give average formulas when f, g, h all have smallweights ( ≤ ) with the constants calculated more explicitly. See (32) (33) and (34) .In particular, when k = 2 and N is a prime with N = 11 or N > , our result (32) reproves the main theorem of [FW10] .4) Furthermore, in Section 6.1 we obtain an exact average formula (Corollary6.2) of L fin (3 k − , f × g × h ) as all three forms run through F k ( N ) . One can finda more explicit result of small-weight cases in Example 6.3. In Section 6.2 we apply the above theorem to the nonvanishing problem. Inparticular we prove the following result.
Corollary 1.3.
Let N be a square-free integer with an odd number of prime factors.Then, for any h ∈ F k ( N ) , { ( f, g ) ∈ F k ( N ) × F k ( N ) : L fin (3 k − , f × g × h ) = 0 } ≫ k,ǫ N / − ǫ . The RTF, originally introduced by Jacquet to study periods integrals (see [Jac05]for an overview), along with Ichino’s period formula [Ich08], plays an important rolein the proof of the adelic version of the Main Theorem (Theorem 2.15) in the caseof general weight and level. This method could also be applied to the case of tripleproduct L -functions attached to Hilbert modular forms over a totally real numberfield. We now give an overview of these tools.1.2. Ichino’s period formula.
From an adelic point of view, one can considerthe triple product L -function L ( s, π ⊗ π ⊗ π ) (defined in Section 2.5) associatedto three irreducible unitary cuspidal automorphic representations of PGL(2 , A ),where A is the adele ring over Q . Harris and Kudla [HK04] proved a conjecture ofJacquet, that the central value L (1 / , π ⊗ π ⊗ π ) = 0 if and only if there existsa quaternion algebra D over Q such that the period integral Z A × D × ( Q ) \ D × ( A ) φ ( x ) φ ( x ) φ ( x ) d × x = 0for some φ i ∈ π ′ i , where A × is diagonally embedded in D × ( A ) as its center, and π ′ is the irreducible unitary automorphic representation of D × ( A ) associated to π bythe Jacquet–Langlands correspondence.Moreover, Gross and Kudla [GK92] established an explicit identity relating cen-tral L -values and period integrals (which are finite sums in their case), when thecusp forms are of prime levels and weight 2. This Gross–Kudla period formula isthe key ingredient when [FW10] proves their average formula for the case of weight2 and a prime level p . B¨ocherer, Schulze–Pillot [BSP96] and Watson [Wat02] gen-eralized this identity to more general levels and weights. At last, Ichino [Ich08]proved an adelic version of this period formula which would work for all the cases: (cid:12)(cid:12)(cid:12)R [ D × ] φ ( h ) φ ( h ) φ ( h ) dh (cid:12)(cid:12)(cid:12) Q i =1 R [ D × ] φ i ( h ) φ i ( h ) dh ∼ L ( , π ⊗ π ⊗ π ) L (1 , π ⊗ π ⊗ π , Ad) . Here [ D × ] := A × D × ( Q ) \ D × ( A ). The exact formula can be found in Theorem 3.7. VERAGES AND NONVANISHING OF TRIPLE PRODUCT CENTRAL L -VALUES 5 Many authors have derived several explicit versions of Ichino’s formula in variouscases. An incomplete list includes [Woo11, Nel11, NPS14, Hu17, CC19, HK20,Col20, Hsi21]. In this paper, we continue the work of Woodbury [Woo11] andChen–Cheng [CC19] to establish Lemma 3.8, an explicit version of Ichino’s formulafor L (1 / , π ⊗ π ⊗ π ), in terms of the period integrals appearing in the spectralside of the RTF, which we use to prove the adelic Main Theorem 2.15.1.3. Jacquet’s relative trace formula.
Here we consider a general version ofthe RTF (cf. [FW09, Zha18]). Let G be an anisotropic algebraic group definedover a global field F and H , H be closed subgroups of G . Let f ∈ C ∞ c ( G ( A F )).Integrating f against the action of G ( A F ) gives a linear map R ( f ) : L ( G ( F ) \ G ( A F )) → L ( G ( F ) \ G ( A F ))defined by ( R ( f ) φ )( x ) := Z G ( A F ) f ( g ) φ ( xg ) dg. One sees that R ( f ) is an integral operator with kernel K f ( x, y ) = X γ ∈ G ( F ) f ( x − γy ) , x, y ∈ G ( A F ) . Let A ( G ) denote the set of automorphic representations on G ( A F ). Fixing auto-morphic forms φ H , φ H in π ∈ A ( H ) and π ∈ A ( H ) respectively, we define adistribution I ( f ) = Z H ( F ) \ H ( A F ) Z H ( F ) \ H ( A F ) K f ( h , h ) φ H ( h ) φ H ( h ) dh dh . The RTF for the case H \ G/H gives two expressions of I ( f ).From the spectral decomposition of L ( G ( F ) \ G ( A F )), K f ( x, y ) = X π ∈A ( G ) X φ ∈ON B ( π ) ( π ( f ) φ )( x ) φ ( y ) , where for each π ∈ A ( G ), ON B ( π ) denotes an orthonormal basis of V π . Thespectral expansion for the kernel K f ( x, y ) gives I ( f ) = X π ∈A ( G ) I π ( f ) , where for each π ∈ A ( G ) I π ( f ) := X φ ∈ON B ( π ) Z H ( F ) \ H ( A F ) ( π ( f ) φ )( h ) φ H ( h ) dh Z H ( F ) \ H ( A F ) φ ( h ) φ H ( h ) dh . From the geometric expansion of the RTF, I ( f ) = Z H ( F ) \ H ( A F ) Z H ( F ) \ H ( A F ) X [ γ ] X ( θ ,θ ) f ( h − θ − γθ h ) φ H ( h ) φ H ( h ) dh dh . Here [ γ ] runs through representatives of H ( F ) \ G ( F ) /H ( F ); and ( θ , θ ) runsthrough H ( F ) × H ( F ) / ( H ( F ) × H ( F )) γ , where we define( H ( F ) × H ( F )) γ := { ( θ , θ ) ∈ H ( F ) × H ( F ) : θ − γθ = γ } . BIN GUAN
Let θ i h i be the new h i ( i = 1 , I ( f ) = X [ γ ] ∈ H ( F ) \ G ( F ) /H ( F ) I [ γ ] ( f )where I [ γ ] ( f ) = Z ( H ( F ) × H ( F )) γ \ H ( A F ) × H ( A F ) f ( h − γh ) φ H ( h ) φ H ( h ) d ( h , h ) . As a generalization of the Arthur–Selberg trace formula, Jacquet’s relative traceformula is a powerful tool in the study of period integrals. With a period formulalike Ichino’s, the average of central values of L -functions appears in the spectraldecomposition of a certain distribution. In the compact quotient case one can getan explicit orbital decomposition of the same distribution. For example, Feigon andWhitehouse [FW09] have considered the RTF for the case E × \ D × /E × (where D is a quaternion algebra over a totally real number field F and E/F is a quadraticextension embedded in D ) and, using a period formula of Waldspurger, obtainedan exact formula for averages of central values of twisted quadratic base change L -functions associated to Hilbert modular forms. In this paper an analogous methodis applied to the case D × \ ( D × × D × ) /D × to obtain exact formulas for averages ofcentral values of triple product L -functions.1.4. A sketch of the proof and the structure of the paper.
Let D be thedefinite quaternion algebra over Q with discriminant N . In Section 2.3 we usethe Jacquet–Langlands correspondence to associate a newform f ∈ F k ( N ) with acuspidal automorphic representation π ′ of D × ( A ) with the same level and weight.With the definitions of L -functions in Section 2.5, we translate the Main Theoremto an adelic version, which is Theorem 2.15.In Sections 3 and 4 we apply the RTF for the case D × \ ( D × × D × ) /D × to prove the adelic Theorem 2.15. Briefly, let G ′ be the algebraic group definedover Q with G ′ ( Q ) = Z ( Q ) \ D × ( Q ). We take G ′ × G ′ as the “big” group G as inthe previous section, and take H = H = G ′ embedded in G diagonally. On onehand, we choose a suitable test function f ∈ C ∞ c ( G ′ ( A ) × G ′ ( A )) (as in Section3.1), so that (almost) all terms I π ( f ) on the spectral side of the RTF are associatedto representations of level N and weight 2 k . Moreover I π ( f ), when it is nonzero, isessentially a period integral (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z A × D × ( Q ) \ D × ( A ) φ ( h ) φ ( h ) φ ( h ) dh (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , and it can be written as the central value of corresponding triple product L -functionvia Ichino’s formula [Ich08].On the other hand, there are at most three terms I [ γ ] ( f ) which do not vanishon the geometric side of the RTF (see Theorem 4.1). For a nontrivial orbit [ γ ], thenonzero term I [ γ ] ( f ) is of the form[some congruency condition] × Z A × E × γ \ A × Eγ φ ′ ( t ) φ ′′ ( t ) dt, VERAGES AND NONVANISHING OF TRIPLE PRODUCT CENTRAL L -VALUES 7 where E γ is a quadratic extension of Q . In Section 4.2 we use harmonic analysisand write the integral over A × E × γ \ A × E γ as a sum of period integrals, and the periodintegrals over the torus E × γ can be related to L (1 / , π E γ ⊗ Ω) by Waldspurger’speriod formula [Wal85].Now we go back to Theorem 1.1 and let h ∈ F k ( N ) vary. Feigon and Whitehouse[FW09] have established an exact average formula for central values of twistedquadratic base change L -functions associated to Hilbert modular forms. Using thisresult we can obtain an exact average formula of L fin (3 k − , f × g × h ) as all threeforms run through F k ( N ). See Corollary 6.2. Also, in Section 5 we do more-detailed calculations on the small-weight cases (when three forms have the sameweight 2, 4, or 6). These exact average formulas are (32) (33) (34) and Example6.3.At last in Section 6.2 we apply the Main Theorem to the nonvanishing problems.With the weight 2 k fixed, a lower bound on the number of nonvanishing centralvalues of triple product L -functions is given in Section 6.2, when one of the formsis fixed. We also have a result (Corollary 6.8) on the nonvanishing modulo suitableprimes p of the algebraic part of triple product L -values.2. Notation and Assumptions
Quaternion Algebras.
For any field F of characteristic = 2, and a, b ∈ F × ,let D = (cid:18) a, bF (cid:19) := F { i, j } / ( i − a, j − b, ij + ji ) , denote the quaternion algebra with F -basis 1 , i, j, k (or 1 D , i D , j D , k D ) such that i = a , j = b and ij = − ji = k (so k = − ab ). We know that either D ∼ = M (2 , F )is split or D is a division algebra. Let Ram( D ) be a set of places v in F such that D is ramified at v , i.e. such that D v := D ⊗ F F v is not split.Denote by Tr D and N D the (reduced) trace and norm in D . We recall a lemmaabout the norm group N D ( D × ). Lemma 2.1 ([Voi20] Lemma 13.4.9) . For any quaternion algebra D over a localfield F , N D ( D × ) = ( R × > , if D ∼ = H := ( − , − R ) ,F × , otherwise. It is well known that, in a non-split quaternion algebra, elements are conjugateto each other if and only if they have the same trace and norm. That is to say,conjugacy classes in D × can be parametrized by traces and norms. Instead of D × ,in this paper we deal with the group G ′ ( F ) := F × \ D × , for which we have thefollowing two lemmas. Lemma 2.2.
Fix a set Σ of representatives in F × / ( F × ) . (For example, when F = Q , Σ can be the set of square-free integers.) Then [¯ x ] (Tr D ( x ) , N D ( x )) isa well-defined injection from the set of conjugacy classes of G ′ ( F ) := F × \ D × to ( {± }\ F ) × Σ .Proof. For any two representatives x , x ∈ D × of ¯ x ∈ G ′ ( F ), there exists λ ∈ F × so that x = λx , and we have Tr D ( x ) = λ Tr D ( x ), N D ( x ) = λ N D ( x ).Fixing Σ, moreover, with N D ( x ) fixed, we can only take λ = 1. So the traces ofrepresentatives in ¯ x might differ by a sign. (cid:3) BIN GUAN
For a fixed quaternion, one can check the following result about its centralizerby direct calculation.
Lemma 2.3.
Suppose D = ( a,bF ) is a division algebra. Consider the centralizer of x ∈ G ′ ( F ) := F × \ D × given by G ′ x ( F ) := { g ∈ G ′ ( F ) : gx = xg } . Then • G ′ x ( F ) = G ′ ( F ) when x = 1 ; • when x = 1 + βi + γj + δk = 1 , G ′ x ( F ) is the image of { λ + µx ∈ D × : λ, µ ∈ F } ; • and when x = βi + γj + δk , G ′ x ( F ) is the image of { λ + µx ∈ D × : λ, µ ∈ F }∪{ x i + x j + x k ∈ D × : x , x , x ∈ F ; aβx + bγx = abδx } . When F = Q , a quaternion algebra over Q is called definite if D ∞ = D ⊗ Q R is not split (i.e. isomorphic to the algebra H of Hamilton quaternions). We definethe discriminant of D by disc( D ) := Q p ∈ Ram( D ) p . The quaternion algebra D corresponding to a fixed square-free discriminant can be constructed explicitly. Inthis paper we only consider the following two kinds of discriminants. Lemma 2.4.
Let N be a square-free integer with an odd number of prime divisors.(1) If N has no prime divisor of the form n +1 , ( − , − N Q ) is the definite quaternionalgebra over Q with discriminant N ;(2) If N has no prime divisor of the form n +1 , ( − , − N Q ) is the definite quaternionalgebra over Q with discriminant N .Proof. It is easy to prove this lemma for p ∤ N and p = 2 ,
3, since ( a,b Q p ) is split if p is unramified in Q ( √ a ) and v p ( b ) = 0 (see [Voi20] Corollary 13.4.1). For p | N and p = 2 , a,̟ Q p ) is the only non-split quaternionalgebra over Q p (up to isomorphism) if ̟ is a uniformizer of Q p and a ∈ Z × p is anelement such that Q p ( √ a ) is the unramified quadratic extension of Q p (see [Voi20]Theorem 13.3.10).A more detailed proof of this lemma using the Hilbert symbol can be found inAppendix A.1. (cid:3) Let D be a quaternion algebra over F = Q . The maximal orders of quaternionalgebras D p = D ⊗ Q Q p can be described as following. Proposition 2.5 ([Voi20]) . • When D p = M (2 , Q p ) , the maximal orders of D p are the GL(2 , Q p ) -conjugatesof M (2 , Z p ) ; • When D p is not split, there is a unique maximal order O p = { x ∈ D p : N D p ( x ) ∈ Z p } which contains all Z p -integral elements of D p . In this paper we fix a maximal order O of D such that O p = O ⊗ Z Z p is theunique maximal order for p | disc( D ), and O p is the preimage of M (2 , Z p ) under acertain isomorphism O p ∼ −→ M (2 , Q p ) for p ∤ disc( D ) (see Sections 4.4 and 4.5). VERAGES AND NONVANISHING OF TRIPLE PRODUCT CENTRAL L -VALUES 9 Normalization of measures.
Let F be a number field. For a finite place v of F , the ring of integers in F v is denoted by O F v . Let ̟ v denote a uniformizer in F v and q v := O F v / ( ̟ v )).Fix an additive character ψ of F \ A F . For a place v of F we take the additiveHaar measure dx v on F v which is self-dual with respect to ψ v . On F × v we take themeasure d × x v := ζ F v (1) dx v | x | v = dx | x | , F v = R ; π − dx dx x ¯ x , F v = C , x = x + x i ;(1 − q − v ) − dx v | x | v , v < ∞ . Let E be a quadratic extension over F . We define measures on E v = F v ⊗ F E and E × v similarly with respect to the additive character ψ ◦ Tr E/F .We note that with these choices of measures we have, for v < ∞ ,vol( O × F v ; d × x v ) = vol( O F v ; dx v ) = | d v | / with d v ∈ F v such that d v O F v is the different of F v over Q p ; and for a quadraticfield extension E v /F v , we havevol( F × v \ E × v ) = , if F v = R , E v = C ; | d v | / , if E v /F v is the unramified field extension;2 | D v d v | / , if E v /F v is ramified , with D v ∈ O F v such that D v O F v is the relative discriminant of E v /F v .For a quaternion algebra D defined over a number field F , fix a maximal order O ⊂ D . For a finite place v of F we take the Haar measure dg v on D × v as dg v := ζ F v (1) | N D v ( g v ) | − dµ v ( g v )where µ v is the additive Haar measure on D v which is self-dual with respect to ψ v .Let K v be the image of Z ( F v ) O × v in G ′ v := Z ( F v ) \ D × v . Then, with the quotientmeasure defined on G ′ v , for v < ∞ , we havevol( K v ) = | d v | ζ F v (2) − · ( ( q v − − , if v ∈ Ram( D ) , , if v / ∈ Ram( D ) . For v | ∞ and a definite quaternion algebra D , D v ∼ = H and G ′ v = Z ( R ) \ H × ∼ = {± }\ SU(2). We parametrize h = (cid:18) α − β ¯ β ¯ α (cid:19) ∈ SU(2) by setting α = re iθ cos γ, β = re iϕ sin γ, r > , ≤ γ ≤ π , ≤ θ, ϕ < π. So dαdβ = 2 r sin 2 γ drdγdθdϕ (notice that dαdβ is the self-dual additive measureon H ), and for a function Φ ∈ L (SU(2)), Z SU(2) Φ( h ) dh = Z π Z π Z π Φ( γ, θ, ϕ ) · γ dγ dθ dϕ. This choice of Haar measure on SU(2) implies vol( G ′ v ) = vol( G ′∞ ) = 4 π .Globally we take the product of these local measures and give discrete subgroupsthe counting measures, and define the Tamagawa measure on[ E × ] := A × F E × \ A × E and [ D × ] := Z ( A F ) D × ( F ) \ D × ( A F ) as the quotient measure. In this way we get(4) vol([ E × ]) = 2 L (1 , η E/F ) and vol([ D × ]) = 2 . Here η E/F is the quadratic character of F × \ A × F associated to E/F by class fieldtheory. For example, when E = Q ( √ d ) such that d is a fundamental discriminant, η E/F is the Hecke character corresponding to the Dirichlet character χ d such that χ d ( p ) := (cid:18) dp (cid:19) = , p splits in E ; − , p remains prime in E ;0 , p ramifies in E .We recall a useful lemma about quotient measure. Lemma 2.6 ([KL06] Corollary 7.14) . Let G be a unimodular group and suppose G = KH for closed unimodular subgroups H and K . Suppose further that K ∩ H is unimodular. Let dh denote a right H -invariant measure on ( K ∩ H ) \ H . Then Z ( K ∩ H ) \ H Z K f ( kh ) dk dh ( f ∈ C c ( KH )) defines a Haar measure on G = KH . Moreover, with this measure on G , Z K \ G f ( g ) dg = Z ( K ∩ H ) \ H f ( h ) dh for all f ∈ C c ( K \ G ) . Jacquet–Langlands correspondence.
Let D be a quaternion algebra over Q . Define an algebraic group D × over Q by D × ( A ) = ( A ⊗ Q D ) × for a Q -algebra A . Thus D × is a reductive algebraic group and we therefore have a theory ofautomorphic forms and representations of D × . We will be more interested in theforms that correspond to some automorphic forms on GL(2 , Q ) via the Jacquet–Langlands correspondence.Let N be a square-free integer with an odd number of prime factors. Fix apositive integer k . Denote by F ( N, k ) the set of cuspidal automorphic represen-tations of PGL(2 , A ) of level N and weight 2 k . The following theorem shows the1-1 correspondence between F k ( N ) and F ( N, k ). Theorem 2.7 ([Gel75, LW12]) . Suppose N = Q p i is a product of distinct primes.If f ( z ) = P n ≥ a n ( f ) e πinz ∈ F k ( N ) (normalized such that a = 1 ), then itscorresponding cuspidal automorphic representation π f = ⊗ v π v of PGL(2 , A ) can bedescribed as follows: • π ∞ ∼ = π k dis = σ ( | · | k − / , | · | − ( k − / ) is the discrete series representation ofweight k ; • if p ∤ N , π p is the spherical representation π ( µ , µ ) such that µ µ is trivialand a p ( f ) = p k − ( µ ( p ) + µ ( p )) ; and • if p | N , π p is the special representation σ δ of GL(2 , Q p ) with trivial cen-tral character, where δ is the unramified character of Q × p with δ ( p ) = a p ( f ) p − ( k − = ± . Let D be the definite quaternion algebra with discriminant N (i.e. the quaternionalgebra defined over Q which is ramified precisely at the infinite place of Q and theprimes dividing N ). We have taken G ′ to be the algebraic group defined over Q VERAGES AND NONVANISHING OF TRIPLE PRODUCT CENTRAL L -VALUES 11 with G ′ ( Q ) = Z ( Q ) \ D × ( Q ). Denote by A ( G ′ ) the set of irreducible automorphicrepresentations of G ′ ( A ). Since the quotient G ′ ( Q ) \ G ′ ( A ) is compact, we have thedecomposition L ( G ′ ( Q ) \ G ′ ( A )) = dM π ′ ∈A ( G ′ ) V π ′ , where V π ′ denotes the space of π ′ .Clearly A ( G ′ ) contains all the characters of G ′ ( Q ) \ G ′ ( A ), which are of the form δ ◦ N D where δ : Q × \ A × → {± } is a Hecke character. Let A res ( G ′ ) be the setof these characters. Then its complement, denoted by A cusp ( G ′ ), contains all theinfinite-dimensional irreducible automorphic representations of G ′ ( A ). Every rep-resentation π ′ ∈ A cusp ( G ′ ), according to Jacquet–Langlands [JL70], can correspondto a cuspidal automorphic representation of PGL(2 , A ).Let F ′ ( N, k ) be the set of representations π ′ ∈ A cusp ( G ′ ) which map to repre-sentations in F ( N, k ) under the Jacquet–Langlands correspondence. The compat-ibility between the local and global Jacquet–Langlands correspondence gives thefollowing theorem, which describes explicitly π ′ = ⊗ π ′ v ∈ F ′ ( N, k ). Theorem 2.8 ([JL70] Jacquet–Langlands correspondence, as recalled in [FW09]Fact 3.1) . Under the Jacquet–Langlands correspondence
JL : A ( G ′ ) ֒ → A (PGL(2)) ,the image of A cusp ( G ′ ) is equal to the set of cuspidal automorphic representa-tions π = ⊗ v π v of PGL(2 , A ) such that π v is a discrete series representation of PGL(2 , Q v ) at all places v where D is ramified. In particular, when D is definiteand has discriminant N , for π ∈ F ( N, k ) , there exists π ′ = ⊗ π ′ v ∈ A cusp ( G ′ ) suchthat JL( π ′ ) = π and(1) π ∞ ∼ = π k dis , π ′∞ ∼ = π ′ k is a (2 k − -dimensional irreducible representationof G ′∞ = Z ( R ) \ H × (which is defined in Section 2.4);(2) for v = p | N , π p is the special representation σ δ p where δ p : Q × p → {± } is an unramified character, π ′ p ∼ = δ p ◦ N D p is a character of G ′ p ; and(3) for all the other v , π v is unramified, π ′ v ∼ = π v . For π ′ ∈ F ′ ( N, k ), the following lemma defines a new-line vector φ ∈ V π ′ . Lemma 2.9.
Fix a maximal order
O ⊂ D such that O p = M (2 , Z p ) whenever D splits at p . For any p < ∞ , let K p be the image of Z ( Q p ) O × p in G ′ p = G ′ ( Q p ) , and K fin := Q p K p be an open subgroup of G ′ fin := Q p G ′ p . Then (with X k − ∈ V π ′ k being the unit highest weight vector defined in Section 2.4) C X k − ⊗ ( π ′ fin ) K fin is a one-dimensional subspace of V π ′ for π ′ ∈ F ′ ( N, k ) . We call any nonzerovector φ in this subspace a new-line vector in π ′ , and write it as φ = ⊗ v φ v with φ ∞ := k φ k X k − and φ p being the unit spherical vector in π ′ p (we fix a G ′ p -invariant bilinear form on π ′ p ⊗ ˜ π ′ p ).Proof. When p | N , the non-ramification of δ p implies that δ p ◦ N D p is K p -invariant.Then this Lemma is a direct result from Theorem 2.8. (cid:3) In particular, when 2 k = 2, π ′∞ ∼ = Sym V ⊗ det is trivial. So every automorphicform in π ′ ∈ F ′ ( N, G ′ ( A fin ). Representation theory of
SU(2) . Notice that G ′∞ = R × \ H × ∼ = SU(2) / {± } .Set π ′ k := Sym k − V ⊗ det − k +1 , where V denotes the irreducible 2-dimensional representation of G ′∞ coming fromthe isomorphism D × ( R ) ∼ −→ GL(2 , C ) (see (26)). We have dim π ′ k = 2 k −
1. Moreexplicitly, π ′ k can be realized on the space of homogeneous polynomials in X, Y ofdegree 2 k −
2, i.e. V π ′ k := k − M n =0 C X n Y k − − n with π ′ k ( g ) P ( X, Y ) := P (( X, Y ) g ) det( g ) − k for g ∈ (cid:26)(cid:18) α − β ¯ β ¯ α (cid:19) ∈ GL(2 , C ) (cid:27) ∼ = D × ( R ) . Recall that(5) h X i Y k − − i , X j Y k − − j i k = ((cid:0) k − i (cid:1) − , if i = j ;0 , otherwisedefines a G ′∞ -invariant inner product on V π ′ k .One can check that π ′ k is an irreducible representation with highest weight2 k −
2, and X k − is a highest weight vector. As in Lemma 2.9, let φ ∞ be thehighest weight vector k φ k X k − . We have h φ, φ i = Q v h φ v , φ v i v since the length of φ v is assumed to be 1 for any v < ∞ .Denote by ∆ (resp. ∆ ) the diagonal embedding from G ′∞ to two (resp. three)copies of it. One can view π ′ k ⊗ ◦ ∆ and π ′ k ⊗ ◦ ∆ as representations of G ′∞ .Denote by { X i Y k − − i ⊗ X j Y k − − j } (resp. { X i Y k − − i ⊗ X j Y k − − j ⊗ X r Y k − − r } )a basis of π ′ k ⊗ = π ′ k ⊗ π ′ k (resp. π ′ k ⊗ ). [CC19] shows that, P k = det (cid:18) X X Y Y (cid:19) k − ⊗ det (cid:18) X X Y Y (cid:19) k − ⊗ det (cid:18) X X Y Y (cid:19) k − is the only G ′∞ -invariant vector in π ′ k ⊗ ◦ ∆ up to a constant multiple.Let h· , ·i be the D × ( Q ) or D × ( Q ) -invariant pairing on π ′ k ⊗ or π ′ k ⊗ givenby(6) h· , ·i = h· , ·i k ⊗ h· , ·i k or h· , ·i k ⊗ h· , ·i k ⊗ h· , ·i k . We can calculate the lengths of some particular vectors.
Lemma 2.10.
Let w ◦ k := (cid:18) − Y Y (cid:12)(cid:12)(cid:12)(cid:12) X X Y Y (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) k − ; P k := (cid:12)(cid:12)(cid:12)(cid:12) X X Y Y (cid:12)(cid:12)(cid:12)(cid:12) k − ⊗ (cid:12)(cid:12)(cid:12)(cid:12) X X Y Y (cid:12)(cid:12)(cid:12)(cid:12) k − ⊗ (cid:12)(cid:12)(cid:12)(cid:12) X X Y Y (cid:12)(cid:12)(cid:12)(cid:12) k − . Then k w ◦ k k = h w ◦ k , w ◦ k i = Γ( k ) Γ(3 k − k − Γ(2 k ) , k P k k = h P k , P k i = Γ( k ) Γ(3 k − k − . In particular k w ◦ k k = k P k k / (2 k − . VERAGES AND NONVANISHING OF TRIPLE PRODUCT CENTRAL L -VALUES 13 Proof.
The length of P k is shown in [CC19] Proposition 5.1.By definition h w ◦ k , w ◦ k i = * k − X r =0 k − r ! X r ( − Y ) k − − r X k − − r Y k − r , k − X r ′ =0 k − r ′ ! X r ′ ( − Y ) k − − r ′ X k − − r ′ Y k − r ′ + = k − X r =0 k − r ! h X r Y k − − r , X r Y k − − r i k h X k − − r Y k − r , X k − − r Y k − r i k = k − X r =0 k − r ! k − r ! − k − k − − r ! − . One can verify that (cid:18) k − r (cid:19) (cid:18) k − r (cid:19) − (cid:18) k − k − − r (cid:19) − = (cid:18) k − k − (cid:19) − (cid:18) k − rk − (cid:19)(cid:18) k − − rk − (cid:19) . Therefore, with Lemma 2.11 we have h w ◦ k , w ◦ k i = (cid:18) k − k − (cid:19) − k − X r =0 (cid:18) k − rk − (cid:19)(cid:18) k − − rk − (cid:19) = (cid:18) k − k − (cid:19) − (cid:18) k − k − (cid:19) = Γ( k ) Γ(3 k − k − Γ(2 k ) . (cid:3) Lemma 2.11.
For any m, n ≥ n X r =0 (cid:18) n + rn (cid:19)(cid:18) m + n − rn (cid:19) = (cid:18) n + m + 1 m (cid:19) . Proof.
Assume that a point moves from (0 ,
0) to (2 n + 1 , m ) by moving up or to theright by one unit each time. Then the point has (cid:0) n + m +1 m (cid:1) possible paths. But thepath can intersect the vertical line x = n + 1 / n, r ) and( n + 1 , r ) where 0 ≤ r ≤ m . Then (cid:0) n + rn (cid:1)(cid:0) n + m − rn (cid:1) is the number of all possible pathsthat the point moves from (0 ,
0) to ( n, r ) and then from ( n + 1 , r ) to (2 n + 1 , m ).While r varies from 0 to m , all possible paths are counted. (cid:3) Now we recall two lemmas in the representation theory of compact groups. Thefirst lemma can be obtained through direct calculation.
Lemma 2.12.
Let K be a compact topological group with Haar measure dk , Π bea unitary representation (might not be irreducible) of K . Define P Π ( v ) := 1vol( K ; dk ) Z K Π( k ) v dk, v ∈ V Π . Then P Π is the projection map from V Π to its K -invariant subspace V K Π , and Z K h Π( k ) u, v i dk = vol( K ; dk ) h P Π ( u ) , P Π ( v ) i where h , i is a K -invariant inner product defined on V Π . Lemma 2.13 ([Kna01] Schur Orthogonality Relations) . Let K be a compact Liegroup, π, π ′ be two finite-dimensional irreducible unitary representations of K , h , i be a K -invariant inner product of π or π ′ . Then, for u, v ∈ V π , u ′ , v ′ ∈ V π ′ , Z K h π ( k ) u, v ih π ′ ( k ) u ′ , v ′ i dk = ( , if π = π ′ ;vol( K ; dk ) h u,u ′ ih v,v ′ i deg π , if π = π ′ . At last we prove a lemma which we use to motivate our choice of the test functionin Section 3.1 (essentially the choice of w ◦ k ). The following lemma implies that Z G ′∞ h π ′ k ⊗ π ′ k ( g, g ) w ◦ k , w ◦ k i π ′ k ( g ) X k − dg is a constant multiple of X k − ; more explicitly, Z G ′∞ h π ′ k ⊗ π ′ k ( g, g ) w ◦ k , w ◦ k i π ′ k ( g ) X k − dg = vol( G ′∞ ) k w ◦ k k k P k k X k − . One can understand the above identity as follows: the representation π ′ k ⊗ ◦ ∆ of G ′∞ can be decomposed as π ′ k − ⊕ π ′ k − ⊕ · · · ⊕ π ′ ⊕ π ′ (with highest weight 4 k − , k − , . . . , , π ′ k in this decomposition, with w ◦ k being its lowest weightvector. Lemma 2.14.
With w ◦ k and P k defined in Lemma 2.10, we have(1) h w ◦ k ⊗ X k − − i Y i , P k i = ( h w ◦ k , w ◦ k i , i = 0;0 , i = 0 . (2) *Z G ′∞ h π ′ k ⊗ π ′ k ( g, g ) w ◦ k , w ◦ k i π ′ k ( g ) X k − dg, X k − − i Y i + = ( vol( G ′∞ ) k w ◦ k k k P k k , i = 0;0 , i = 0 . Proof. (1) Here w ◦ k has nothing to do with X , Y . So only the terms in P k with X k − − i Y i contribute to the inner product h w ◦ k ⊗ X k − − i Y i , P k i , i.e. the innerproduct is equal to *(cid:18) − Y Y (cid:12)(cid:12)(cid:12)(cid:12) X X Y Y (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) k − X k − − i Y i , (cid:12)(cid:12)(cid:12)(cid:12) X X Y Y (cid:12)(cid:12)(cid:12)(cid:12) k − X k − − i Y i X r + r = i (cid:18) k − r (cid:19) ( X ) r ( − Y ) k − − r (cid:18) k − r (cid:19) ( Y ) k − − r ( − X ) r + = *(cid:18) − Y Y (cid:12)(cid:12)(cid:12)(cid:12) X X Y Y (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) k − , (cid:12)(cid:12)(cid:12)(cid:12) X X Y Y (cid:12)(cid:12)(cid:12)(cid:12) k − X r + r = i (cid:18) k − r (cid:19) ( X ) r ( − Y ) k − − r (cid:18) k − r (cid:19) ( Y ) k − − r ( − X ) r + · h X k − − i Y i , X k − − i Y i i . VERAGES AND NONVANISHING OF TRIPLE PRODUCT CENTRAL L -VALUES 15 Notice that the sum of exponents of X , X in the first term w ◦ k = (cid:0) − Y Y ( X Y − X Y ) (cid:1) k − is always k −
1, while that in the second term is always k − i . Sothe inner product is 0 unless i = 0.When i = 0, we see that h w ◦ k ⊗ X k − , P k i = *(cid:18) − Y Y (cid:12)(cid:12)(cid:12)(cid:12) X X Y Y (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) k − , (cid:12)(cid:12)(cid:12)(cid:12) X X Y Y (cid:12)(cid:12)(cid:12)(cid:12) k − X r + r =0 k − r ! ( X ) r ( − Y ) k − − r k − r ! ( Y ) k − − r ( − X ) r + · h X k − , X k − i = *(cid:18) − Y Y (cid:12)(cid:12)(cid:12)(cid:12) X X Y Y (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) k − , (cid:12)(cid:12)(cid:12)(cid:12) X X Y Y (cid:12)(cid:12)(cid:12)(cid:12) k − ( − Y ) k − ( Y ) k − + · h w ◦ k , w ◦ k i . (2) We have that *Z G ′∞ h π ′ k ⊗ π ′ k ( g, g ) w ◦ k , w ◦ k i π ′ k ( g ) X k − dg, X k − − i Y i + = Z G ′∞ h π ′ k ⊗ π ′ k ( g, g ) w ◦ k , w ◦ k ih π ′ k ( g ) X k − , X k − − i Y i i dg = Z G ′∞ h π ′ k ⊗ π ′ k ⊗ π ′ k ( g, g, g ) w ◦ k ⊗ X k − , w ◦ k ⊗ X k − − i Y i i dg = Z G ′∞ h π ′ k ⊗ ◦ ∆ ( g ) w ◦ k ⊗ X k − , w ◦ k ⊗ X k − − i Y i i dg. Recall that P k is the only G ′∞ -invariant vector in π ′ k ⊗ ◦ ∆ up to a constantmultiple. Hence, by Lemma 2.12, the above integral is equal tovol( G ′∞ ) h w ◦ k ⊗ X k − , P k k P k k ih w ◦ k ⊗ X k − − i Y i , P k k P k k i . The value of h w ◦ k ⊗ X k − − i Y i , P k i completes the proof. (cid:3) L -functions and adelic version of Main Theorem. Let F be a local field.According to the local Langlands correspondence, for every irreducible admissiblerepresentation π of GL(2 , F ), there is a representation ρ : W F → GL(2 , C ) of theWeil group such that L ( s, ρ ) = L ( s, π ). The triple product local L -factor can bedefined by L ( s, π ⊗ π ⊗ π ) = L ( s, ρ ⊗ ρ ⊗ ρ ) ,L ( s, π ⊗ π ⊗ π , Ad) = L ( s, ⊕ i Ad( ρ i )) = Y i L ( s, Ad( ρ i )) = Y i L ( s, π i , Ad) , where Ad( ρ i ) : W F → GL(3 , C ) is the adjoint representation. For the cases athand, we can define the local L -factors more explicitly.Let F = R or C be an Archimedean local field. Recall that, for s ∈ C , ζ R ( s ) := π − s/ Γ( s/ , ζ C ( s ) := 2(2 π ) − s Γ( s ) , where Γ( s ) is the standard Γ-function. For a character µ : F × → C × , define L ( s, µ ) = ( ζ R ( s + r + m ) , when F = R , µ ( x ) = | x | r R sgn m ( x ) , r ∈ C , m ∈ { , } ; ζ C ( s + r + | m | ) , when F = C , µ ( z ) = | z | r C ( z/ ¯ z ) m , r ∈ C , m ∈ Z . (In this paper we denote sgn ( z ) = z/ ¯ z for z ∈ C .) For a discrete series represen-tation π k dis of GL(2 , R ) with weight 2 k , one can define L ( s, π k dis ) = ζ C ( s + k −
12 ) , L ( s, π k dis , Ad) = ζ R ( s + 1) ζ C ( s + 2 k − L ( s, π k dis ⊗ π k dis ⊗ π k dis ) = ζ C ( s + 3 k −
32 ) ζ C ( s + k −
12 ) . For a quadratic extension E/ Q , L ( s, ( π k dis ) E ⊗ sgn m ) = ζ C ( s + | k + m − | ) ζ C ( s + | − k + m + 12 | ) . Now let F be a non-Archimedean local field with uniformizer ̟ , and let q = O F / ( ̟ )) be the order of the residue field. For an unramified character µ (perhapswith superscripts and subscripts), L ( s, µ ) = (1 − µ ( ̟ ) q − s ) − , ζ F ( s ) = L ( s, F ) = (1 − q − s ) − . For a spherical representation π ( µ , µ ) with µ , µ unramified, L ( s, π ( µ , µ )) = L ( s, µ ) L ( s, µ ) = (1 − µ ( ̟ ) q − s ) − (1 − µ ( ̟ ) q − s ) − ; L ( s, π ( µ , µ ) , Ad) = ζ F ( s ) L ( s, µ µ − ) L ( s, µ − µ ); L ( s, π ( µ (1)1 , µ (1)2 ) ⊗ π ( µ (2)1 , µ (2)2 ) ⊗ π ( µ (3)1 , µ (3)2 )) = Y i ,i ,i ∈{ , } L ( s, µ (1) i µ (2) i µ (3) i ) . For a special representation σ µ with µ unramified, L ( s, σ µ ) = L ( s + 12 , µ ) = (1 − µ ( ̟ ) q − s − / ) − ; L ( s, σ µ , Ad) = ζ F ( s + 1) = (1 − q − s − ) − ; L ( s, σ µ ⊗ σ µ ⊗ σ µ ) = L ( s + 32 , µ µ µ ) L ( s + 12 , µ µ µ ) . In this case the local root numbers are ε ( 12 , σ µ ) = − µ ( ̟ ) , ε ( 12 , σ µ ⊗ σ µ ⊗ σ µ ) = − µ µ µ ( ̟ ) . For a quadratic extension
E/F , the base change L -factors can be defined in thesame way as above, noticing that ([GG12] Appendix E.6)( π ( µ , µ )) E = π ( µ ◦ N E/F , µ ◦ N E/F ) , ( σ µ ) E = σ µ ◦ N E/F . Globally, for a number field F , a Hecke character µ on A × F , automorphic repre-sentations π, π , π , π of GL(2 , A F ), a quadratic extension E/F and a characterΩ : E × \ A × E / A × F → C × , we define the completed L -functions ζ ∗ F ( s ) , L ( s, µ ) , L ( s, π ) , L ( s, π ⊗ π ⊗ π ) ,L ( s, π E ⊗ Ω) , L ( s, π, Ad) , L ( s, π ⊗ π ⊗ π , Ad)as Euler products of corresponding local L -factors over all places of F .These L -functions can also be defined classically associated to cuspidal modularforms. For example, normalizing f ( z ) = P n ≥ a n ( f ) e πinz , g, h ∈ F k ( N ) such VERAGES AND NONVANISHING OF TRIPLE PRODUCT CENTRAL L -VALUES 17 that a = 1, the local factors of triple product L -functions are defined as follows.When p ∤ N , L p ( s, f × g × h ) := Y i ,i ,i ∈{ , } (1 − α ( i ) p ( f ) α ( i ) p ( g ) α ( i ) p ( h ) p − s ) − where α (1) p ( f ), α (2) p ( f ) are defined to be the roots of X − a p ( f ) X + p k − = 0;when p | N , noticing that a p ( f ) p − ( k − = ±
1, we define L p ( s, f × g × h ) := (1 + ε p p k − p − s ) − (1 + ε p p k − p − s ) − where ε p := − a p ( f ) a p ( g ) a p ( h ) p − k − . Then L fin ( s, f × g × h ) is absolutely con-vergent in the half plane ℜ ( s ) > k − .We recall some facts in order to translate Theorem 1.1 to adelic language. Let π f , π g , π h be the cuspidal automorphic representations of GL(2 , A ) generated by f, g, h respectively (see Theorem 2.7). One can check by direct calculation that(7) L fin ( 12 , π f ⊗ π g ⊗ π h ) = L fin (3 k − , f × g × h ) , and(8) L fin ( 12 , ( π h ) E ⊗ Ω) = ( L fin ( k, h ) L fin ( k, h ⊗ χ − d ) , Ω = ,L fin ( k + | m | + , h × Θ Ω ) , Ω = ( χ − d , m and Θ Ω defined as in Theorem 1.1). Moreover, there is an identity thatrelates a special value of the adjoint L -function and the Petersson norm of a new-form:(9) L (1 , π f , Ad) = 2 k N ( f, f ) . Here ( · , · ) is the Petersson inner product on F k ( N ) defined in (2). The proof canbe found in [Wat02] or [CST14].With the above identities and the definition of Archimedean L -factors, one caneasily check that the following theorem is equivalent to Theorem 1.1. Recall thatfor h ∈ F k ( N ), when p | N , ( π h ) p ∼ = σ δ p is a special representation with δ p ( p ) = a p ( h ) p − ( k − = − ε p ( 12 , π h ) = ± . Theorem 2.15 (Main Theorem, adelic version) . Let N be a square-free integer withan odd number of prime factors, and F ( N, k ) be the set of cuspidal automorphic representations of PGL(2 , A ) of level N and weight k . For any π ∈ F ( N, k ) , (10) 12 N X π ,π ∈F ( N, k ) ε p = − , ∀ p | N L ( , π ⊗ π ⊗ π ) L (1 , π ⊗ π ⊗ π , Ad)= (cid:18) − δ ( k ) ϕ ( N ) (cid:19) Γ( k ) Γ(3 k − ω ( N ) Γ(2 k )Γ(2 k − + (2 π ) − k Γ(2 k − N · L (1 , π , Ad) · · ord ( N ) Y p | N − χ − ( p )2 X Ω ∈ d [ E × ] I (Ω) · L fin ( 12 , ( π ) χ − ⊗ Ω)+6 √ · ord ( N ) Y p | N − χ − ( p )2 X Ω ∈ d [ E × ] I (Ω) · L fin ( 12 , ( π ) χ − ⊗ Ω) , where I and I (defined in Theorem 4.3) depend only on k and Ω . In the following two sections we are going to prove Theorem 2.15 using therelative trace formula (RTF).3.
Spectral Side of the RTF
Let D be the definite quaternion algebra over Q with discriminant N . Let G ′ bethe algebraic group defined over Q with G ′ ( Q ) = Z ( Q ) \ D × ( Q ). We consider theRTF introduced in Section 1.3 for the case G ′ \ ( G ′ × G ′ ) /G ′ .Let f ∈ C ∞ c ( G ′ ( A ) × G ′ ( A )). Integrating f against the action of G ′ ( A ) × G ′ ( A )gives a linear map R ( f ) : L ( G ′ ( Q ) × G ′ ( Q ) \ G ′ ( A ) × G ′ ( A )) → L ( G ′ ( Q ) × G ′ ( Q ) \ G ′ ( A ) × G ′ ( A ))defined by( R ( f )Φ)( x , x ) = Z G ′ ( A ) Z G ′ ( A ) f ( g , g )Φ( x g , x g ) dg dg . From the spectral decomposition of L ( G ′ ( Q ) × G ′ ( Q ) \ G ′ ( A ) × G ′ ( A )) one sees that R ( f ) is an integral operator with kernel K f ( x , x ; y , y ) = X π ′ ⊗ π ′ ∈A ( G ′ × G ′ ) X Φ ∈ON B ( π ′ ⊗ π ′ ) (( π ′ ⊗ π ′ )( f )Φ)( x , x )Φ( y , y ) , where ON B ( π ) denotes an orthonormal basis of V π .Having fixed the diagonal embedding G ′ ֒ → G ′ × G ′ we get an injection G ′ ( A ) ֒ → G ′ ( A ) × G ′ ( A ). Let π ′ ∈ F ′ ( N, k ) be an automorphic representation of G ′ . Fixingan automorphic form φ ∈ C X k − ⊗ ( π ′ , fin ) K fin on G ′ (a new-line vector definedin Lemma 2.9), we define a distribution(11) I ( f ) := Z G ′ ( Q ) \ G ′ ( A ) Z G ′ ( Q ) \ G ′ ( A ) K f ( h , h ; h , h ) φ ( h ) φ ( h ) dh dh . The spectral expansion for the kernel K f ( x , x ; y , y ) gives I ( f ) = X π ′ ⊗ π ′ I π ′ ,π ′ ( f ) VERAGES AND NONVANISHING OF TRIPLE PRODUCT CENTRAL L -VALUES 19 where for each π ′ , π ′ ∈ A ( G ′ ) we define I π ′ ,π ′ ( f ) := X Φ ∈ON B ( π ′ ⊗ π ′ ) Z G ′ ( Q ) \ G ′ ( A ) (( π ′ ⊗ π ′ )( f )Φ)( h , h ) φ ( h ) dh · Z G ′ ( Q ) \ G ′ ( A ) Φ( h , h ) φ ( h ) dh . Recall that A ( G ′ ) = A cusp ( G ′ ) ⊔A res ( G ′ ) where A res ( G ′ ) is the set of charactersof G ′ ( Q ) \ G ′ ( A ). The automorphic representations of G ′ × G ′ are of the form π ′ ⊗ π ′ where • π ′ , π ′ ∈ A cusp ( G ′ ); • one in A cusp ( G ′ ) and another in A res ( G ′ ); or • π ′ , π ′ ∈ A res ( G ′ ).Correspondingly we can decompose I ( f ) as(12) I ( f ) = X π ′ ,π ′ ∈A cusp ( G ′ ) I π ′ ,π ′ ( f )+2 X π ′ ∈A res ( G ′ ) X π ′ ∈A cusp ( G ′ ) I π ′ ,π ′ ( f )+ X π ′ ,π ′ ∈A res ( G ′ ) I π ′ ,π ′ ( f ) . Our goal is to choose a suitable test function f ∈ C ∞ c ( G ′ ( A ) × G ′ ( A )) such that R ( f ) kills all π ′ ⊗ π ′ ∈ A cusp ( G ′ ) ⊗ A cusp ( G ′ ) unless π ′ , π ′ ∈ F ′ ( N, k ).3.1. Test function.
For v = ∞ , recall that π ′ k (defined in Section 2.4) correspondsto π k dis , via the local Jacquet–Langlands correspondence. Let h , i be a G ′∞ -invariantinner product of π ′ k ⊗ π ′ k . By Lemma 2.9 we fix φ , ∞ = k φ k X k − , and Lemma2.14 shows that there exists a vector w ◦ k ∈ π ′ k ⊗ π ′ k such that Z G ′∞ h π ′ k ⊗ π ′ k ( g, g ) w ◦ k , w ◦ k i π ′ k ( g ) X k − dg is a constant multiple of X k − . We fix such a nonzero vector w ◦ k ∈ π ′ k ⊗ π ′ k asin Lemma 2.10 and define f ∞ ∈ C ∞ c ( G ′∞ × G ′∞ ) by f ∞ ( g , g ) := h π ′ k ⊗ π ′ k ( g , g ) w ◦ k , w ◦ k i / h w ◦ k , w ◦ k i . For v = p < ∞ , fix a maximal order O p of D p = D ( Q p ). In particular, fix O p = M (2 , Z p ) and O × p = GL(2 , Z p ) when p ∤ N . Let K p be the image of Z ( Q p ) O × p in G ′ p . Clearly f := Y p< ∞ f p where f p = K p is a function in C ∞ c ( G ′ ( A fin )). We define the test function on G ′ ( A ) × G ′ ( A ) by f := f ∞ × ( f ⊗ f ), i.e.(13) f ( g , g ) := f ∞ ( g , ∞ , g , ∞ ) Y p< ∞ K p ( g ,p ) K p ( g ,p ) . In particular when 2 k = 2, π ′ k is trivial and so is f ∞ . In this case the test functionis simply f ( g , g ) = f ⊗ f = Y p< ∞ K p ( g ,p ) K p ( g ,p ) . When choosing Φ ∈ ON B ( π ′ ⊗ π ′ ) we take Φ = Φ ∞ · φ , fin ⊗ φ , fin , where Φ ∞ is aunit vector in π ′ , ∞ ⊗ π ′ , ∞ , and φ , fin , φ , fin are functions in π ′ , fin , π ′ , fin respectively.With this test function we have( R ( f )Φ)( x , x ) =(( π ′ ⊗ π ′ )( f ∞ · f ⊗ f )Φ)( x , x )=( π ′ , ∞ ⊗ π ′ , ∞ ( f ∞ )Φ ∞ ) · ( R ( f ⊗ f )( φ , fin ⊗ φ , fin ))( x , x )and( R ( f ⊗ f )( φ , fin ⊗ φ , fin ))( x , x )= Z G ′ ( A fin ) Z G ′ ( A fin ) f ( g ) f ( g ) φ , fin ( x g ) φ , fin ( x g ) dg dg =( R ( f ) φ , fin )( x ) · ( R ( f ) φ , fin )( x ) = ( π ′ , fin ( f ) φ , fin )( x ) · ( π ′ , fin ( f ) φ , fin )( x ) . Then the distribution I π ′ ,π ′ becomes I π ′ ,π ′ ( f ∞ · f ⊗ f )= X Φ ∈ONB ( π ′ ⊗ π ′ )Φ=Φ ∞ · φ , fin ⊗ φ , fin Z G ′ ( Q ) \ G ′ ( A ) (cid:16) π ′ , ∞ ⊗ π ′ , ∞ ( f ∞ )Φ ∞ (cid:17)(cid:16) π ′ , fin ( f ) φ , fin (cid:17)(cid:16) π ′ , fin ( f ) φ , fin (cid:17) φ ( h ) dh · Z G ′ ( Q ) \ G ′ ( A ) Φ( h , h ) φ ( h ) dh . In the next three sections we will answer the question that, with the test functiondefined by (13), which representations would contribute to the spectral decompo-sition (12).3.2.
Cusp ⊗ Cusp.
For an irreducible admissible representation σ of G ′ v actingon the space V σ and for f v ∈ C ∞ c ( G ′ v ), we define σ ( f v ) : V σ → V σ by σ ( f v ) w := Z G ′ v f v ( g v ) σ ( g v ) w dg v . Lemma 3.1 ([FW09] Lemma 3.2, 3.3) . For v = p < ∞ , let f p = K p as above. Let σ be an irreducible unitary representation of G ′ p . Then σ ( f p ) kills the orthogonalcomplement of σ K p in V σ , and σ ( f p ) w = vol( K p ) w for w ∈ σ K p .Proof. By definition, σ ( f p ) w = Z G ′ p K p ( g ) σ ( g ) w dg = Z K p σ ( g ) w dg for w ∈ V σ . The rest can be shown by Lemma 2.12. (cid:3)
On the Archimedean place we can prove a similar result as [FW09] Lemma 3.4.
Lemma 3.2.
Let σ = σ ⊗ σ be an irreducible unitary representation of G ′∞ × G ′∞ .Then, with the definition of f ∞ as above, σ ( f ∞ ) kills the space V σ unless σ ∼ = π ′ k ⊗ π ′ k . Furthermore for σ = π ′ k ⊗ π ′ k , σ ( f ) kills the orthogonal complement of C w ◦ k in V σ , and π ′ k ⊗ π ′ k ( f ∞ ) w ◦ k = (cid:18) vol( G ′∞ )2 k − (cid:19) w ◦ k . VERAGES AND NONVANISHING OF TRIPLE PRODUCT CENTRAL L -VALUES 21 Proof.
Since f ∞ is a matrix coefficient, Schur Orthogonality Relations (Lemma2.13) show that σ ( f ∞ ) kills the space V σ unless σ ∼ = π ′ k ⊗ π ′ k , and h π ′ k ⊗ π ′ k ( f ∞ ) w , w i = Z G ′∞ × G ′∞ f ∞ ( g ) h π ′ k ⊗ π ′ k ( g ) w , w i dg = Z G ′∞ × G ′∞ h π ′ k ⊗ π ′ k ( g ) w , w i h π ′ k ⊗ π ′ k ( g ) w ◦ k , w ◦ k ih w ◦ k , w ◦ k i dg = vol( G ′∞ × G ′∞ )dim π ′ k ⊗ π ′ k h w , w ◦ k ih w , w ◦ k ih w ◦ k , w ◦ k i = (cid:16) vol( G ′∞ )2 k − (cid:17) k w ◦ k k when w = w = w ◦ k ;0 when w or w ∈ ( C w ◦ k ) ⊥ . (cid:3) Now we apply the lemmas for π ′ ,v ⊗ π ′ ,v and work on π ′ ⊗ π ′ ( f )Φ for Φ ∈ON B ( π ′ ⊗ π ′ ). Lemma 3.3.
For cuspidal representations π ′ , π ′ of G ′ , X Φ ∈ON B ( π ′ ⊗ π ′ ) (cid:16) π ′ ⊗ π ′ ( f )Φ (cid:17) ( x ) = Φ π ′ ⊗ π ′ ( x ) (cid:16) vol( K ′ )2 k − (cid:17) , π ′ , π ′ ∈ F ′ ( N, k );0 , otherwise . Here K ′ := G ′∞ Q p< ∞ K p is an open subgroup of G ′ ( A ) , Φ π ′ ⊗ π ′ is the orthonormalbasis of the 1-dimensional subspace W π ′ ⊗ π ′ := C w ◦ k ⊗ ( π ′ , fin ) K fin ( π ′ , fin ) K fin .Proof. It follows from the previous two lemmas that R ( f ) kills the orthogonalcomplement of Φ π ′ ⊗ π ′ in V π ′ ⊗ π ′ and (cid:16) π ′ ⊗ π ′ ( f )Φ π ′ ⊗ π ′ (cid:17) ( x , x ) = Φ π ′ ⊗ π ′ ( x , x ) vol( G ′∞ )2 k − Y p< ∞ vol( K p ) ! . (cid:3) This lemma implies that, for π ′ ∈ F ′ ( N, k ), X π ′ ,π ′ ∈A cusp ( G ′ ) I π ′ ,π ′ ( f ) = (cid:18) vol( K ′ )2 k − (cid:19) X π ′ ,π ′ ∈F ′ ( N, k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z G ′ ( Q ) \ G ′ ( A ) Φ π ′ ⊗ π ′ ( h ) φ ( h ) dh (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Res ⊗ Res.
Every π ′ ∈ A res ( G ′ ) is a character δ ◦ N D for some δ : Q × \ A × →{± } . We can take φ ∈ ON B ( π ′ ) to be the normalization of δ ◦ N D . Lemma 3.4. R ( f )( δ ◦ N D ⊗ δ ◦ N D ) ≡ unless k = 2 and δ , δ are unramifiedeverywhere, in which case (with K ′ := G ′∞ Q p< ∞ K p ) R ( f )( δ ◦ N D ⊗ δ ◦ N D )( x , x ) = δ ( N D ( x )) δ ( N D ( x )) vol( K ′ ) . Proof.
By definition we have R ( f )( δ ◦ N D ⊗ δ ◦ N D )( x , x )= Z G ′ ( A ) × G ′ ( A ) f ( g , g ) δ ( N D ( x g )) δ ( N D ( x g )) dg dg = Z G ′∞ × G ′∞ f ∞ ( g , g ) δ , ∞ ( N D ∞ ( x , ∞ g )) δ , ∞ ( N D ∞ ( x , ∞ g )) dg dg · Y p< ∞ Z G ′ p K p ( g ) δ v ( N D p ( x ,p g )) dg Z G ′ p K p ( g ) δ p ( N D p ( x ,p g )) dg. When p < ∞ , since the norm map N D p : O × p → Z × p is surjective (whether D p issplit or not), we have Z G ′ p K p ( g ) δ p ( N D p ( xg )) dg = δ p ( N D p ( x )) Z K p δ p ( N D p ( g )) dg = ( , δ p ramified; δ p ( N D p ( x )) vol( K p ) , δ p unramified.When v = ∞ , N D ∞ ( D ×∞ ) = R × > by Lemma 2.1. Then δ ∞ ( N D ∞ ( g )) = 1 for all g ∈ D ×∞ since δ ∞ is quadratic. Thus Z G ′∞ × G ′∞ f ∞ ( g , g ) δ , ∞ ( N D ∞ ( x g )) δ , ∞ ( N D ∞ ( x g )) dg dg = δ , ∞ ( N D ∞ ( x )) δ , ∞ ( N D ∞ ( x )) R G ′∞ × G ′∞ h π ′ k ⊗ π ′ k ( g , g ) w ◦ k , w ◦ k i d ( g , g ) h w ◦ k , w ◦ k i = ( , if 2 k > δ , ∞ ( N D ( x )) δ , ∞ ( N D ( x )) vol( G ′∞ ) , if π ′ k ⊗ π ′ k ∼ = id . , i.e. 2 k = 2 . Putting these local calculations together shows the statement. (cid:3)
For any global field F , we define X un ( F ) to be the set of Hecke characters δ : F × \ A × F → {± } that are unramified everywhere. Notice that k δ ◦ N D k =vol( G ′ ( Q ) \ G ′ ( A )) / . Then P π ′ ,π ′ ∈A res ( G ′ ) I π ′ ,π ′ ( f )vol( K ′ ) = P δ ,δ ∈ X un ( Q ) (cid:12)(cid:12)(cid:12)R G ′ ( Q ) \ G ′ ( A ) δ δ ( N D ( h )) φ ( h )vol( G ′ ( Q ) \ G ′ ( A )) dh (cid:12)(cid:12)(cid:12) , if 2 k = 2;0 , otherwise . Recall that vol( G ′ ( Q ) \ G ′ ( A )) = vol([ D × ]) = 2. The following lemma shows thatany character δ : Q × \ A × → {± } that is unramified everywhere can only be trivial,i.e. X un ( Q ) = { } . Then P π ′ ,π ′ ∈A res ( G ′ ) I π ′ ,π ′ ( f ⊗ f )vol( K ′ ) = 14 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z G ′ ( Q ) \ G ′ ( A ) φ ( h ) dh (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 14 |h φ , i| . By orthogonality, h φ , i = 0. So X π ′ ,π ′ ∈A res ( G ′ ) I π ′ ,π ′ ( f ) = 0 . VERAGES AND NONVANISHING OF TRIPLE PRODUCT CENTRAL L -VALUES 23 Lemma 3.5.
Let F be Q or an imaginary quadratic extension of Q with classnumber 1. Then the set X un ( F ) of all characters δ : F × \ A × F → C × which areunramified everywhere is parameterized by \ F ×∞ / O × F , the set of characters of F ×∞ which are invariant under O × F . Here O × F is the group of units in the ring O F ofalgebraic integers, which is actually the set of roots of unity in F .Proof. This is a direct corollary of the strong approximation theorem: O × F \ F ×∞ × Y v< ∞ O × F v ∼ = F × \ A × F ;and the Dirichlet’s unit theorem. (cid:3) Res ⊗ Cusp.
In the same way as in the previous section we can show that
Lemma 3.6.
For π ′ ∈ A cusp ( G ′ ) , R ( f )( δ ◦ N D ⊗ φ ) ≡ unless δ is unramifiedeverywhere, k = 2 , π ′ ∈ F ′ ( N, and φ ∈ ( π ′ ) K fin , in which case R ( f )( δ ◦ N D ⊗ φ )( x , x ) = δ ( N D ( x )) φ ( x ) vol( K ′ ) , where K ′ := G ′∞ Q p< ∞ K p .Proof. The proof on the non-Archimedean places has been done in the proof ofLemma 3.1 and Lemma 3.4. On the Archimedean place, one can apply a similarproof as in Lemma 3.4 to show that, for π ′ , ∞ = δ ∞ ◦ N D ∞ , π ′ , ∞ = π ′ k ′ , π ′ , ∞ ⊗ π ′ , ∞ ( f ∞ )( δ ∞ ◦ N D ∞ ⊗ φ , ∞ ) = 0unless 2 k = 2 k ′ = 2, in which case φ , ∞ = and π ′ , ∞ ⊗ π ′ , ∞ ( f ∞ )( δ ∞ ◦ N D ∞ ⊗ φ , ∞ ) = ( δ ∞ ◦ N D ∞ ⊗ φ , ∞ ) vol( G ′∞ ) . (cid:3) For π ′ ∈ A cusp ( G ′ ), let φ π ′ ∈ C X k − ⊗ ( π ′ fin ) K fin be the normalized new-linevector as defined in Lemma 2.9. Then we can write P π ′ ∈A res ( G ′ ) P π ′ ∈A cusp ( G ′ ) I π ′ ,π ′ ( f )vol( K ′ ) = P δ ∈ X un ( Q ) P π ′ ∈F ′ ( N, (cid:12)(cid:12)(cid:12)R G ′ ( Q ) \ G ′ ( A ) δ ( N D ( h ))vol( G ′ ( Q ) \ G ′ ( A )) / φ π ′ ( h ) φ ( h ) dh (cid:12)(cid:12)(cid:12) , if 2 k = 2;0 , otherwise . We already know that X un ( Q ) = { } . So when 2 k = 2, P π ′ ∈A res ( G ′ ) P π ′ ∈A cusp ( G ′ ) I π ′ ,π ′ ( f ⊗ f )vol( K ′ ) = 12 X π ′ ∈F ′ ( N, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z G ′ ( Q ) \ G ′ ( A ) φ π ′ ( h ) φ ( h ) dh (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 12 X π ′ ∈F ′ ( N, (cid:12)(cid:12) h φ , φ π ′ i (cid:12)(cid:12) . By orthogonality, h φ , φ i 6 = 0 only when φ π ′ = φ / k φ k up to a constant multiple.So P π ′ ∈A res ( G ′ ) P π ′ ∈A cusp ( G ′ ) I π ′ ,π ′ ( f ⊗ f )vol( K ′ ) = 12 (cid:12)(cid:12)(cid:12)(cid:12) h φ , φ k φ k i (cid:12)(cid:12)(cid:12)(cid:12) = 12 h φ , φ i . Application of Ichino’s formula.
In summary, for π ′ ∈ F ′ ( N, k ),(14) P π ′ ,π ′ ∈A cusp ( G ′ ) I π ′ ,π ′ ( f )vol( K ′ ) = (cid:18) k − (cid:19) X π ′ ,π ′ ∈F ′ ( N, k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z G ′ ( Q ) \ G ′ ( A ) Φ π ′ ⊗ π ′ ( h ) φ ( h ) dh (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . P π ′ ∈A res ( G ′ ) π ′ ∈A cusp ( G ′ ) I π ′ ,π ′ ( f )vol( K ′ ) = ( h φ , φ i , if 2 k = 2;0 , otherwise . P π ′ ,π ′ ∈A res ( G ′ ) I π ′ ,π ′ ( f )vol( K ′ ) = 0 . Now we can apply Ichino’s triple product formula to the sum P π ′ ,π ′ ∈A cusp ( G ′ ) I π ′ ,π ′ ( f ). Theorem 3.7 ([Ich08] Ichino’s period formula) . Let F be a number field, E = F × F × F , π i be an cuspidal automorphic representation of GL(2 , A F ) for i = 1 , , .Assume that the product of the central characters of π i is trivial. D is a quaternionalgebra over F so that there exists an irreducible unitary automorphic representation Π ′ = π ′ ⊗ π ′ ⊗ π ′ of ( D × ( A F )) associated to Π = π ⊗ π ⊗ π by the Jacquet–Langlands correspondence. Then, for Φ = ⊗ v Φ v ∈ Π ′ , (cid:12)(cid:12)(cid:12)R G ′ ( F ) \ G ′ ( A F ) Φ( g ) dg (cid:12)(cid:12)(cid:12) h Φ , Φ i = ζ ∗ F (2) L ( , π ⊗ π ⊗ π ) L (1 , π ⊗ π ⊗ π , Ad) Y v I v , where dg = Q v dg v is the Tamagawa measure on G ′ ( A F ) , I v = 1 ζ F v (2) L v (1 , Π , Ad) L v (1 / , Π) Z G ′ v B v (Π ′ v ( g v )Φ v , Φ v ) B v (Φ v , Φ v ) dg v , the B v ’s are ( D × ( A F )) -invariant pairings between Π ′ v and its contragredient ˜Π ′ v sothat Y v B v (Φ v , Φ ′ v ) = h Φ , Φ ′ i := y ( G ′ ( F ) \ G ′ ( A F )) Φ( g , g , g )Φ ′ ( g , g , g ) dg dg dg and B v (Φ v , Φ ′ v ) = 1 for almost all v . We calculate the local factors I v for the cases at hand. Assume F = Q , D isdefinite, and N = disc( D ) is square-free with an odd number of prime factors. Let Φbe of the form Φ π ′ ⊗ π ′ ⊗ φ which contributes to the sum P π ′ ,π ′ ∈A cusp ( G ′ ) I π ′ ,π ′ ( f ).Then(15) Φ ∞ = w ◦ k k w ◦ k k ⊗ k φ k X k − and on the non-Archimedean places Φ p are tensor products of three K p -invariantunit vectors.When p ∤ N , according to [Ich08] Lemma 2.2, I p = vol( K p ; dg p ) = ζ p (2) − , p ∤ N. When p | N , ( π i ) p is the special representation σ δ i for some unramified character δ i : Q × p → {± } . The corresponding φ i , up to a constant multiple, is δ i ◦ N D p . VERAGES AND NONVANISHING OF TRIPLE PRODUCT CENTRAL L -VALUES 25 [Woo11] Proposition 5.5 shows that I p = (1 − ε p ) 1 p (1 − p ) ζ p (2) − , p | N with ε p = − ( δ δ δ )( p ) = ε p ( , Π). (Note that the measure in [Woo11] differs by afactor ζ p (2).)On the Archimedean place we have I ∞ = 1 ζ R (2) ( L (1 , π k dis , Ad)) L (1 / , π k dis ⊗ π k dis ⊗ π k dis ) Z G ′∞ B ∞ (Π ′∞ ( g ∞ )Φ ∞ , Φ ∞ ) B ∞ (Φ ∞ , Φ ∞ ) dg ∞ . Here B ∞ ( · , ¯ · ) = h· , ·i is the inner product defined in (6). By (15), h Φ ∞ , Φ ∞ i ∞ = k φ k , and, applying Lemma 2.12, we have Z G ′∞ B ∞ (Π ′∞ ( g )Φ ∞ , Φ ∞ ) dg = k φ k k w ◦ k k Z G ′∞ h π ′ k ⊗ π ′ k ⊗ π ′ k ◦ ∆ ( g ) w ◦ k ⊗ X k − , w ◦ k ⊗ X k − i dg = k φ k k w ◦ k k vol( G ′∞ ) h w ◦ k ⊗ X k − , P k k P k k ih w ◦ k ⊗ X k − , P k k P k k i = k φ k k w ◦ k k π |h w ◦ k ⊗ X k − , P k i| k P k k . Here P k , as defined in Lemma 2.10, is the only D × ( R )-invariant vector in π ′ k ⊗ π ′ k ⊗ π ′ k ◦ ∆ up to a constant multiple. Lemma 2.14 shows that h w ◦ k ⊗ X k − , P k i = k w ◦ k k . Moreover, ζ R (2) = π − , L (1 , π k dis , Ad) = ζ C (2 k ) ζ R (2) = 2 (2 π ) − − k Γ(2 k ) ,L (1 / , π k dis ⊗ π k dis ⊗ π k dis ) = ζ C ( k ) ζ C (3 k −
1) = 2 (2 π ) − k Γ( k ) Γ(3 k − . Therefore, with Lemma 2.10 one can check that I ∞ = Γ(2 k ) Γ(2 k − k ) Γ(3 k − . Now we see that Q v I v = 0 unless ε p = − p | N , in which case Y v I v = ζ ∗ Q (2) − Γ(2 k ) Γ(2 k − k ) Γ(3 k −
1) 2 ω ( N ) ϕ ( N ) πN . Theorem 3.7 explicitly becomes
Lemma 3.8.
With Φ π ′ ⊗ π ′ defined in Lemma 3.3 and φ a new-line vector definedin Lemma 2.9, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z G ′ ( Q ) \ G ′ ( A ) Φ π ′ ⊗ π ′ ( h ) φ ( h ) dh (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = k φ k Γ(2 k ) Γ(2 k − k ) Γ(3 k −
1) 2 ω ( N ) ϕ ( N )48 N L ( , π ⊗ π ⊗ π ) L (1 , π ⊗ π ⊗ π , Ad) if ε p = − for every p | N ; otherwise it vanishes. Recall that vol( K ′ ) = vol( G ′∞ ; dg ∞ ) Y p vol( K p ; dg p ) = 24 ϕ ( N ) . Combining (12) (14) and the above lemma, we get that:
Theorem 3.9 (Main Theorem, spectral side) . Let N be a square-free product ofan odd number of primes. For π ′ ∈ F ′ ( N, k ) , I ( f ) h φ , φ i vol( K ′ ) = Γ(2 k − Γ( k ) Γ(3 k −
1) 2 ω ( N ) N X π ,π ∈F ( N, k ) ε p = − , ∀ p | N L ( , π ⊗ π ⊗ π ) L (1 , π ⊗ π ⊗ π , Ad)+ ( h φ , φ i ϕ ( N ) , if k = 2;0 , otherwise. Geometric Side of the RTF
Recall that G ′ = Z \ D × , K p is the image of Z p O × p in G ′ p ; f ( g , g ) = f ∞ ( g , g ) Y p< ∞ K p ( g ) K p ( g ) ,f ∞ ( g , g ) = h π ′ k ⊗ π ′ k ( g , g ) w ◦ k , w ◦ k i / h w ◦ k , w ◦ k i , w ◦ k = (cid:18) − Y Y (cid:12)(cid:12)(cid:12)(cid:12) X X Y Y (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) k − ; P k = (cid:12)(cid:12)(cid:12)(cid:12) X X Y Y (cid:12)(cid:12)(cid:12)(cid:12) k − ⊗ (cid:12)(cid:12)(cid:12)(cid:12) X X Y Y (cid:12)(cid:12)(cid:12)(cid:12) k − ⊗ (cid:12)(cid:12)(cid:12)(cid:12) X X Y Y (cid:12)(cid:12)(cid:12)(cid:12) k − ; φ is a new-line vector in π ′ ∈ F ′ ( N, k ) with φ , ∞ a highest weight vector of π ′ k ,as defined in Lemma 2.9. In Section 4.1 we will proof the following theorem whichgives the orbital expansion of the distribution I ( f ) = Z G ′ ( Q ) \ G ′ ( A ) Z G ′ ( Q ) \ G ′ ( A ) K f ( h , h ; h , h ) φ ( h ) φ ( h ) dh dh . Theorem 4.1.
Let N be a square-free integer which has an odd number of primefactors. Let D be the quaternion algebra Q which is ramified precisely at ∞ andthe primes dividing N . Then I ( f ) = I [1] + I [ γ ] + I [ γ ] where γ , γ ∈ D × ( Q ) such that Tr D ( γ ) = 0 , N D ( γ ) = 1; Tr D ( γ ) = N D ( γ ) = 1 , with I [1] = h φ , φ i vol( K ′ )2 k − ,I [ γ ] = 12 vol( K ′ ) Z A × E \ D × ( A ) ϕ γ ( h ) Z A × E × \ A × E ( R ( h ) φ ∗ )( t )( R ( h ) φ ∗∗ )( t ) dt dh,I [ γ ] = vol( K ′ ) Z A × E \ D × ( A ) ϕ γ ( h ) Z A × E × \ A × E ( R ( h ) φ ∗ )( t )( R ( h ) φ ∗∗ )( t ) dt dh. Here K ′ = G ′∞ Q p< ∞ K p ; E = Q ( γ ) is the quadratic extension of Q which can beembedded in D when γ exists (in particular E = Q ( γ ) = Q ( √− , E = Q ( γ ) = VERAGES AND NONVANISHING OF TRIPLE PRODUCT CENTRAL L -VALUES 27 Q ( √− ); φ ∗ , φ ∗∗ ∈ π ′ such that φ ∗ = φ ; φ ∗∗ = ⊗ φ ∗∗ v , φ ∗∗∞ = k φ kk P k k e γ , φ ∗∗ p = φ ,p ;(16) e γ = k − X i =0 (cid:18) k − i (cid:19) h π ′ k ⊗ π ′ k ⊗ π ′ k ( h − γh, , P k , w ◦ k ⊗ X k − − i Y i i X k − − i Y i ;(17) ϕ γ is the characteristic function of the set { h ∈ G ′ γ ( A ) \ G ′ ( A ) : h − p γh p ∈ K p for all primes p } . And • I [ γ ] = 0 if N has a prime factor of the form n + 1 (in this case γ doesnot exist); • I [ γ ] = 0 if N has a prime factor of the form n + 1 (in this case γ doesnot exist).(These two primes do not have to be distinct.) For some particular N only the trivial orbit appears in the orbital decomposition.In fact we have: Corollary 4.2.
With assumptions and notations as before, if in addition N has aprime factor ≡ and one ≡ , we have I ( f )vol( K ′ ) = h φ , φ i k − . With Theorem 3.9 we have that ω ( N ) N X π ,π ∈F ( N, k ) ε p = − , ∀ p | N L ( , π ⊗ π ⊗ π ) L (1 , π ⊗ π ⊗ π , Ad) = ( − ϕ ( N ) , if k = 2; Γ( k ) Γ(3 k − k − Γ(2 k ) = (cid:0) k − k − (cid:1) − (cid:0) k − k − (cid:1) , otherwise. This corollary and (7) (8) together give a proof of (3).After proving Theorem 4.1 we will use Waldspurger’s formula (Theorem 4.9)to compute I [ γ ] and I [ γ ] in Section 4.2. This combined with the calculations inSections 4.3, 4.4 and 4.6 gives a proof of the following theorem. Theorem 4.3 (Main Theorem, geometric side) . Let N be a square-free product ofan odd number of primes. For π ′ ∈ F ′ ( N, k ) , I ( f ) = I [1] + I [ γ ] + I [ γ ]8 BIN GUAN with I [1] h φ , φ i vol( K ′ ) = 12 k − I [ γ ] h φ , φ i vol( K ′ ) = 4 Γ(2 k − (2 π ) k Γ( k ) Γ(3 k −
1) 2 ω ( N ) N · L (1 , π , Ad) · ord ( N ) Y p | N − χ − ( p )2 X Ω ∈ d [ E × ] I (Ω) · L fin ( 12 , ( π ) E ⊗ Ω) ,I [ γ ] h φ , φ i vol( K ′ ) = 6 √ k − (2 π ) k Γ( k ) Γ(3 k −
1) 2 ω ( N ) N · L (1 , π , Ad) · ord ( N ) Y p | N − χ − ( p )2 X Ω ∈ d [ E × ] I (Ω) · L fin ( 12 , ( π ) E ⊗ Ω) . Here E = Q ( √− , E = Q ( √− , I and I are constants depending only on k and Ω (and on a ( h ) , a ( h ) when or | N respectively). More precisely, I and I vanish unless Ω satisfies the restrictions in Lemma 4.10 respectively, in whichcase I (Ω) := I ( m )2 k ( γ ) , I (Ω) := I ( m )2 k ( γ ) as described in Lemma 4.18. In particular for Ω = the trivial character, I ( ) = I (0)2 k ( γ ) = Γ( k ) Γ(2 k −
1) 2 k − P i =0 γ k − − i )0 (cid:0) k − i (cid:1) − | C i, k − − i,k − | , if ∤ N, (1 − ε ( , π )) I (0)2 k ( γ ) , if | N ; I ( ) = I (0)2 k ( γ ) = Γ( k ) Γ(2 k −
1) 2 k − P i =0 γ k − − i )1 (cid:0) k − i (cid:1) − | C i, k − − i,k − | , if ∤ N, (1 − ε ( , π )) I (0)2 k ( γ ) , if | N. Here C i,j,r is the coefficient of X k − − i Y i ⊗ X k − − j Y j ⊗ X k − − r Y r in P k := (cid:12)(cid:12)(cid:12)(cid:12) X X Y Y (cid:12)(cid:12)(cid:12)(cid:12) k − ⊗ (cid:12)(cid:12)(cid:12)(cid:12) X X Y Y (cid:12)(cid:12)(cid:12)(cid:12) k − ⊗ (cid:12)(cid:12)(cid:12)(cid:12) X X Y Y (cid:12)(cid:12)(cid:12)(cid:12) k − . Theorem 3.9 and 4.3 together imply the Main Theorem 2.15.4.1.
Orbital decomposition.
We apply the geometric side of the RTF to thedistribution I ( f ). When G = G ′ × G ′ and H = H = G ′ ( H , H ֒ → G diagonally),the representatives [( γ , γ )] in G ′ ( Q ) \ G ′ × G ′ ( Q ) /G ′ ( Q ) can be chosen such that γ = 1 and [ γ ] runs through all conjugacy classes of G ′ ( Q ). For θ , θ ∈ G ′ ( Q ), θ − ( γ, θ = ( γ,
1) if and only if θ = θ ∈ G ′ γ ( Q ), the centralizer of γ in G ′ ( Q ).By the orbital expansion of the RTF, we have I ( f ) = Z G ′ ( Q ) \ G ′ ( A ) Z G ′ ( Q ) \ G ′ ( A ) K f ( h , h ; h , h ) φ ( h ) φ ( h ) dh dh = x ( G ′ ( Q ) \ G ′ ( A )) X γ ,γ ∈ G ′ ( Q ) f ( h − γ h , h − γ h ) φ ( h ) φ ( h ) dh dh = x ( G ′ ( Q ) \ G ′ ( A )) X [ γ ,γ ]=[ γ, γ ∈ [ G ′ ( Q )] X ( θ ,θ ) f ( h − θ − γθ h , h − θ − θ h ) φ ( h ) φ ( h ) dh dh L -VALUES 29 Here ( θ , θ ) runs through G ′ ( Q ) × G ′ ( Q ) / { ( θ , θ ) : θ = θ ∈ G ′ γ ( Q ) } ∼ = ( G ′ γ ( Q ) \ G ′ ( Q )) × G ′ ( Q ) , i.e. for a fixed θ ∈ G ′ γ ( Q ) \ G ′ ( Q ), θ runs through G ′ ( Q ). So we have I ( f ) = X [ γ ] X θ ,θ x ( G ′ ( Q ) \ G ′ ( A )) f (cid:0) ( θ h ) − γ ( θ h ) , ( θ h ) − ( θ h ) (cid:1) φ ( θ h ) φ ( θ h ) dθ h dθ h where [ γ ] runs through conjugacy classes of G ′ ( Q ). Let θ i h i be the new h i ( i = 1 , I ( f ) = X [ γ ] Z G ′ γ ( Q ) \ G ′ ( A ) Z G ′ ( A ) f ( h − γh , h − h ) φ ( h ) dh ! φ ( h ) dh . We split h = th with h ∈ G ′ γ ( A ) \ G ′ ( A ) and t ∈ G ′ γ ( Q ) \ G ′ γ ( A ), and let g = t − h .Then I ( f ) = X [ γ ] Z G ′ γ ( Q ) \ G ′ γ ( A ) Z G ′ γ ( A ) \ G ′ ( A ) Z G ′ ( A ) f (( th ) − γ ( tg ) , ( th ) − tg ) φ ( tg ) d ( tg ) ! φ ( th ) dh dt = X [ γ ] Z G ′ γ ( Q ) \ G ′ γ ( A ) Z G ′ γ ( A ) \ G ′ ( A ) Z G ′ ( A ) f ( h − γg, h − g ) φ ( tg ) dg ! φ ( th ) dh dt. Denote the summand as I [ γ ] ( f ).Recall that the new-line vector φ = ⊗ φ ,v ∈ C X k − ⊗ π ′ K fin , fin can be writtensuch that φ , ∞ = k φ k X k − and φ ,p ’s are K p -invariant unit vectors for p < ∞ (see Lemma 2.9). Let f = f ∞ · ( f ⊗ f ) be the test function defined in (13). Then Lemma 4.4. R G ′ ( A ) f ( h − γg, h − g )( R ( g ) φ ) dg is a pure tensor in π ′ .Proof. For f = f ∞ · ( f ⊗ f ), Z G ′ ( A ) f ( h − γg, h − g )( R ( g ) φ ) dg = k φ k Z G ′∞ f ∞ ( h − ∞ γg ∞ , h − ∞ g ∞ )( π ′ k ( g ∞ ) X k − ) dg ∞ · Y p< ∞ Z G ′ p K p ( h − p γg p ) K p ( h − p g p ) R ( g p ) φ ,p dg p . For v = p < ∞ the local test function K p ( h − p γh ,p ) K p ( h − p h ,p ) is nonzeroonly if both h − p γg p , h − p g p ∈ K p , and hence Z G ′ p K p ( h − p γg p ) K p ( h − p g p ) R ( g p ) φ ,p dg p = Z h p K p ∩ γ − h p K p R ( g p ) φ ,p dg p . These two left cosets either coincide or are disjoint, and h p K p = γ − h p K p if andonly if h − p γh p ∈ K p . Let ϕ γ = Q ϕ γ,p and ϕ γ,p : G ′ γ ( Q p ) \ G ′ p → C be thecharacteristic function of the set { h p : h − p γh p ∈ K p } . Since φ ,p is K p -invariant,we have Z G ′ p K p ( h − p γg p ) K p ( h − p g p ) R ( g p ) φ ,p dg p = ϕ γ,p ( h p ) Z h p K p R ( g p ) φ ,p dg p = vol( K p ) ϕ γ,p ( h p ) R ( h p ) φ ,p . For v = ∞ , by the following lemma we have Z G ′∞ f ∞ ( h − ∞ γg ∞ , h − ∞ g ∞ ) π ′ k ( g ∞ ) X k − dg ∞ = vol( G ′∞ ) k P k k π ′ k ( h ∞ ) e γ where e γ is defined in (16). (cid:3) Lemma 4.5.
For γ ∈ G ′ ( Q ) , h ∈ G ′ γ ( R ) \ G ′∞ , with e γ defined in (16) , Z G ′∞ h π ′ k ⊗ π ′ k ( h − γg, h − g ) w ◦ k , w ◦ k i π ′ k ( g ) X k − dg = vol( G ′∞ ) k w ◦ k k k P k k π ′ k ( h ) e γ . Proof.
First we consider the inner product *Z G ′∞ h π ′ k ⊗ π ′ k ( h − γg, h − g ) w ◦ k , w ◦ k i π ′ k ( g ) X k − dg, π ′ k ( h ) X k − − i Y i + = Z G ′∞ h π ′ k ⊗ π ′ k ( g, g ) w ◦ k , π ′ k ⊗ π ′ k ( γ − h, h ) w ◦ k ih π ′ k ( g ) X k − , π ′ k ( h ) X k − − i Y i i dg = Z G ′∞ h π ′ k ⊗ π ′ k ⊗ π ′ k ( g, g, g ) w ◦ k ⊗ X k − , π ′ k ⊗ π ′ k ⊗ π ′ k ( γ − h, h, h ) w ◦ k ⊗ X k − − i Y i i dg. By Lemma 2.12 the above integral is equal tovol( G ′∞ ) h w ◦ k ⊗ X k − , P k k P k k ih π ′ k ⊗ π ′ k ⊗ π ′ k ( γ − h, h, h ) w ◦ k ⊗ X k − − i Y i , P k k P k k i . Recall that P k is G ′∞ -invariant, and we have h w ◦ k ⊗ X k − , P k i = k w ◦ k k byLemma 2.14. Now the above integral is equal tovol( G ′∞ ) k w ◦ k k h π ′ k ⊗ π ′ k ⊗ π ′ k ( γ, , P k , π ′ k ⊗ π ′ k ⊗ π ′ k ( h, h, h ) w ◦ k ⊗ X k − − i Y i i = vol( G ′∞ ) k w ◦ k k k P k k h π ′ k ⊗ π ′ k ⊗ π ′ k ( γh, h, h ) P k , π ′ k ⊗ π ′ k ⊗ π ′ k ( h, h, h ) w ◦ k ⊗ X k − − i Y i i = vol( G ′∞ ) k w ◦ k k k P k k h π ′ k ⊗ π ′ k ⊗ π ′ k ( h − γh, , P k , w ◦ k ⊗ X k − − i Y i i . Recall that, for a fixed h , { π ′ k ( h ) X k − − i Y i } forms an orthogonal basis of V π ′ k ,with the inner product defined as (5). So we have Z G ′∞ h π ′ k ⊗ π ′ k ( h − γg, h − g ) w ◦ k , w ◦ k i π ′ k ( g ) X k − dg = k − X i =0 DR G ′∞ h π ′ k ⊗ π ′ k ( h − γg, h − g ) w ◦ k , w ◦ k i π ′ k ( g ) X k − dg, π ′ k ( h ) X k − − i Y i E h π ′ k ( h ) X k − − i Y i , π ′ k ( h ) X k − − i Y i i π ′ k ( h ) X k − − i Y i = k − X i =0 vol( G ′∞ ) k w ◦ k k k P k k h π ′ k ⊗ π ′ k ⊗ π ′ k ( h − γh, , P k , w ◦ k ⊗ X k − − i Y i ih X k − − i Y i , X k − − i Y i i π ′ k ( h ) X k − − i Y i = vol( G ′∞ ) k w ◦ k k k P k k k − X i =0 k − i ! h π ′ k ⊗ π ′ k ⊗ π ′ k ( h − γh, , P k , w ◦ k ⊗ X k − − i Y i i π ′ k ( h ) X k − − i Y i . (cid:3) Take φ ∗ , φ ∗∗ ∈ π ′ such that φ ∗ = φ ; φ ∗∗ = ⊗ φ ∗∗ v , φ ∗∗ v = ( k φ kk P k k e γ , v = ∞ ; φ ,p , v = p < ∞ . VERAGES AND NONVANISHING OF TRIPLE PRODUCT CENTRAL L -VALUES 31 We can write I [ γ ] ( f ) = Z G ′ γ ( Q ) \ G ′ γ ( A ) Z G ′ γ ( A ) \ G ′ ( A ) vol( K ′ )( R ( h ) φ ∗∗ )( t ) Y p< ∞ ϕ γ,p ( h p )( R ( h ) φ ∗ )( t ) dh dt = vol( K ′ ) Z G ′ γ ( A ) \ G ′ ( A ) ϕ γ ( h ) Z G ′ γ ( Q ) \ G ′ γ ( A ) ( R ( h ) φ ∗ )( t )( R ( h ) φ ∗∗ )( t ) dt dh, with K ′ , e γ , ϕ γ defined in Theorem 4.1.When γ = 1, the centralizer G ′ γ = G ′ . According to Lemma 2.14, e γ = k w ◦ k k X k − . Then by Lemma 2.10 I [1] ( f )vol( K ′ ) = Z G ′ ( Q ) \ G ′ ( A ) φ ∗ ( t ) φ ∗∗ ( t ) dt = h φ ∗ , φ ∗∗ i = Y v h φ ∗ v , φ ∗∗ v i v = k φ k h X k − , k P k k e γ i ∞ = k φ k k w ◦ k k k P k k = k φ k k − . Now we study the property for the other [ γ ]’s such that ϕ γ is not identically 0.Instead of Q , we consider it over an arbitrary number field F . Lemma 4.6. ϕ γ = 0 unless Tr D ( γ ) ∈ {± }\O F and N D ( γ ) ∈ O × F / ( O × F ) . Inparticular, when F = Q , ϕ γ = 0 unless Tr D ( γ ) ∈ {± }\ Z ∼ = Z ≥ and N D ( γ ) = ± .Proof. Fix a set Σ ⊂ O F − { } of representatives in F × / ( F × ) . We can chooseΣ to be the set of “square-free” integers. More precisely, the factorization of theprincipal ideal of O F generated by any number in Σ has exactly one factor for eachprime ideal that appears in it. Then as in Lemma 2.2 we can fix a representativeof γ in D × ( F ) (also denoted γ ) so that N D ( γ ) ∈ Σ. Under this assumption we seethat ord v ( N D ( γ )) is either 0 or 1.Suppose that ϕ γ ( h ) = 0. This means h − v γh v ∈ K v for all v < ∞ .When v / ∈ Ram( D ), K v = PGL(2 , O F v ). We can say that, fixing a representativeof γ in GL(2 , F v ) there is an h v ∈ GL(2 , F v ) such that λ v h − v γh v ∈ GL(2 , O F v ) forsome λ v ∈ F × v . As a matrix in GL(2 , O F v ), we have thatTr D v ( λ v h − v γh v ) ∈ O F v , N D v ( λ v h − v γh v ) ∈ O × F v . Conjugate matrices have the same norm and trace, soord v (Tr D v ( λ v γ )) = ord v ( λ v Tr D ( γ )) ≥ , ord v ( N D v ( λ v γ )) = ord v ( λ v N D ( γ )) = 0 . With the assumption of Σ, one can imply that ord v ( N D ( γ )) = 0, λ v ∈ O × F v is aunit, and then ord v (Tr D ( γ )) ≥ v ∈ Ram( D ) and v < ∞ , D v is a division algebra and it has only onemaximal order O v = { x ∈ D v : N D v ( x ) ∈ O F v } . We still know that, there is an h v ∈ D × v such that λ v h − v γh v ∈ O × v for some λ v ∈ F × v . With the assumption of Σ, as in the previous case, one can implythat ord v ( N D ( γ )) = 0, i.e. γ ∈ O × v . Moreover, Proposition 2.5 shows that γ is O F v -integral and then ord v (Tr D ( γ )) ≥ γ such that N D ( γ ) ∈ Σ, for any place v of F , N D ( γ )cannot be divisible by v , and the order of v in the ideal decomposition of Tr D ( γ ) ∈ F has to be non-negative. In other words N D ( γ ) has to be a unit and Tr D ( γ ) in O F .With Lemma 2.2 we get the conclusion. (cid:3) With this lemma, we only need to consider the summand I [ γ ] ( f ) with N D ( γ ) a“square-free” unit in O × F and Tr D ( γ ) ∈ {± }\O F . Going back to the case when F = Q , we only need to consider the summand I [ γ ] ( f ) with N D ( γ ) = 1 and Tr D ( γ ) ∈ Z ≥ (obviously N D ( γ ) cannot be − D is definite). Now we consider the infiniteplace Q ∞ = R . When D is definite, D ( R ) ∼ = H is non-split. That means thecharacteristic polynomial X − Tr D ( x ) X + N D ( x )of any x ∈ D × ( Q ) is irreducible over R if and only if x / ∈ R ∩ D × ( Q ) = Q × . For γ ∈ G ′ ( Q ) = Z ( Q ) \ D × ( Q ), if γ = 1, X − Tr D ( γ ) X + N D ( γ ) is irreducible over R ,which implies that Tr D ( γ ) < N D ( γ ). Now N D ( γ ) = 1, we only need Tr D ( γ ) = 0or 1. Proposition 4.7.
Denote N and D as in Theorem 4.1.(1) There is an x ∈ D with Tr D ( x ) = 0 , N D ( x ) = 1 if and only if N has no primefactor of the form n + 1 .(2) There is an x ∈ D with Tr D ( x ) = 1 , N D ( x ) = 1 if and only if N has no primefactor of the form n + 1 .Proof. The sufficiency can be seen directly from Lemma 2.4. If N has no primefactor of the form 4 n +1, D has a presentation D = ( − , − N Q ) and clearly Tr D ( i ) = 0, N D ( i ) = 1. If N has no prime factor of the form 3 n + 1, D has a presentation D = ( − , − N Q ) and one can check Tr D ( i ) = 1, N D ( i ) = 1.See Appendix A.2 for the proof of the necessity. (cid:3) Lemma 4.6, Proposition 4.7, and the following lemma together imply Theorem4.1.
Lemma 4.8.
With notations in Theorem 4.1, I [ γ ] ( f )vol( K ′ ) = 12 Z A × E \ D × ( A ) ϕ i ( h ) Z A × E × \ A × E ( R ( h ) φ ∗ )( t )( R ( h ) φ ∗∗ )( t ) dt ! dh,I [ γ ] ( f )vol( K ′ ) = Z A × E \ D × ( A ) ϕ i ( h ) Z A × E × \ A × E ( R ( h ) φ ∗ )( t )( R ( h ) φ ∗∗ )( t ) dt ! dh. Here E = Q [ X ] / ( X + 1) = Q ( √− , E = Q [ X ] / ( X + 3) = Q ( √− arequadratic extensions of Q which can be embedded in D when γ , γ exist respectively.Proof. (1) When N has no prime factor of the form 4 n + 1, we can write D =( − , − N Q ) by Lemma 2.4 and take γ = i D (the i in the Q -basis { , i, j, k } of D ).Then I [ γ ] = I [ i ] = vol( K ′ ) Z G ′ i ( A ) \ G ′ ( A ) ϕ i ( h ) Z G ′ i ( Q \ A ) ( R ( h ) φ ∗ )( t )( R ( h ) φ ∗∗ )( t ) dt ! dh, where ϕ i is the characteristic function of the set { h ∈ G ′ i ( A ) \ G ′ ( A ) : h − p i D h p ∈ K p for all primes p } . Lemma 2.3 shows that G ′ i ( Q ) is the image in G ′ ( Q ) of { λ + µi ∈ D × ( Q ) : λ, µ ∈ Q }∪{ λj + µk ∈ D × ( Q ) : λ, µ ∈ Q } = Q ( i ) × ·{ , j } = { , j }· Q ( i ) × . VERAGES AND NONVANISHING OF TRIPLE PRODUCT CENTRAL L -VALUES 33 Instead of using G ′ i we can simply consider a subgroup T in G ′ such that T ( Q ) isthe image in G ′ ( Q ) of Q ( i ) × . By Lemma 2.6, for f ∈ C ∞ c ( G ′ i ( Q ) \ G ′ i ( A )), Z G ′ i ( Q ) \ G ′ i ( A ) f ( g ) dg = Z G ′ i ( Q ) \ G ′ i ( Q ) T ( A ) f ( g ) dg = Z ( G ′ i ( Q ) ∩ T ( A )) \ T ( A ) f ( t ) dt = Z T ( Q ) \ T ( A ) f ( t ) dt and hence Z G ′ i ( Q \ A ) ( R ( h ) φ ∗ )( t )( R ( h ) φ ∗∗ )( t ) dt = Z T ( Q \ A ) ( R ( h ) φ ∗ )( t )( R ( h ) φ ∗∗ )( t ) dt,I [ i ] vol( K ′ ) = Z G ′ i ( A ) \ G ′ ( A ) ϕ i ( h ) Z T ( Q \ A ) ( R ( h ) φ ∗ )( t )( R ( h ) φ ∗∗ )( t ) dt ! dh = 12 Z T ( A ) \ G ′ ( A ) ϕ i ( h ) Z T ( Q \ A ) ( R ( h ) φ ∗ )( t )( R ( h ) φ ∗∗ )( t ) dt ! dh. (2) When N has no prime factor of the form 3 n + 1, we can write D = ( − , − N Q )by Lemma 2.4 and take γ = (1 + i D ). Then I [ γ ] = I [1+ i ] = vol( K ′ ) Z G ′ i ( A ) \ G ′ ( A ) ϕ i ( h ) Z G ′ i ( Q \ A ) ( R ( h ) φ ∗ )( t )( R ( h ) φ ∗∗ )( t ) dt ! dh, where ϕ i is the characteristic function of the set { h ∈ G ′ i ( A ) \ G ′ ( A ) : h − p (1 + i ) h p ∈ K p for all primes p } . Lemma 2.3 shows that G ′ i ( Q ) is the image in G ′ ( Q ) of { λ + µ (1 + i ) ∈ D × ( Q ) : λ, µ ∈ Q } = Q ( i ) × = T ( Q )(notice that this i = √− i = √− I [1+ i ] vol( K ′ ) = Z T ( A ) \ G ′ ( A ) ϕ i ( h ) Z T ( Q \ A ) ( R ( h ) φ ∗ )( t )( R ( h ) φ ∗∗ )( t ) dt ! dh. (cid:3) Nontrivial orbits and Waldspurger’s formula.
For any two forms φ ′ , φ ′′ ∈ π ′ , as functions in L ([ D × ]), we already have a Petersson inner product h· , ·i definedas(18) h φ ′ , φ ′′ i := Z [ D × ] φ ′ ( g ) φ ′′ ( g ) dg. We can also consider them as functions in L ([ E × ]), in which we have an innerproduct h φ ′ , φ ′′ i E := Z [ E × ] φ ′ ( t ) φ ′′ ( t ) dt. Here E/ Q is a quadratic field extension as in the above lemma, embedded in D by E = Q ( i D ). We can choose the set of all characters on E × \ A × E (whose restrictionson A × are trivial) as a basis of L ([ E × ]) and decompose φ | [ E × ] by φ | [ E × ] = X Ω ∈ [ [ E × ] h φ | [ E × ] , Ω i E h Ω , Ω i E Ω = X Ω ∈ [ [ E × ] P Ω − ( φ ) h Ω , Ω i E Ω . Here P Ω : π ′ → C is a period integral defined by P Ω ( φ ) := Z [ E × ] φ ( t )Ω( t ) dt, φ ∈ π ′ , where the Haar measure gives total volume 2 L (1 , η E/ Q ) on [ E × ], with η E/ Q : Q × \ A × → {± } the quadratic character associated to E/ Q by class field theory.Then Z [ E × ] φ ′ ( t ) φ ′′ ( t ) dt = h φ ′ | [ E × ] , φ ′′ | [ E × ] i E = * X Ω ∈ [ [ E × ] P Ω − ( φ ′ ) h Ω , Ω i E Ω , X Ω ∈ [ [ E × ] P Ω − ( φ ′′ ) h Ω , Ω i E Ω + E = X Ω ∈ [ [ E × ] P Ω − ( φ ′ ) h Ω , Ω i E (cid:18) P Ω − ( φ ′′ ) h Ω , Ω i E (cid:19) h Ω , Ω i E . Noticing h Ω , Ω i E = vol([ E × ]), we have(19) Z [ E × ] φ ′ ( t ) φ ′′ ( t ) dt = 1vol([ E × ]) X Ω ∈ [ [ E × ] P Ω − ( φ ′ ) P Ω ( φ ′′ ) . The product of two period integrals is related via a period formula of Waldspurgerto the central value of a base change L -function. Theorem 4.9 ([Wal85, YZZ13] Waldspurger’s formula) . For any φ ′ , φ ′′ ∈ π ′ , wehave P Ω ( φ ′ ) · P Ω − ( φ ′′ ) h φ ′ , φ ′′ i = ζ ∗ F (2) L ( , π E ⊗ Ω)2 L (1 , π, Ad) Y v α v ( φ ′ v , φ ′′ v ; Ω v ) , with α v := L v (1 , η E/F ) L v (1 , π, Ad) ζ F v (2) L v ( , π E ⊗ Ω) Z E × v /F × v B v ( π ′ v ( t v ) φ ′ v , φ ′′ v ) B v ( φ ′ v , φ ′′ v ) Ω v ( t v ) dt v . Here the B v ’s are D × v -invariant bilinear forms on π ′ v ⊗ ˜ π ′ v such that Q v B v ( · , ¯ · ) = h· , ·i as defined in (18) . The integral converges absolutely if both π ′ v and Ω v areunitary. Moreover, for v < ∞ , α v ( φ ′ v , φ ′′ v ; Ω v ) = vol( O × E v / O × F v ; dt v ) when D v ∼ = M (2 , F v ) , E v /F v is unramified, π ′ v ∼ = π v and Ω v are both unramified,and φ ′ v ∈ π v , φ ′′ v ∈ ˜ π v are unit spherical vectors. Recall that I [ γ ] ( f )vol( K ′ ) = c γ Z A × E \ D × ( A ) ϕ γ ( h ) Z A × E × \ A × E ( R ( h ) φ ∗ )( t )( R ( h ) φ ∗∗ )( t ) dt ! dh by Lemma 4.8. Here c γ = when γ = γ , and c γ = 1 when γ = γ . Apply (19)and Theorem 4.9 with φ ′ = R ( h ) φ ∗ , φ ′′ = R ( h ) φ ∗∗ , and we have R [ E × ] ( R ( h ) φ ∗ )( t )( R ( h ) φ ∗∗ )( t ) dt h R ( h ) φ ∗ , R ( h ) φ ∗∗ i = 1vol([ E × ]) ζ ∗ Q (2)2 L (1 , π , Ad) X Ω ∈ [ [ E × ] L ( 12 , ( π ) E ⊗ Ω) Y v α v ( π ′ ,v ( h v ) φ ∗ v , π ′ ,v ( h v ) φ ∗∗ v ; Ω v ) . Now we give a condition for Ω such that Q v α v does not vanish. VERAGES AND NONVANISHING OF TRIPLE PRODUCT CENTRAL L -VALUES 35 In the following three sections we will focus on the finite places. By Lemma 4.11and 4.15, α p = 0 unless Ω p is unramified for any finite p .Lemma 3.5 shows that, if Ω : E × \ A × E → C × is unramified everywhere, it isdetermined by its Archimedean factor, i.e. a character of C × which is invariantunder O × E . Here O × E = {± , ± i } , E = Q ( γ ) = Q ( √− , i = √− O × E = { ζ r } r =0 , E = Q ( γ ) = Q ( √− , ζ = (1 + √− . Then the sum over Ω ∈ [ [ E × ] becomes that over Ω ∞ ∈ \ O × E \ C × / R × . Recall thatevery character of C × / R × is of the form z ( z/ ¯ z ) m . Denote it by sgn m wheresgn( z ) := z/ | z | is the “sign” of z ∈ C × on the complex unit circle (and hencesgn( z ) = z/ ¯ z ). Obviously(20) sgn m is O × E -invariant ⇔ m ∈ Z , | m ;sgn m is O × E -invariant ⇔ m ∈ Z , | m. These are the Ω ∞ that may appear in the sum.Moreover, by Lemma 4.11, for p | disc( D ), α p ( φ ′ p , φ ′′ p ; Ω p ) = 0 unless Ω p = δ p ◦ N D p , where δ p is the character such that π p = σ δ p . This is to say, besides theunramification of Ω p , we also need thatΩ p ( ̟ E p ) = δ p ( N D p ( ̟ E p )) = δ p ( N E p / Q p ( ̟ E p )) . This gives no extra information when E p / Q p is unramified. But when E p / Q p isramified, this leads to more restriction for Ω ∞ , noticing that Ω is trivial on E × : • For E = Q ( √−
1) and p = 2, 1 + i is a uniformizer of E p , and thenΩ ∞ (1 + i )Ω (1 + i ) = 1 ⇒ Ω ∞ (1 + i ) = Ω (1 + i ) − = δ ( N E / Q (1 + i )) = δ (2) . For Ω ∞ = sgn m with 2 | m , this shows thatsgn m (1 + i ) = δ (2) = ( ⇒ m ≡ , − ⇒ m ≡ . • For E = Q ( √−
3) and p = 3, √− E p . And similarly forΩ ∞ = sgn m with 3 | m , this shows thatsgn m ( √−
3) = δ (3) = ( ⇒ m ≡ , − ⇒ m ≡ . At last, Lemma 4.17 shows that α ∞ ∼ Z C × / R × B ∞ ( π ′∞ ( t ∞ ) φ ′∞ , φ ′′∞ ) sgn m ( t ∞ ) dt ∞ = 0unless | m | ≤ k −
1. In conclusion we have the following restrictions on Ω:
Lemma 4.10.
With notations in Theorem 4.1 and Theorem 4.9, Y v α v ( π ′ v ( h v ) φ ∗ v , π ′ v ( h v ) φ ∗∗ v ; Ω v ) = 0 unless Ω v is unramified for all finite v and Ω ∞ ( z ) = sgn m ( z ) = ( z/ ¯ z ) m where m ∈ p Z , − ( k − ≤ m ≤ k − , and ( − ord ( m/p ) = − ε p ( 12 , π ) if p | N. Here p = 2 or is the ramified place of E = E or E . In particular, when theweight is k = 2 or , m can only be , i.e. Ω can only be a trivial character. Now we can write I [ γ ] ( f ) as I [ γ ] ( f )vol( K ′ ) = h φ ∗ , φ ∗∗ i c γ vol([ E × ]) ζ ∗ Q (2)2 L (1 , π , Ad) · X Ω L ( 12 , ( π ) E ⊗ Ω) Y v Z E × v \ D × v α v ( π ′ ,v ( h v ) φ ∗ v , π ′ ,v ( h v ) φ ∗∗ v ; Ω v ) ϕ γ,v ( h v ) dh v , where the sum is over all Ω satisfying the conditions in Lemma 4.10. Denote φ ′ = R ( h ) φ ∗ , φ ′′ = R ( h ) φ ∗∗ . Notice that B ∞ ( φ ′∞ , φ ′′∞ ) = h φ ∗ , φ ∗∗ i . We can write I [ γ ] ( f )vol( K ′ ) = c γ ζ ∗ Q (2)2 L (1 , π , Ad) vol([ E × ]) · X Ω L ( 12 , ( π ) E ⊗ Ω) Z C × \ D ×∞ B ∞ ( φ ′∞ , φ ′′∞ ) α ∞ dh ∞ Y p Z E × p \ D × p ϕ γ,p ( h p ) α p dh p , (21)with B ∞ ( φ ′∞ , φ ′′∞ ) α ∞ = L (1 , sgn) L (1 , π k dis , Ad) ζ R (2) L ( , ( π k dis ) E ⊗ sgn m ) Z C × / R × B ∞ ( π ′ k ( t ) φ ′∞ , φ ′′∞ ) sgn m ( t ) dt = 2 (2 π ) − − k Γ(2 k ) L ∞ ( , ( π ) E ⊗ Ω) Z C × / R × B ∞ ( π ′ k ( t ) φ ′∞ , φ ′′∞ ) sgn m ( t ) dt. (22)We will show in the next sections the calculations of the above local integrals.Lemma 4.18 together with (22) shows Z C × \ D ×∞ B ∞ ( φ ′∞ , φ ′′∞ ) α ∞ dh ∞ = vol( G ′∞ ) k φ k L ∞ ( , ( π ) E ⊗ Ω) 2 Γ(2 k − (2 π ) k +1 Γ( k ) Γ(3 k − I ( m )2 k ( γ )for some constant I ( m )2 k ( γ ) depending only on γ , k and m . For the integrals at finiteplaces, Lemma 4.14, Lemma 4.15 and (24) imply that Z E × p \ D × p ϕ γ,p α p dh p = vol( K p ) · ( , p ∤ disc( D );2(1 − p − ) , p | disc( D )when α p does not vanish. Recall that ζ ∗ Q (2) = π/ , vol([ E × ]) = 2 L (1 , η ) . With the above results, we can rewrite (21) as I [ γ ] ( f )vol( K ′ ) = c γ k φ k vol( K ′ ) π/ L (1 , π , Ad) L (1 , η ) 2 ω ( N ) ϕ ( N ) N Γ(2 k − (2 π ) k +1 Γ( k ) Γ(3 k − · X Ω L fin ( 12 , ( π ) E ⊗ Ω) I ( m )2 k ( γ ) . Here we havevol( K ′ ) = 24 ϕ ( N ) , c γ = ( , η corresponds to ( χ − , γ = γ ; χ − , γ = γ . VERAGES AND NONVANISHING OF TRIPLE PRODUCT CENTRAL L -VALUES 37 By the Dirichlet class number formula we have that L (1 , χ − ) = 14 , L (1 , χ − ) = 13 √ . Therefore I [ γ ] ( f )vol( K ′ ) = 2 c γ L (1 , η ) k φ k L (1 , π , Ad) 2 ω ( N ) N Γ(2 k − (2 π ) k Γ( k ) Γ(3 k − · X Ω L fin ( 12 , ( π ) E ⊗ Ω) I ( m )2 k ( γ );in particular we have I [ γ ] ( f )vol( K ′ ) = 4 k φ k L (1 , π , Ad) 2 ω ( N ) N Γ(2 k − (2 π ) k Γ( k ) Γ(3 k − · X Ω L fin ( 12 , ( π ) E ⊗ Ω) I ( m )2 k ( γ ) ,I [ γ ] ( f )vol( K ′ ) = 6 √ k φ k L (1 , π , Ad) 2 ω ( N ) N Γ(2 k − (2 π ) k Γ( k ) Γ(3 k − · X Ω L fin ( 12 , ( π ) E ⊗ Ω) I ( m )2 k ( γ ) . (23)With Theorem 4.1 and (23), we can get Theorem 4.3, the orbital decompositionof I ( f ).4.3. Local calculation on ramified quaternion algebras.
In the following twosections we will explicitly calculate the local integrals in (21) at non-Archimedeanplaces.When p | disc( D ), with Proposition 4.7, we can assume χ − d ( p ) = 1, i.e. E p / Q p is non-split. Recall that E = Q ( √− d ) (where − d is a fundamental discriminant)corresponds to a Dirichlet character χ − d . When E p / Q p is unramified and non-split, χ − d ( p ) = −
1, the ramification index is e = 1, and the inertia degree is f = 2; when E p / Q p is ramified, χ − d ( p ) = 0, e = 2, f = 1.In this case, π p = σ δ p is the special representation of GL(2 , Q p ) and π ′ p = δ p ◦ N D p is a character of G ′ p , where δ p is an unramified character of Q × p of order at most 2.We can take φ ′ p = φ ′′ p = π ′ p ( h p ) φ p , φ p := δ p ◦ N D p with B p ( φ p , φ p ) defined to be 1. Lemma 4.11.
When p | disc( D ) , π p = σ δ p , we have α p ( φ ′ p , φ ′′ p ; Ω p ) = 0 unless Ω p = δ p ◦ N D p , in which case α p ( φ ′ p , φ ′′ p ; δ p ◦ N D p ) = (1 − p − ) vol( E × p / Q × p ) . In particular, α p ( φ ′ p , φ ′′ p ; ) = (1 − p − ) vol( E × p / Q × p ) · ( , χ − d ( p ) = − (1 − ε ( , ( π ) p )) , χ − d ( p ) = 0 . Proof.
We have that Z E × p / Q × p B p ( π ′ p ( t p ) π ′ p ( h p ) φ p , π ′ p ( h p ) φ p ) B p ( π ′ p ( h p ) φ p , π ′ p ( h p ) φ p ) Ω p ( t p ) dt p = Z E × p / Q × p B p ( δ p ( N D p ( t p )) π ′ p ( h p ) φ p , π ′ p ( h p ) φ p ) B p ( π ′ p ( h p ) φ p , π ′ p ( h p ) φ p ) Ω p ( t p ) dt p = Z E × p / Q × p δ p ( N D p ( t p ))Ω p ( t p ) dt p = ( vol( E × p / Q × p ) , if Ω p = δ p ◦ N D p ;0 , otherwise. So α p ( φ ′ p , φ ′′ p ; Ω p ) = 0 unless Ω p = δ p ◦ N D p , in which case α p ( φ ′ p , φ ′′ p ; δ p ◦ N D p ) := L p (1 , η E/ Q ) L p (1 , π, Ad) ζ Q p (2) L p ( , π E ⊗ Ω) Z E × p / Q × p B p ( π ′ p ( t p ) φ ′ p , φ ′′ p ) B p ( φ ′ p , φ ′′ p ) δ p ( N D p ( t p ))) dt p = (1 − ( − p − ) − (1 − p − ) − L p (1 ,π, Ad) L p ( ,π E ⊗ Ω) vol( E × p / Q × p ) , if E p / Q p is unramified and non-split − p − ) − L p (1 ,π, Ad) L p ( ,π E ⊗ Ω) vol( E × p / Q × p ) , if E p / Q p is ramified= vol( E × p / Q × p )(1 − p − e ) L p (1 , π, Ad) L p ( , π E ⊗ Ω)= vol( E × p / Q × p )(1 − p − e ) 1 − p − f − p − = vol( E × p / Q × p )(1 − p − ) . Recall that for π p = σ δ p , one has ( π E ) p = σ δ p ◦ N Ep/ Q p and L p ( s, π, ξ ) = (1 − δ p ( ̟ p ) ξ p ( ̟ p ) | ̟ p | s +1 / ) − ,L p ( 12 , π E ⊗ Ω) = (1 − δ p ( N E p / Q p ( ̟ E p ))Ω p ( ̟ E p ) | ̟ E p | ) − = (1 − p − f ) − ; L p ( s, π, Ad) = ζ p ( s + 1) , L p (1 , π, Ad) = (1 − p − ) − . For the rest of the lemma, notice that δ p is unramified and of order at most2. So when E p / Q p is unramified and non-split, N D p ( ̟ E p ) = N E p / Q p ( p ) = p and therefore δ p ◦ N D p = E × p always holds. But if E p / Q p is ramified we know N D p ( ̟ E p ) = N E p / Q p ( ̟ E p ) = p ; so δ p ◦ N D p ( ̟ E p ) = δ p ( p ) , and then α p ( φ ′ p , φ ′′ p ; ) = 0 unless δ p ( p ) = 1. Notice that we have ε ( , π p ) = − δ p ( p )in this case. (cid:3) Next we study R E × p \ D × p ϕ γ,p ( h p ) dh p . Lemma 4.12.
For p | disc( D ) , Z E × p \ D × p ϕ γ,p ( h p ) dh p = vol( E × p \ D × p ) . Proof.
When p | disc( D ), D p is a division algebra. Proposition 2.5 shows O × p = { x ∈ D × p : v p ( N D p ( x )) = 0 } . Clearly for any h p ∈ D p , N D p ( h − p γh p ) = N D p ( γ ) = 1for γ = γ or γ . So the condition h − p γh p ∈ O × p is trivial, i.e. ϕ γ,p ( h p ) ≡ (cid:3) The above two lemmas show that, when α p = 0, Z E × p \ D × p ϕ γ,p ( h p ) α p dh p = (1 − p − ) vol( Q × p \ E × p ) vol( E × p \ D × p ) = (1 − p − ) vol( G ′ p ) . Notice that N D p ( Q × p ) = ( Q × p ) while N D p ( D × p ) = Q × p by Lemma 2.1 . Since in this case O × p = { g p ∈ D × p : v p ( N D p ( g p )) = 0 } , we can write G ′ p = Q × p \ D × p = Q × p \ (cid:16) { g p : 2 | N D p ( g p ) } ⊔ { g p : 2 ∤ N D p ( g p ) } (cid:17) = Q × p \ Q × p O × p ⊔ Q × p \ j Q × p O × p = K p ⊔ jK p . VERAGES AND NONVANISHING OF TRIPLE PRODUCT CENTRAL L -VALUES 39 (Here v p ( N D p ( j )) = v p ( N ) = 1 for j ∈ D p = ( − q, − N Q p ). Recall that D p = ( − q, − N Q p )for q = 1 or 3, by Lemma 2.4.) And thereforevol( G ′ p ) = vol( Q × p \ D × p ) = 2 vol( K p ) . In conclusion, for p | N and χ − d ( p ) = 1,(24) α p Z E × p \ D × p ϕ γ,p ( h p ) dh p = ( − p − ) vol( K p ) , Ω p = δ p ◦ N D p ;0 , otherwise . In particular if Ω p = then(25) α p Z E × p \ D × p ϕ γ,p ( h p ) dh p = (1 − χ − d ( p ))(1 − p − ) vol( K p ) · ( , χ − d ( p ) = − − ε ( , ( π ) p )) , χ − d ( p ) = 0 . Local calculation on split quaternion algebras.
When p ∤ disc( D ) we fixan isomorphism D p ∼ = M (2 , Q p ) and take the maximal order O p to be the preimageof M (2 , Z p ). Lemma 4.13.
Under the natural embedding E p = Q p ( γ ) ֒ → D p ∼ = M (2 , Q p ) , we have O E p = E p ∩ O p . The isomorphism D p ∼ = M (2 , Q p ) and the proof of the above lemma can be foundlater case by case in this section.In this case, π ′ p ∼ = π p and they are spherical. We can write π p = π ( µ p , µ − p ) = Ind G p B p ( µ p ⊗ µ − p ) , where µ p is an unramified quasicharacter of Q × p such that µ p = | · | . A sphericalvector in the principal series representation can be given as a constant multiple of φ p ( (cid:18) a bd (cid:19) k p ) = µ p (cid:16) ad (cid:17) (cid:12)(cid:12)(cid:12) ad (cid:12)(cid:12)(cid:12) / for any k p ∈ GL(2 , Z p ) . We can fix φ ′ p = φ ′′ p = π p ( h p ) φ p .Notice that here h p satisfies h − p γh p ∈ K p for all p . (Our choice of γ has N D ( γ ) =1, so h − p γh p ∈ GL(2 , Z p ).) For φ ′ p = φ ′′ p = π p ( h p ) φ p we care about • R E × p / Q × p B p ( π p ( t p ) φ ′ p , φ ′′ p )Ω p ( t p ) dt p for h − p γh p ∈ K p ; and • R E × p \ D × p K p ( h − p γh p ) dh p .For the second integral we have the following lemma. Lemma 4.14.
Assume that p ∤ disc( D ) , γ = γ = √− or = γ = √− .(1) For h ∈ D × p ,(i) h ∈ E × p GL(2 , Z p ) if and only if (ii) h − γh ∈ GL(2 , Z p ) . (2) Z E × p \ D × p K p ( h − p γh p ) dh p = vol( O × p )vol( O × E p ) . Proof.
The second statement can be shown with Lemma 2.6: Z E × p \ D × p K p ( h − p γh p ) dh p = Z E × p \ E × p O × p dh p = vol(( O × p ∩ E × p ) \O × p ) = vol( O × p )vol( O × E p ) . For the first statement, (i) ⇒ (ii) is obvious since γ ∈ GL(2 , Z p ) and γ commuteswith E × p , and (ii) ⇒ (i) will be proved case by case later in this section. (cid:3) For the first integral we have
Lemma 4.15.
For h p ∈ E × p GL(2 , Z p ) , φ ′ p = φ ′′ p = π p ( h p ) φ p , φ p a unit sphericalvector in π p , we have α ( φ ′ p , φ ′′ p ; Ω p ) = vol( O × Ep )vol( Z × p ) , Ω p unramified; , Ω p ramified.Proof. Write h = t h u h with t h ∈ E × p , u h ∈ O × p . Then h − th = u − h t − h tt h u h = u − h tu h . Recall that the GL(2 , Q p )-invariant bilinear form on π p ⊗ ˜ π p can be definedby B p ( φ, φ ′ ) := Z GL(2 , Z p ) φ ( k ) φ ′ ( k ) dk, φ ∈ π p , φ ′ ∈ ˜ π p . Then B p ( π p ( t p ) φ ′ p , φ ′′ p ) = B p ( π p ( h − th ) φ p , φ p ) = Z O × p π p ( h − th ) φ p ( k ) φ p ( k ) dk = Z O × p φ p ( kh − th ) φ p ( k ) dk = Z O × p φ p ( ku − h tu h ) φ p ( k ) dk = Z O × p φ p ( ku − h t ) φ p ( ku − h ) d ( ku − h ) = Z O × p φ p ( kt ) φ p ( k ) dk = B p ( π p ( t ) φ p , φ p ) . So Z E × p / Q × p B p ( π p ( t p ) φ ′ p , φ ′′ p )Ω p ( t p ) dt p = Z E × p / Q × p B p ( π p ( t p ) φ p , φ p )Ω p ( t p ) dt p and therefore, by Theorem 4.9, α ( φ ′ p , φ ′′ p ; Ω p ) = α ( φ p , φ p ; Ω p ) = vol( O × E p ) / vol( Z × p )when Ω p is unramified and E p / Q p is unramified.The rest of this lemma (when Ω p is ramified or E p / Q p is ramified) is provedlater in this section. (cid:3) When E p / Q p is split. For D ( Q p ) = (cid:16) − , − N Q p (cid:17) or (cid:16) − , − N Q p (cid:17) we fix the isomor-phism D ( Q p ) ∼ = M (2 , Q p ) with1 (cid:18) (cid:19) , i (cid:18) α N ββ − α (cid:19) , j (cid:18) − N (cid:19) , k (cid:18) N β − N α − α − N β (cid:19) where ( α, β ) is a solution of α + N β = − q in Q p , q = 1 or 3. When p splits in E , − q is a square in Q p and we can take i (cid:18) √− q −√− q (cid:19) VERAGES AND NONVANISHING OF TRIPLE PRODUCT CENTRAL L -VALUES 41 where √− q is a fixed solution of x = − q in Q p . Then E p = Q p ⊕ Q p is embeddedin M (2 , Q p ) as the set of all the diagonal matrices. With this embedding Lemma4.13 is obvious. Proof of Lemma 4.14 when E p / Q p is split. The Iwasawa decomposition gives that h = c h (cid:18) p r (cid:19) (cid:18) x h (cid:19) k h , c h ∈ Q × p , r ∈ Z , x h ∈ Q p , k h ∈ GL(2 , Z p ) . Then its inverse is h − = ( c h p r ) − k − h (cid:18) − x h p r p r (cid:19) . Let ¯ t = x − yi D for t = x + yi D ∈ E p = Q p ( i D ). Then, for γ = i D = √− i D = √− ), γ can be embedded in GL(2 , Q p ) as (cid:18) γ ¯ γ (cid:19) , and h − γh = ( c h p r ) − k − h (cid:18) − x h p r p r (cid:19) (cid:18) γ ¯ γ (cid:19) c h (cid:18) p r x h p r (cid:19) k h = k − h (cid:18) γ ( γ − ¯ γ ) x h ¯ γ (cid:19) k h . Therefore h − γh ∈ GL(2 , Z p ) if and only if ( γ − ¯ γ ) x h ∈ Z p . This implies v p ( x h ) ≥ p = 2 (or p = 3 respectively). Then h ∈ t h (cid:18) p r (cid:19) GL(2 , Z p ). This provesthe statement when E p / Q p is split. (cid:3) Proof of Lemma 4.15 when E p / Q p is split. Now E × p = Q × p × Q × p . Notice that E × p / Q × p ∼ = { ( x,
1) : x ∈ Q × p } , We can write Z E × p / Q × p B p ( π p ( t p ) φ p , φ p )Ω p ( t p ) dt p = Z Q × p B p ( π p (cid:18) x (cid:19) φ p , φ p )Ω p ( x, d × x. For Ω p ( t , t ) = ν p ( t ) ν − p ( t ) we have Z E × p / Q × p B p ( π p ( t p ) φ p , φ p )Ω p ( t p ) dt p = X r ∈ Z Z v p ( x )= r B p ( π p (cid:18) x (cid:19) φ p , φ p )Ω p ( x, d × x = X r ∈ Z Z u ∈ Z × p B p ( π p (cid:18) p r (cid:19) (cid:18) u (cid:19) φ p , φ p ) ν p ( p r u ) du = X r ∈ Z B p ( π p (cid:18) p r (cid:19) φ p , φ p ) ν p ( p ) r Z Z × p ν p ( u ) du = 0 when ν p is ramified . (cid:3) When E p / Q p is not split. As before, we fix the isomorphism D ( Q p ) ∼ = M (2 , Q p ) with i (cid:18) α N ββ − α (cid:19) , j (cid:18) − N (cid:19) where ( α, β ) is a solution of α + N β = − q in Q p , q = 1 or 3. In particular we canchoose α, β ∈ Z p : When p = 2, we can find a solution satisfying α, β ∈ Z × p ∪ { } . With Hensel’sLemma we only need to prove α + N β ≡ − q (mod p ) has a solution in Z /p Z ,that means, there is at least one quadratic residue (including 0) mod p in {− q, − q − N, − q − N, . . . , − q − (( p − / N } , which is a set of p +12 congruency classes mod p (here p, q, N are relatively prime toeach other). If not, the p +12 quadratic residues can only be found in the complementof the above set, which has p − numbers, a contradiction.When p = q = 3 ∤ N , N is the product of an odd number of distinct primes ofthe form 3 n + 2. Then − − N ≡ − − N is a square in Q × . We havea solution α = − − N, β = 1 . When p = 2 ∤ N , actually α + N β = − q has no solution in Z × ∪ { } , But onecan get a solution such that one of α and β is in Z × . For example, when q = 1,according to Corollary 4.1, N is the product of an odd number of distinct primesof the form 4 n + 3 and we have the following solution: α = 2 , β = 1 − N − ( N −
38 + 1) for N ≡ ,α = 4 , β = 1 − N − ( N −
78 + 3) for N ≡ q = 3, N is the product of an odd number of distinct primes of the form6 n + 5: α = 1 − N + 12 , β = 2 . Recall that a unit u ∈ Z × is a square if and only if u ≡ Proof of Lemma 4.13.
When E p / Q p is non-split, the isomorphism D ( Q p ) ∼ = M (2 , Q p )maps λ + µ √− q to (cid:18) λ + αµ N βµβµ λ − αµ (cid:19) . Then O E p ⊇ E p ∩ O p is obvious since the determinant of a matrix in M (2 , Z p ) isalways in Z p and det (cid:18) λ + αµ N βµβµ λ − αµ (cid:19) = λ + qµ is exactly the norm over E p / Q p of λ + µ √− q .Now assume λ + µ √− q ∈ O × E p . Then with our choice of α, β and the next lemma,both λ and µ are in Z p for most cases, and therefore (cid:18) λ + αµ N βµβµ λ − αµ (cid:19) ∈ GL(2 , Z p ). This also holds in the exceptional case ( q = 3, p = 2) since12 + 12 √− (cid:18) α Nβ β − α (cid:19) ∈ GL(2 , Z ) (recall that α ∈ Z × , β = 2) . (cid:3) Lemma 4.16.
For γ = √− or (1 + √− , O Q p ( γ ) = Z p [ γ ] . Equivalently onecan say, when E p = Q p ( √− q ) ( q = 1 or ) is non-split, λ + µ √− q ∈ O × E p if andonly if • λ, µ ∈ Z p and λ + qµ ∈ Z × p ; or • λ, µ ∈ + Z for q = 3 , p = 2 . VERAGES AND NONVANISHING OF TRIPLE PRODUCT CENTRAL L -VALUES 43 Proof.
With the normalized valuation on E p , λ + µ √− q ∈ O E p if and only if λ + qµ ∈ Z p . When p = 2, both λ and µ are in Z p because − q is not a quadraticresidue mod p . When p = 2, assume that λ and µ are not together in Z p . Then λ + qµ ∈ Z implies v ( λ ) = v ( µ ) = − r < r . Say λ = 2 − r u , µ = 2 − r v for some u, v ∈ Z × . Then λ + qµ = 2 − r ( u + qv ) ∈ Z . Recall that u , v ≡ u + qv ≡ q (mod 8), i.e. v ( λ + qµ ) = − r + v ( u + qv ) = ( − r + 1 , q = 1; − r + 2 , q = 3 . The only possibility such that v ( λ + qµ ) ≥ q = 3 and r = 1,which implies λ, µ ∈ + Z . (cid:3) Now we prove Lemma 4.14 and 4.15.
Proof of Lemma 4.14 when E p / Q p is non-split (i.e. when χ − d ( p ) = 1 ). We claim that, h − γh ∈ O × p if and only if h ∈ Z p O × p ⊔ ∅ , p inert in E = Q ( γ ); ! Z p O × p , γ = √− , p = 2; αβ − ! Z p O × p , γ = √− , p = 3 . Notice that E × p = ( ⊔ r ∈ Z p r O × E p = Q × p O × E p , p inert , Q × p O × E p ⊔ ̟ E p Q × p O × E p , p ramified , and O × E p ⊂ O × p by Lemma 4.13. If the above statement holds, we only need toshow ̟ E ∈ (cid:18) (cid:19) GL(2 , Z ) , ̟ E ∈ (cid:18) αβ − (cid:19) GL(2 , Z )just for the ramified case, which can be easily checked by the following calculation,taking ̟ E = 1 + √− ̟ E = √− (cid:18) (cid:19) − ̟ E = (cid:18) (cid:19) − (cid:18) α Nββ − α (cid:19) = (cid:18) − β + α Nβ − α β − α (cid:19) ∈ GL(2 , Z ); (cid:18) αβ − (cid:19) − ̟ E = (cid:18) αβ − (cid:19) − (cid:18) α Nββ − α (cid:19) = (cid:18) − β − β − α (cid:19) ∈ GL(2 , Z ) . Now we prove the claim. The sufficiency is easy to check. To show the necessity,assume that h − γh ∈ GL(2 , Z p ) with h = (cid:18) y x (cid:19) k h for some k h ∈ Z p GL(2 , Z p )using Iwasawa decomposition.When γ = √− h − γh ∈ GL(2 , Z p ) implies (cid:18) y x (cid:19) − (cid:18) α N ββ − α (cid:19) (cid:18) y x (cid:19) = (cid:18) X − ( X + 1) Y − Y − X (cid:19) ∈ GL(2 , Z p ) , where X = α − βx , Y = βy . Our assumption on α and β says α ∈ Z p , β ∈ Z × p , inwhich case X = α − βx ∈ Z p ⇔ x ∈ Z p ; Y = βy ∈ Z p ⇔ y ∈ Z p . We will only consider the case when h / ∈ GL(2 , Z p ), i.e. when v p ( X + 1) ≥ v p ( Y ) > . This cannot happen when χ − ( p ) = − − p and v p ( X + 1) = 0. But if χ − ( p ) = 0 i.e. p = 2, we have v ( X + 1) > ⇔ X ∈ Z × ⇔ X ∈ Z ⇔ v ( X + 1) = 1and then v ( Y ) = 1. So (recall that in this case v ( α ) ≥ x ∈ Z × and y ∈ Z × ,which is equivalent to say that (cid:18) y x (cid:19) ∈ (cid:18) (cid:19) GL(2 , Z ) . When γ = √− , h − γh ∈ GL(2 , Z p ) implies (cid:18) y x (cid:19) − (cid:18) α N ββ − α (cid:19) (cid:18) y x (cid:19) = 12 (cid:18) X − ( X + 3) Y − Y − X (cid:19) ∈ GL(2 , Z p )where X = α − βx , Y = βy . When p = 2, we can do similar deduction as above,considering only the case when h / ∈ GL(2 , Z p ), and get v p ( X + 3) ≥ v p ( Y ) > p = 3. But we know that v ( X + 3) > ⇔ X ∈ Z ⇔ v ( X + 3) = 1and then v ( Y ) = 1. Recall that we choose α, β ∈ Z × . Therefore we have x ∈ αβ − + 3 Z , y ∈ Z × , which is equivalent to say that (cid:18) y x (cid:19) ∈ (cid:18) αβ − (cid:19) GL(2 , Z ) . When γ = √− and p = 2, we can choose α ∈ Z × and β = 2. Then thecondition h − γh ∈ GL(2 , Z ) becomes (cid:18) y x (cid:19) − (cid:18) α N − α (cid:19) (cid:18) y x (cid:19) = α − x − ( α − x ) +34 y y − α + x ! ∈ GL(2 , Z p ) . We still can get x, y ∈ Z , but ( α − x ) +34 y ∈ Z implies y ∈ Z × , noticing that( α − x ) ∈ Z for α ∈ Z × . So in this case we can only have (cid:18) y x (cid:19) ∈ GL(2 , Z ). (cid:3) Proof of Lemma 4.15 when E p / Q p is unramified and non-split. Assume that Ω p has conductor c . In this case E × p = ⊔ r ∈ Z p r O × E p = Q × p O × E p .Lemma 2.6 implies, for any integrable function f defined on E × p / Q × p , Z E × p / Q × p f ( t ) dt = Z O × Ep / Z × p f ( t ) dt. VERAGES AND NONVANISHING OF TRIPLE PRODUCT CENTRAL L -VALUES 45 Recall that O E p = E p ∩ O p and φ p is spherical. So we have Z E × p / Q × p B p ( π p ( t p ) φ p , φ p )Ω p ( t p ) dt p = Z O × Ep / Z × p B p ( π p ( t p ) φ p , φ p )Ω p ( t p ) dt p = B p ( φ p , φ p ) Z O × Ep / Z × p Ω p ( t p ) dt p = ( B p ( φ p , φ p ) vol( O × E p / Z × p ) , c = 0;0 , c > . (cid:3) Proof of Lemma 4.15 when E p / Q p is ramified. Now E × p = Q × p O × E p ⊔ ̟ E p Q × p O × E p . Then for any integrable function f on E × p / Q × p , Z E × p / Q × p f ( t ) dt = Z Q × p O × Ep / Q × p f ( t ) dt + Z Q × p O × Ep / Q × p f ( ̟ E p t ) dt. By Lemma 2.6 Z Q × p O × Ep / Q × p f ( t ) dt = Z O × Ep / ( Q × p ∩O × Ep ) f ( t ) dt = Z O × Ep / Z × p f ( t ) dt, and hence we can write Z E × p / Q × p f ( t ) dt = Z O × Ep / Z × p ( f ( t ) + f ( ̟ E p t )) dt. Recall that O × E p ⊂ O × p by Lemma 4.13. Now we have Z E × p / Q × p B p ( π p ( t p ) φ p , φ p )Ω p ( t p ) dt p = Z O × Ep / Z × p B p ( π p ( t p ) φ p , φ p )Ω p ( t p ) dt p + Z O × Ep / Z × p B p ( π p ( ̟ E p t p ) φ p , φ p )Ω p ( ̟ E p t p ) dt p = B p ( φ p , φ p ) Z O × Ep / Z × p Ω p ( t p ) dt p + B p ( π p ( ̟ E p ) φ p , φ p )Ω p ( ̟ E p ) Z O × Ep / Z × p Ω p ( t p ) dt p = ( vol( O × E p / Z × p ) (cid:16) B p ( φ p , φ p ) + B p ( π p ( ̟ E p ) φ p , φ p )Ω p ( ̟ E p ) (cid:17) , Ω p unramified;0 , Ω p ramified.By the proof of Lemma 4.14, in this case, ̟ E p ∈ (cid:18) p ∗ (cid:19) O × p . The Macdonaldformula ([Bum97] Theorem 4.6.6) implies that B p ( π p (cid:18) p (cid:19) φ p , φ p ) B p ( φ p , φ p ) = 11 + p − p − / (cid:18) µ p ( p ) 1 − p − µ p ( p ) − − µ p ( p ) − + µ p ( p ) − − p − µ p ( p ) − µ p ( p ) (cid:19) = p − / p − ( µ p ( p ) + µ p ( p ) − ) . For unramified Ω p , α p ( φ ′ p , φ ′′ p ; Ω p ) := L p (1 , η E/ Q ) L p (1 , π, Ad) ζ Q p (2) L p ( , π E ⊗ Ω) Z E × p / Q × p B p ( π ′ p ( t p ) φ ′ p , φ ′′ p ) B p ( φ ′ p , φ ′′ p ) Ω p ( t p ) dt p = L p (1 , η E/ Q ) L p (1 , π, Ad) ζ Q p (2) L p ( , π E ⊗ Ω) vol( O × E p / Z × p ) (cid:18) p − / p − ( µ p ( p ) + µ p ( p ) − )Ω p ( ̟ E p ) (cid:19) . Here L p (1 , η E/ Q ) = 1, ζ Q p (2) = (1 − p − ) − , L p (1 , π ( µ , µ ) , Ad) = L p (1 , )(1 − µ ( p ) µ − ( p ) p − )(1 − µ − ( p ) µ ( p ) p − )= (1 − p − ) − (1 − µ ( p ) p − )(1 − µ − ( p ) p − ) . For π p = π ( µ p , µ − p ), one has e = 2, f = 1, ( π E ) p = π ( µ p ◦ N E p /F p , µ − p ◦ N E p /F p )and L p ( 12 , π E ⊗ Ω) =(1 − µ p ( N E p /F p ( ̟ ))Ω p ( ̟ ) q − / ) − (1 − µ − p ( N E p /F p ( ̟ ))Ω p ( ̟ ) q − / ) − =(1 − µ p ( p )Ω p ( ̟ ) p − / ) − (1 − µ − p ( p )Ω p ( ̟ ) p − / ) − (here ̟ = ̟ E p ) . Then α p ( φ ′ p , φ ′′ p ; Ω p )vol( O × E p / Z × p ) = (1 + p − ) (1 − µ ( p ) p − ) − (1 − µ − ( p ) p − ) − (1 − µ p ( p )Ω p ( ̟ ) p − / ) − (1 − µ − p ( p )Ω p ( ̟ ) p − / ) − · (cid:18) p − / p − ( µ p ( p ) + µ p ( p ) − )Ω p ( ̟ ) (cid:19) . When Ω is unramified everywhere, Ω p ( ̟ ) = 1 and α p ( φ ′ p , φ ′′ p ; Ω p )vol( O × E p / Z × p )= 1 + p − (1 + µ p ( p )Ω p ( ̟ ) p − / )(1 + µ − p ( p )Ω p ( ̟ ) p − / ) (cid:18) p − / p − ( µ p ( p ) + µ p ( p ) − )Ω p ( ̟ ) (cid:19) = 1 + p − + p − / ( µ p ( p ) + µ p ( p ) − )Ω p ( ̟ )(1 + µ p ( p )Ω p ( ̟ ) p − / )(1 + µ − p ( p )Ω p ( ̟ ) p − / ) = 1 . (cid:3) Compatibility of two nontrivial orbits.
When N has no prime factor ≡ ≡ N are either 2, 3, or ≡
11 (mod 12)), γ and γ both appear in the quaternion algebra D . We applieddifferent presentations of D in this case, which, when p ∤ N , lead to local maximalorders that differ by conjugation by an element in GL(2 , Q p ). But the sphericalvector φ ,p in the definition of the distribution I ( f ) should be determined by a fixedmaximal order of D , independent of the different ways to represent D .Say D ( F ) = (cid:16) − , − NF (cid:17) and D ′ ( F ) = (cid:16) − , − NF (cid:17) . For the case at hand, by Lemma2.4, both D ( Q ) and D ′ ( Q ) have discriminant N . Quaternion algebras over Q withthe same discriminant are isomorphic to each other; more explicitly the isomorphism D ′ ∼ −→ D can be given by i ′ xi + yk , j ′ j with x + N y = 3 for some x, y ∈ Q . VERAGES AND NONVANISHING OF TRIPLE PRODUCT CENTRAL L -VALUES 47 We consider the choice of φ in the definition of the distribution I ( f ) (see (11)): φ ∈ C X k − ⊗ ( π ′ , fin ) K fin is a new-line vector in π ′ as defined in Lemma 2.9. It depends on the definition of X k − (i.e. the way we construct π ′ , ∞ ) and of K fin = Q p< ∞ K p (i.e. the choice ofmaximal orders O p at every finite place). For p | disc( D ) there is a unique maximalorder O p of D p and φ ,p is a constant multiple of δ p ◦ N D p for both cases.For each p ∤ disc( D ), in the previous section we have fixed an isomorphism D p = (cid:16) − , − N Q p (cid:17) ∼ −→ M (2 , Q p ), under which the preimage of M (2 , Z p ) gives a maximal order O p . We choose φ ,p to be the normalized O × p -invariant vector in the sphericalrepresentation π ,p . Lemma 4.14 and 4.15 show that Z E × p \ D × p α p ϕ γ,p ( h p ) dh p = vol( K p ) , p ∤ N holds for γ = γ . We will show it still holds for γ = γ , i.e. for D ′ p .Analogously we fix an isomorphism D ′ p = (cid:16) − , − N Q p (cid:17) ∼ −→ M (2 , Q p ). Let O ′ p bethe preimage of M (2 , Z p ), which is the a maximal order we fixed in the previoussection. The endomorphism M (2 , Q p ) ∼ −→ (cid:18) − , − N Q p (cid:19) = D p ∼ = D ′ p = (cid:18) − , − N Q p (cid:19) ∼ −→ M (2 , Q p )is a conjugation A T AT − for some T ∈ GL(2 , Q p ). This endomorphism givesanother maximal order: the image of O p under the isomorphism D p ∼ = D ′ p . Itsimage under the isomorphism D ′ p ∼ −→ M (2 , Q p ) becomes T · M (2 , Z p ) · T − and we stilldenote it by O p . Then the condition for φ ,p becomes that it is T · GL(2 , Z p ) · T − -invariant. It’s easy to check that π p ( T − ) φ p is GL(2 , Z p )-invariant i.e. ( O ′ p ) × -invariant.With the new notations, Lemma 4.14 shows that h ∈ E × p ( O ′ p ) × if and only if h − γh ∈ ( O ′ p ) × and therefore Z E × p \ D × p K ′ p ( h − p γh p ) dh p = vol(( O ′ p ) × )vol( O × E p ) ;Lemma 4.15 shows that for h ∈ E × p ( O ′ p ) × , α ( π p ( h ) π p ( T − ) φ p , π p ( h ) π p ( T − ) φ p ; Ω p ) = vol( O × Ep )vol( Z × p ) , Ω p unramified;0 , Ω p ramified.We work on another maximal order O p = T O ′ p T − . Now we have h − γh ∈ O × p ⇔ T − h − γhT ∈ O ′ p ⇔ hT ∈ E × p ( O ′ p ) × ⇔ h ∈ E × p ( O ′ p ) × T − ;and then Z E × p \ D × p K p ( h − p γh p ) dh p = Z E × p \ D × p E × p ( O ′ p ) × T − ( h p ) dh p = vol(( O ′ p ) × T − )vol( O × E p ) = vol(( O ′ p ) × )vol( O × E p ) ; under the condition that h − γh ∈ O × p , by Lemma 4.15, α ( π p ( hT )( π p ( T − ) φ p ) , π p ( hT )( π p ( T − ) φ p ); Ω p ) = vol( O × Ep )vol( Z × p ) , Ω p unramified,0 , Ω p ramified,because that hT ∈ E × p ( O ′ p ) × . Thus we have Z E × p \ D × p α p ϕ γ,p ( h p ) dh p = vol( K p ) , p ∤ N also holds for γ = γ .4.6. Local calculation: Archimedean.
First we give a condition when α ∞ van-ishes. Recall that, when v = ∞ , π ′ v ∼ = π ′ k = Sym k − V ⊗ det − k +1 can be realizedon the space of homogeneous polynomials in X, Y of degree 2 k −
2, i.e. V π ′ k = k − M n =0 C X k − − n Y n with π ′ k ( g ) P ( X, Y ) = P (( X, Y ) g ) det( g ) − k for g ∈ (cid:26)(cid:18) α − β ¯ β ¯ α (cid:19) ∈ GL(2 , C ) (cid:27) ∼ = D × ( R ) . (See Section 2.4.) This representation is determined by the last isomorphism. Tomake the action on E × and E × consistent (so that the two nontrivial orbits arecompatible as we mentioned in the previous section), we fix the following isomor-phism for q = 1 and 3:(26) D ( R ) = (cid:18) − q, − N R (cid:19) ∼ −→ (cid:26)(cid:18) α − β ¯ β ¯ α (cid:19) ∈ M (2 , C ) (cid:27) ,i D (cid:18) √− q −√− q (cid:19) , j D (cid:18) −√ N √ N (cid:19) . Then the image of E ∞ = R ( γ ) are the same if (cid:16) − , − N Q (cid:17) and (cid:16) − , − N Q (cid:17) express thesame quaternion algebra; and for t ∈ E ×∞ = C × , we have t (cid:18) t ¯ t (cid:19) and π ′ k ( t ) X k − − n Y n = (cid:0) t/t (cid:1) k − − n X k − − n Y n . In general(27) π ′ k ( t ) k − X n =0 c n X k − − n Y n ! = k − X n =0 sgn k − − n ) ( t ) c n X k − − n Y n , where sgn m ( t ) = (cid:0) t/t (cid:1) m . We fix the bilinear form B ∞ such that B ∞ ( · , · ) = h· , ·i k is the inner product on π ′ k defined as (5). Lemma 4.17.
For any φ ′ v , φ ′′ v ∈ V π ′ k , Z C × / R × B ∞ ( π ′ k ( t ) φ ′ v , φ ′′ v ) sgn m ( t ) dt = 0 unless m ∈ Z , − ( k − ≤ m ≤ k − . VERAGES AND NONVANISHING OF TRIPLE PRODUCT CENTRAL L -VALUES 49 Proof.
Write φ ′ v = k − X r =0 c ′ r X k − − r Y r , φ ′′ v = k − X r =0 c ′′ r X k − − r Y r . Recall that { X k − − r Y r } forms an orthogonal basis of V π ′ k which are eigenvectorsunder the action of C × . Then Z C × / R × B ∞ ( π ′ k ( t ) φ ′ v , φ ′′ v ) sgn m ( t ) dt = Z C × / R × B ∞ k − X r =0 sgn k − − r ) ( t ) c ′ r X k − − r Y r , φ ′′ v ! sgn m ( t ) dt = k − X r =0 c ′ r c ′′ r B ∞ ( X k − − r Y r , X k − − r Y r ) Z C × / R × sgn k − − r ) ( t ) sgn m ( t ) dt = ( c ′ k − m c ′′ k − m k X k − − m Y k − m k vol( C × / R × ) , if m ∈ Z , − ( k − ≤ m ≤ k − , otherwise. (cid:3) The last lemma gives a formula to calculate the integral in (21) at infinity. Inparticular, it is a number depending only on γ , k and m . Lemma 4.18.
For φ ′∞ = k φ k π ′ k ( h ) X k − , φ ′′∞ = k φ kk P k k π ′ k ( h ) e γ ∈ π ′ k , | γ | = 1 ,define I ( m )2 k ( γ ) by Z C × \ D ×∞ Z C × / R × B ∞ ( π ′ k ( t ) φ ′∞ , φ ′′∞ ) sgn m ( t ) dt dh = vol( G ′∞ ) k φ k k − k − Γ( k ) Γ(3 k − · I ( m )2 k ( γ ) . Then we have,(i) for ≤ r ≤ k − , I ( r − k +1)2 k ( γ ) = (cid:18) k − r (cid:19) − X ≤ i,j ≤ k − i + j =3( k − − r γ k − − i ) (cid:18) k − i (cid:19) − (cid:18) k − j (cid:19) − | C i,j,r | , where C i,j,r is the coefficient in P k of X k − − i Y i ⊗ X k − − j Y j ⊗ X k − − r Y r .In particular I (0)2 k ( γ ) = Γ( k ) Γ(2 k − k − X i =0 γ k − − i ) (cid:18) k − i (cid:19) − | C i, k − − i,k − | . (ii) I ( m )2 k ( γ ) does not depend on N . Moreover k − X m = − ( k − | I ( m )2 k ( γ ) | ≤ k P k k . Proof. (i) By the proof of Lemma 4.17 we have(28) Z C × / R × h π ′ k ( t ) φ ′ v , φ ′′ v i k sgn − k − − r ) ( t ) dt = c ′ r c ′′ r k X k − − r Y r k vol( C × / R × )with(29) c ′ r c ′′ r = h φ ′∞ , X k − − r Y r i k h φ ′′∞ , X k − − r Y r i k k X k − − r Y r k . We first deal with h φ ′′∞ , X k − − r Y r i . Recall that in Lemma 4.5, e γ is defined by Z G ′∞ h π ′ k ⊗ π ′ k ( h − γg, h − g ) w ◦ k , w ◦ k i π ′ k ( g ) X k − dg = vol( G ′∞ ) k w ◦ k k k P k k π ′ k ( h ) e γ . We write φ ′′∞ back as an integral: φ ′′∞ = k φ k vol( G ′∞ ) k w ◦ k k Z G ′∞ h π ′ k ⊗ π ′ k ( h − γg, h − g ) w ◦ k , w ◦ k i π ′ k ( g ) X k − dg. Then we have h φ ′′∞ , X k − − r Y r i vol( G ′∞ ) k w ◦ k k k φ k = h Z G ′∞ h π ′ k ⊗ π ′ k ( h − γg, h − g ) w ◦ k , w ◦ k i π ′ k ( g ) X k − dg, X k − − r Y r i = Z G ′∞ h π ′ k ⊗ π ′ k ( γg, g ) w ◦ k , π ′ k ⊗ π ′ k ( h, h ) w ◦ k ih π ′ k ( g ) X k − , X k − − r Y r i dg. (30)Noticing that { X k − − i Y i ⊗ X k − − j Y j } forms an orthogonal basis of π ′ k ⊗ π ′ k ,the first matrix coefficient h π ′ k ⊗ π ′ k ( γg, g ) w ◦ k , π ′ k ⊗ π ′ k ( h, h ) w ◦ k i is equal to X ≤ i,j ≤ k − h π ′ k ⊗ π ′ k ( γg, g ) w ◦ k , X k − − i Y i ⊗ X k − − j Y j k X k − − i Y i kk X k − − j Y j k i· h π ′ k ⊗ π ′ k ( h, h ) w ◦ k , X k − − i Y i ⊗ X k − − j Y j k X k − − i Y i kk X k − − j Y j k i = X ≤ i,j ≤ k − k X k − − i Y i k − k X k − − j Y j k − · h π ′ k ⊗ π ′ k ( g, g ) w ◦ k , π ′ k ⊗ π ′ k ( γ − , X k − − i Y i ⊗ X k − − j Y j ) i· h π ′ k ⊗ π ′ k ( h, h ) w ◦ k , X k − − i Y i ⊗ X k − − j Y j i . By (27), when | γ | = 1, π ′ k ( γ − ) X k − − i Y i = γ − k − − i ) X k − − i Y i . VERAGES AND NONVANISHING OF TRIPLE PRODUCT CENTRAL L -VALUES 51 Therefore h π ′ k ⊗ π ′ k ( γg, g ) w ◦ k , π ′ k ⊗ π ′ k ( h, h ) w ◦ k i = X ≤ i,j ≤ k − k X k − − i Y i k − k X k − − j Y j k − γ − k − − i ) · h π ′ k ⊗ π ′ k ( g, g ) w ◦ k , X k − − i Y i ⊗ X k − − j Y j i· h π ′ k ⊗ π ′ k ( h, h ) w ◦ k , X k − − i Y i ⊗ X k − − j Y j i . (31)Combining (28) (29) (30) (31) we have Z C × / R × h π ′ k ( t ) φ ′∞ , φ ′′∞ i k sgn − k − − r ) ( t ) dt = vol( C × / R × )vol( G ′∞ ) k φ k k w ◦ k k k X k − − r Y r k − X ≤ i,j ≤ k − γ k − − i ) k X k − − i Y i k − k X k − − j Y j k − · h π ′ k ( h ) X k − , X k − − r Y r i k h π ′ k ⊗ π ′ k ( h, h ) w ◦ k , X k − − i Y i ⊗ X k − − j Y j i· Z G ′∞ h π ′ k ⊗ π ′ k ( g, g ) w ◦ k , X k − − i Y i ⊗ X k − − j Y j ih π ′ k ( g ) X k − , X k − − r Y r i k dg = vol( C × / R × )vol( G ′∞ ) k φ k k w ◦ k k k X k − − r Y r k − X ≤ i,j ≤ k − γ k − − i ) k X k − − i Y i k − k X k − − j Y j k − · h (cid:0) π ′ k ⊗ ◦ ∆ ( h ) (cid:1) w ◦ k ⊗ X k − , X k − − i Y i ⊗ X k − − j Y j ⊗ X k − − r Y r i· Z G ′∞ h (cid:0) π ′ k ⊗ ◦ ∆ ( g ) (cid:1) w ◦ k ⊗ X k − , X k − − i Y i ⊗ X k − − j Y j ⊗ X k − − r Y r i dg. Here we denote by ∆ the diagonal embedding from G ′∞ to three copies of it. Bythe definition of quotient measure on C × \ D ×∞ ∼ = ( R × \ C × ) \ G ′∞ , we have Z C × \ D ×∞ Z C × / R × h π ′ k ( t ) φ ′∞ , φ ′′∞ i k sgn m ( t ) dt dh = 1vol( C × / R × ) Z G ′∞ Z C × / R × h π ′ k ( t ) φ ′∞ , φ ′′∞ i k sgn m ( t ) dt dh, and then Z C × \ D ×∞ Z C × / R × h π ′ k ( t ) φ ′∞ , φ ′′∞ i k sgn − k − − r ) ( t ) dt dh = 1vol( G ′∞ ) k φ k k w ◦ k k k X k − − r Y r k − X ≤ i,j ≤ k − γ k − − i ) k X k − − i Y i k − k X k − − j Y j k − · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z G ′∞ h (cid:0) π ′ k ⊗ ◦ ∆ ( g ) (cid:1) w ◦ k ⊗ X k − , X k − − i Y i ⊗ X k − − j Y j ⊗ X k − − r Y r i dg (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Recall that P k is the only G ′∞ -invariant vector in ( π ′ k ) ⊗ · ∆ up to a constantmultiple. Then Lemma 2.12 gives that Z G ′∞ h (cid:0) π ′ k ⊗ ◦ ∆ ( g ) (cid:1) w ◦ k ⊗ X k − , X k − − i Y i ⊗ X k − − j Y j ⊗ X k − − r Y r i dg = vol( G ′∞ ) h w ◦ k ⊗ X k − , P k k P k k ih X k − − i Y i ⊗ X k − − j Y j ⊗ X k − − r Y r , P k k P k k i . Lemma 2.14 shows that h w ◦ k ⊗ X k − , P k i = k w ◦ k k . So Z C × \ D ×∞ Z C × / R × h π ′ k ( t ) φ ′∞ , φ ′′∞ i k sgn − k − − r ) ( t ) dt dh = vol( G ′∞ ) k φ k k w ◦ k k k P k k k X k − − r Y r k − X ≤ i,j ≤ k − γ k − − i ) k X k − − i Y i k − k X k − − j Y j k − · (cid:12)(cid:12)(cid:12) h P k , X k − − i Y i ⊗ X k − − j Y j ⊗ X k − − r Y r i (cid:12)(cid:12)(cid:12) = vol( G ′∞ ) k φ k k w ◦ k k k P k k (cid:18) k − r (cid:19) − X ≤ i,j ≤ k − i + j =3( k − − r γ k − − i ) (cid:18) k − i (cid:19) − (cid:18) k − j (cid:19) − · (cid:12)(cid:12)(cid:12)(cid:16) the coefficient in P k of X k − − i Y i ⊗ X k − − j Y j ⊗ X k − − r Y r (cid:17)(cid:12)(cid:12)(cid:12) . By Lemma 2.10 we have k w ◦ k k k P k k = 12 k − k − Γ( k ) Γ(3 k − , and this completes the proof.(ii) The above result implies that, for | γ | = 1, k − X m = − ( k − | I ( m )2 k ( γ ) | ≤ X ≤ i,j,r ≤ k − i + j + r =3( k − (cid:18) k − i (cid:19) − (cid:18) k − j (cid:19) − (cid:18) k − r (cid:19) − | C i,j,r | . To show the right hand side is equal to k P k k , we write P k = X i,j,r C i,j,r X k − − i Y i ⊗ X k − − j Y j ⊗ X k − − r Y r . VERAGES AND NONVANISHING OF TRIPLE PRODUCT CENTRAL L -VALUES 53 Recall that { X k − − r Y r } forms an orthogonal basis of V π ′ k which are eigenvectorsunder the action of C × . Analogous to what we have done in Lemma 4.17, Z C × / R × h π ′ k ⊗ π ′ k ⊗ π ′ k ( t ) P k , P k i sgn m ( t ) dt = Z C × / R × h k − X i,j,r sgn k − − i ) ( t ) sgn k − − j ) ( t ) sgn k − − r ) ( t ) · C i,j,r X k − − i Y i ⊗ X k − − j Y j ⊗ X k − − r Y r , P k i sgn m ( t ) dt = X i,j,r | C i,j,r | k X k − − i Y i ⊗ X k − − j Y j ⊗ X k − − r Y r k Z C × / R × sgn k − − i − j − r ) ( t ) sgn m ( t ) dt = P ≤ i,j,r ≤ k − i + j + r =3( k − − m | C i,j,r | (cid:0) k − i (cid:1) − (cid:0) k − j (cid:1) − (cid:0) k − r (cid:1) − vol( C × / R × ) , if m ∈ Z , − k − ≤ m ≤ k − , otherwise.In particular X ≤ i,j,r ≤ k − i + j + r =3( k − (cid:18) k − i (cid:19) − (cid:18) k − j (cid:19) − (cid:18) k − r (cid:19) − | C i,j,r | = 1vol( C × / R × ) Z C × / R × h π ′ k ⊗ π ′ k ⊗ π ′ k ( t ) P k , P k i dt = h P k , P k i . The last equality is because P k is G ′∞ -invariant and hence C × / R × -invariant. (cid:3) Examples of small weights
In this section we simplify our Main Theorem, particularly I and I , for smallweights.5.1. Weight 2 or 4.
In both cases the only character Ω contributing to the sumon the right hand side of the Main Theorem is the trivial character , according toLemma 4.10. When 2 k = 2, the polynomial P = 1 is trivial, and hence I (0)2 ( γ ) = 1as defined in Lemma 4.18. Following the notations in Theorem 4.3, we have I ( ) = ( , if 2 ∤ N, (1 + a ( h )) , if 2 | N ; , I ( ) = ( , if 3 ∤ N, (1 + a ( h )) , if 3 | N. The Main Theorem simplifies to the following identity.
Example 5.1.
For any h ∈ F ( N ) , (32) N π X f,g ∈F ( N ) ε p = − , ∀ p | N L fin (2 , f × g × h )( f, f )( g, g )( h, h ) = 1 − /ϕ ( N )2 ω ( N ) + 116 π ( h, h ) · L · ord ( N ) Y p | N − χ − ( p )2 + 6 √ L · ord ( N ) Y p | N − χ − ( p )2 , where L = L fin (1 , h ) L fin (1 , h ⊗ χ − ) · ( , if ∤ N, (1 + a ( h )) , if | N ; ,L = L fin (1 , h ) L fin (1 , h ⊗ χ − ) · ( , if ∤ N, (1 + a ( h )) , if | N. Recall that when p | N , a p ( h ) p − ( k − = − ε p ( , π h ) = ± h ∈ F k ( N ). Sothe terms (1 + a ( h )) and (1 + a ( h )) can either be 0 or 1.When N is a prime with N = 11 or N >
13, this reproves Theorem 1.1 in [FW10].(The Petersson inner product ( f, f ), defined in (2), is normalized differently in[FW10], so the main formula there is differed by some constants comparing with(32).)When 2 k = 4, P = ( X Y − X Y )( X Y − X Y )( X Y − X Y )= ( − Y X + X Y ) X Y + other terms . The coefficients C i, − i, in P of X − i Y i ⊗ X i Y − i ⊗ X Y are given by C , , = 1 , C , , = 0 , C , , = − . Applying Lemma 4.18, we have I (0)4 ( γ ) = Γ(2) Γ(3) X i =0 γ − i ) (cid:18) i (cid:19) − | C i, − i, | = 12 ( γ + γ − )= ( − , γ = γ = √− − , γ = γ = (1 + √− . Using the notations in Theorem 4.3, we have I ( ) = ( − , if 2 ∤ N, − (1 + a ( h )2 ) , if 2 | N ; , I ( ) = ( − , if 3 ∤ N, − (1 + a ( h )3 ) , if 3 | N. The Main Theorem now becomes the following.
Example 5.2.
For any h ∈ F ( N ) , (33) 3 N π X f,g ∈F ( N ) ε p = − , ∀ p | N L fin (5 , f × g × h )( f, f )( g, g )( h, h ) = 12 ω ( N ) + 1(4 π ) ( h, h ) · L · ord ( N ) Y p | N − χ − ( p )2 + 12 √ L · ord ( N ) Y p | N − χ − ( p )2 , where L = − L fin (2 , h ) L fin (2 , h ⊗ χ − ) · ( , if ∤ N, (1 + a ( h )2 ) , if | N ; ,L = − L fin (2 , h ) L fin (2 , h ⊗ χ − ) · ( , if ∤ N, (1 + a ( h )3 ) , if | N. Notice that L , L are non-positive now, unlike that in the weight 2 case. VERAGES AND NONVANISHING OF TRIPLE PRODUCT CENTRAL L -VALUES 55 Weight 6.
In this case nontrivial Hecke characters Ω of E = Q ( √−
1) mightcontribute to the orbital decomposition. Let ξ be the character on E × \ A × E whichis trivial on A × Q , unramified everywhere, and satisfies ξ ∞ ( z ) = ( z/ ¯ z ) . With thenotations in Theorem 4.1 and (23), Lemma 4.10 shows that only ξ , ξ − and thetrivial character contribute to the nontrivial orbit I [ γ ] , and only contributesto the orbit I [ γ ] .Recall that P = ( X Y − X Y ) ( X Y − X Y ) ( X Y − X Y ) . The coefficient C i,j,r in P of X − i Y i ⊗ X − j Y j ⊗ X − r Y r vanishes unless i + j + r =6, in which case C , , = 1, C , , = − C , , = 1, C , , = − C , , = 2, C , , = 2, C , , = − C , , = 1, C , , = 2, C , , = − C , , = 2, C , , = 1, C , , = − C , , = 2, C , , = 2, C , , = − C , , = 1, C , , = − C , , = 1.Applying Lemma 4.18, we have I (0)6 ( γ ) = (cid:18) (cid:19) − X ≤ i,j ≤ i + j =4 γ − i ) (cid:18) i (cid:19) − (cid:18) j (cid:19) − | C i,j, | = 6 − ( γ + γ + γ − + 4 − ( γ + γ − )) = ( , γ = γ = √− − , γ = γ = (1 + √− . And for γ = γ we have I ( − ( γ ) = (cid:18) (cid:19) − X ≤ i,j ≤ i + j =6 γ − i ) (cid:18) i (cid:19) − (cid:18) j (cid:19) − | C i,j, | = 6 − ( γ + γ − ) + 4 − γ − = 112 ;and I (2)6 ( γ ) = .With the notations in Theorem 4.3, we have I ( ) = ( , if 2 ∤ N, (1 + a ( h )2 ) , if 2 | N ; I ( ) = ( − , if 3 ∤ N, − (1 + a ( h )3 ) , if 3 | N ; I ( ξ ) = I ( ξ − ) = ( , if 2 ∤ N, (1 − a ( h )2 ) , if 2 | N. The Main Theorem now becomes,
Example 5.3.
Let E = Q ( √− , ξ be the character on E × \ A × E which is trivialon A × Q , unramified everywhere, and satisfies ξ ∞ ( z ) = ( z/ ¯ z ) . For any h ∈ F ( N ) , (34) N π X f,g ∈F ( N ) ε p = − , ∀ p | N L fin (8 , f × g × h )( f, f )( g, g )( h, h ) = 12 ω ( N )
135 + 12 π ( h, h ) · L · ord ( N ) Y p | N − χ − ( p )2 + 6 √ L · ord ( N ) Y p | N − χ − ( p )2 , where L = L fin (3 , h ) L fin (3 , h ⊗ χ − ) + (cid:0) L fin ( , h × Θ ξ ) + L fin ( , h × Θ ξ − ) (cid:1) , if ∤ N, L fin (3 , h ) L fin (3 , h ⊗ χ − ) , if | N, a ( h ) = 2 , (cid:0) L fin ( , h × Θ ξ ) + L fin ( , h × Θ ξ − ) (cid:1) , if | N, a ( h ) = − ; L = − L fin (3 , h ) L fin (3 , h ⊗ χ − ) ( , if ∤ N, (1 + a ( h )3 ) , if | N ; and Θ ξ , Θ ξ − (defined before Theorem 1.1) are the CM modular forms arise fromHecke characters ξ , ξ − correspondingly. Applications
Sum over three forms.
Now we go back to Theorem 1.1 and take the sumover h ∈ F k ( N ). The right hand side of (1) now becomes an exact averageformula for central values of twisted quadratic base change L -functions. Using thefollowing result established by Feigon and Whitehouse [FW09], we can obtain anexact average formula of L fin (3 k − , f × g × h ) as all three forms run through F k ( N ). Lemma 6.1 ([FW09]) . Let E be an imaginary quadratic field of fundamental dis-criminant − d < , with associated quadratic character χ − d = ( − d · ) . Let h = h E bethe class number of E , u := O × E / {± } . Let N be a square-free integer which isthe product of an odd number of primes p satisfying χ − d ( p ) = − and N > d , andlet k ≥ be an even integer. Then (2 k − d / u π (4 π ) k − X f ∈F k ( N ) L fin ( k, f ) L fin ( k, f, χ − d )( f, f ) = ( h (1 − huϕ ( N ) ) , if k = 2; h, otherwise.When k > , for a nontrivial character Ω : E × \ A × E / A × → C × which is unramifiedeverywhere, (2 k − d / u π (4 π ) k − X f ∈F k ( N ) L fin ( k + | m | + , f × Θ Ω )( f, f ) = h, with m and Θ Ω defined before Theorem 1.1. VERAGES AND NONVANISHING OF TRIPLE PRODUCT CENTRAL L -VALUES 57 In particular, for d = 4 and N being the product of an odd number of distinctprimes of the form 4 n + 3, we have u = 2 and(2 k − π (4 π ) k − X f ∈F k ( N ) L fin ( k, f ) L fin ( k, f, χ − )( f, f ) = ( − ϕ ( N ) , if 2 k = 2 , , if 2 k > k − π (4 π ) k − X f ∈F k ( N ) L fin ( k + | m | + , f × Θ Ω )( f, f ) = 1 , if 2 k > . For d = 3 and N being the product of an odd number of distinct primes of the form3 n + 2, u = 3 and(2 k − √ π (4 π ) k − X f ∈F k ( N ) L fin ( k, f ) L fin ( k, f, χ − )( f, f ) = ( − ϕ ( N ) , if 2 k = 2 , , if 2 k > k − √ π (4 π ) k − X f ∈F k ( N ) L fin ( k + | m | + , f × Θ Ω )( f, f ) = 1 , if 2 k > . With Theorem 1.1 we can get average formulas of central L -values over all threeforms. Notice that, when 2 , ∤ N , I and I (defined in Theorem 4.3) do not dependon h ∈ F k ( N ). Corollary 6.2.
Let N be a square-free integer with an odd number of prime factors.When , ∤ N , N π X f,g,h ∈F ( N ) ε p = − , ∀ p | N L fin (2 , f × g × h )( f, f )( g, g )( h, h ) = 1 − /ϕ ( N )2 ω ( N ) F ( N )+ − /ϕ ( N )2 Y p | N − χ − ( p )2 + (1 − /ϕ ( N )) Y p | N − χ − ( p )2 ; and for k > , (35) N k − π k − X f,g,h ∈F k ( N ) ε p = − , ∀ p | N L fin (3 k − , f × g × h )( f, f )( g, g )( h, h ) = F k ( N )2 ω ( N ) Γ(2 k )Γ(2 k − + 1Γ( k ) Γ(3 k − X Ω ∈ d [ E × ] I (Ω) Y p | N − χ − ( p )2 + X Ω ∈ d [ E × ] I (Ω) Y p | N − χ − ( p )2 . In particular, if N has a prime factor ≡ and one ≡ , we have N k − π k − X f,g,h ∈F k ( N ) ε p = − , ∀ p | N L fin (3 k − , f × g × h )( f, f )( g, g )( h, h ) = 1 − δ ( k ) /ϕ ( N )2 ω ( N ) Γ(2 k − Γ(2 k ) F k ( N ) , where δ ( k ) = ( , if k = 1 , , otherwise. The size of F k ( N ) is given by [Mar05]: For any integer k ≥ N a square-free integer with an odd number of prime factors, the dimension of the space ofweight 2 k newforms on Γ ( N ) is F k ( N ) = 2 k − ϕ ( N ) − c (2 k ) Y p | N (1 − χ − ( p )) − c (2 k ) Y p | N (1 − χ − ( p )) − δ ( k ) , where δ ( k ) is defined in the above corollary, and c , c are defined by c ( n ) = 14 + ⌊ n ⌋ − n ( / , n ≡ , − / , n ≡ c ( n ) = 13 + ⌊ n ⌋ − n / , n ≡ , , n ≡ , − / , n ≡ . In particular, F ( N ) = ϕ ( N )12 + 14 Y p | N (1 − χ − ( p )) + 13 Y p | N (1 − χ − ( p )) − F ( N ) = ϕ ( N )4 − Y p | N (1 − χ − ( p )); F ( N ) = 512 ϕ ( N ) + 14 Y p | N (1 − χ − ( p )) − Y p | N (1 − χ − ( p ));if N has a prime factor ≡ ≡ F k ( N ) = 2 k − ϕ ( N ) − δ ( k ) . With the above results and calculations in Section 5 we can have the followingexplicit identities.
Example 6.3.
For N square-free with an odd number of prime factors and , ∤ N ,say N is of Type 1 if N has a prime factor ≡ and one ≡ ;Type 5 if all prime factors of N are ≡ ,
11 mod 12 but at least one is ≡ ;Type 7 if all prime factors of N are ≡ ,
11 mod 12 but at least one is ≡ ;Type 11 if all prime factors of N are ≡
11 mod 12 . VERAGES AND NONVANISHING OF TRIPLE PRODUCT CENTRAL L -VALUES 59 (For example, N is of Type 5 if and only if Q p | N − χ − ( p )2 = 0 and Q p | N − χ − ( p )2 = 1 .)Then we have N π X f,g,h ∈F ( N ) ε p = − , ∀ p | N L fin (2 , f × g × h )( f, f )( g, g )( h, h )= ϕ ( N ) − ω ( N )
12 (1 − ϕ ( N ) ) + 34 (1 − ϕ ( N ) ) Y p | N − χ − ( p )2 + 43 (1 − ϕ ( N ) ) Y p | N − χ − ( p )2= ϕ ( N ) − ω ( N )
12 (1 − ϕ ( N ) ) + , if N is of Type 1; (1 − ϕ ( N ) ) , if N is of Type 7; (1 − ϕ ( N ) ) , if N is of Type 5; − ϕ ( N ) , if N is of Type 11. N π X f,g,h ∈F ( N ) ε p = − , ∀ p | N L fin (5 , f × g × h )( f, f )( g, g )( h, h ) = ϕ ( N )2 ω ( N ) − Y p | N − χ − ( p )2 − Y p | N − χ − ( p )2= ϕ ( N )2 ω ( N ) − , if N is of Type 1; , if N is of Type 7; , if N is of Type 5; , if N is of Type 11. N π X f,g,h ∈F ( N ) ε p = − , ∀ p | N L fin (8 , f × g × h )( f, f )( g, g )( h, h ) = ϕ ( N )2 ω ( N ) + 95 Y p | N − χ − ( p )2 − Y p | N − χ − ( p )2= ϕ ( N )2 ω ( N ) + , if N is of Type 1; , if N is of Type 7; − , if N is of Type 5; , if N is of Type 11.In particular, notice that F (5) = F (7) = 1 . By direct calculation we get acorollary: L fin (5 , f × f × f ) = 0 for f ∈ F (5) or F (7) , which essentially means L fin (5 , Sym ( f )) = 0 due to the factorization L ( s, f × f × f ) = L ( s, Sym ( f )) L ( s − k + 1 , f ) . Moreover,
Corollary 6.4.
Let N be a square-free integer with an odd number of prime factors.If N has a prime factor ≡ and one ≡ (i.e. N is of Type 1),we have N π X f,g,h ∈F ( N ) ε p = − , ∀ p | N L fin (2 , f × g × h )( f, f )( g, g )( h, h ) = ( ϕ ( N ) − ϕ ( N ) − ω ( N ) ϕ ( N ) ; N k − π k − X f,g,h ∈F k ( N ) ε p = − , ∀ p | N L fin (3 k − , f × g × h )( f, f )( g, g )( h, h ) = ϕ ( N )2 ω ( N ) k − , if k > . The nonvanishing problems.
When the weight 2 k = 2, we can get a similarresult of the nonvanishing problem as [FW10] Corollary 5.2. Corollary 6.5.
Let N be the product of an odd number of distinct primes such that ϕ ( N ) > . For each h ∈ F ( N ) , there exist f, g ∈ F ( N ) such that L fin (2 , f × g × h ) = 0 ; moreover, { ( f, g ) ∈ F ( N ) × F ( N ) : L fin (2 , f × g × h ) = 0 } ≫ ǫ N / − ǫ . Proof.
The first statement comes from (1) and the non-negativity of L (1 , h ) L (1 , h, χ − d ).For f ∈ F ( N ) we know from Hoffstein–Lockhart [HL94] that ( f, f ) ≫ N (log N ) − .Applying this to (1) together with the non-negativity of L (1 , h ) L (1 , h, χ − d ), we have X f,g ∈F ( N ) L (2 , f × g × h ) ≫ X f,g ∈F ( N ) ε p = − , ∀ p | N L fin (2 , f × g × h ) ≫ − /ϕ ( N )2 ω ( N ) N (log N ) − . Robin [Rob83] shows that, for
N > ω ( N ) := X p | N ≪ log N/ log log N. So 2 ω ( N ) ≪ N C/ log log N . When ϕ ( N ) >
24 we have X f,g ∈F ( N ) L fin (2 , f × g × h ) ≫ N − C/ log log N (log N ) − . Moreover, for any weight 2 k we have the convexity bound L fin (3 k − , f × g × h ) ≪ ǫ ( N k ) / ǫ (see [IK04]). Therefore { ( f, g ) ∈ F ( N ) × F ( N ) : L fin (2 , f × g × h ) = 0 }≫ ǫ N / − ǫ N C/ log log N (log N ) ≫ ǫ N / − ǫ . (cid:3) Analogously we can get a nonvanishing result for higher weight. Now the coef-ficients on the right hand side are not always non-negative (see (33) and (34)), weneed more work on the nontrivial orbits on the geometric side.Recall that, when Ω is unramified everywhere, the convexity bound of centralvalue of base change L -function is L fin ( 12 , π E , Ω) ≪ ǫ ( N k ) / ǫ . Followed by (23) we have I [ γ ] ( f )vol( K ′ ) k φ k = 2 c γ L (1 , η ) 2 ω ( N ) N · L (1 , π , Ad) Γ(2 k − (2 π ) k Γ( k ) Γ(3 k − · X Ω L fin ( 12 , ( π ) E ⊗ Ω) I ( m )2 k ( γ ) ≪ ǫ ω ( N ) N · L (1 , π , Ad) Γ(2 k − (2 π ) k Γ( k ) Γ(3 k − · X Ω ( N k ) / ǫ | I ( m )2 k ( γ ) | . Lemma 4.18 and Lemma 2.10 show that X Ω | I ( m )2 k ( γ ) | ≤ k P k k = Γ( k ) Γ(3 k − k − . VERAGES AND NONVANISHING OF TRIPLE PRODUCT CENTRAL L -VALUES 61 Then(37) I [ γ ] ( f )vol( K ′ ) k φ k ≪ ǫ ω ( N ) N − / ǫ L (1 , π , Ad) k ǫ Γ(2 k − π ) k . Recall that 2 ω ( N ) ≪ N C/ log log N and L ∞ (1 , π , Ad) = π (2 π ) − k Γ(2 k ). For π ∈F ( N, k ) we know from Hoffstein–Lockhart [HL94] and Nelson [Nel11] that L (1 , π , Ad) ≫ Γ(2 k )(2 π ) k log( N k ) . So I [ γ ] ( f )vol( K ′ ) k φ k ≪ N C/ log log N N − / ǫ k ǫ k − N k ) ≪ ǫ N − ( N k ) ǫ . Now we can show that:
Corollary 6.6.
Let N be the product of an odd number of distinct primes. For h ∈ F k ( N ) , { ( f, g ) ∈ F k ( N ) × F k ( N ) : L fin (3 k − , f × g × h ) = 0 } ≫ k,ǫ N / − ǫ . In particular, if N has a prime factor ≡ and one ≡ , then foreach h ∈ F k ( N ) , there exist f, g ∈ F k ( N ) such that L fin (3 k − , f × g × h ) = 0 .Proof. Recall that L (1 / , π k dis ⊗ π k dis ⊗ π k dis ) = 2 (2 π ) − k Γ( k ) Γ(3 k − . For 2 k >
2, rewrite (10) as2 π ω ( N ) N Γ(2 k − X f,g ∈F k ( N ) ε p = − , ∀ p | N L fin (3 k − , f × g × h )(2 π ) k L (1 , π f ⊗ π g ⊗ π h , Ad)= 12 k − I [ γ ] + I [ γ ] k φ k vol( K ′ )using the notations in Theorem 4.3. By the above calculation we know that theright hand side ≫ k,ǫ π ) k L (1 , π , Ad) ≫ Γ(2 k ) log( N k ) − . Therefore X f,g ∈F k ( N ) ε p = − , ∀ p | N L fin (3 k − , f × g × h ) ≫ k,ǫ N ω ( N ) (log N ) − ≫ N − C/ log log N (log N ) − . The rest of proof is the same as in the weight 2 case, while the convexity bound of L fin (3 k − , f × g × h ) does not change with respect to N . (cid:3) Remark 6.7.
A result involving k cannot be shown following the above proof. For k > , rewrite (10) as π ω ( N ) N Γ(2 k − X f,g ∈F k ( N ) ε p = − , ∀ p | N L fin (3 k − , f × g × h )(2 π ) k L (1 , π f ⊗ π g , Ad)= (2 π ) k L (1 , π h , Ad)2 k − π ) k L (1 , π h , Ad) I [ γ ] + I [ γ ] k φ k vol( K ′ ) By (37) we have (2 π ) k L (1 , π h , Ad) I [ γ ] ( f )vol( K ′ ) k φ k ≪ ǫ ω ( N ) N − / ǫ k ǫ Γ(2 k − But the main term (2 π ) k L (1 , π h , Ad)2 k − ≫ Γ(2 k − N k ) is not big enough.But, if N has a prime factor ≡ and one ≡ , we have π ω ( N ) N Γ(2 k − X f,g ∈F k ( N ) ε p = − , ∀ p | N L fin (3 k − , f × g × h )(2 π ) k L (1 , π f ⊗ π g ⊗ π h , Ad) = 12 k − for k > . We can get that X f,g ∈F k ( N ) ε p = − , ∀ p | N L fin (3 k − , f × g × h ) ≫ ǫ N ω ( N ) k − Γ(2 k ) log( N k ) − k − N ω ( N ) (2 k − log( N k ) ≫ N − C/ log log N k log( N k ) . The convexity bound L fin (3 k − , f × g × h ) ≪ ǫ ( N k ) / ǫ gives { ( f, g ) ∈ F k ( N ) × F k ( N ) : L fin (3 k − , f × g × h ) = 0 }≫ ǫ N / − ǫ − C/ log log N k − ǫ log( N k ) ≫ ǫ N / − ǫ k − ǫ . A subconvexity bound of weight aspect for L fin (3 k − , f × g × h ) might have someapplication. More precisely, if L fin (3 k − , f × g × h ) ≪ N,ǫ k − δ + ǫ for some δ > ,we can show that { ( f, g ) ∈ F k ( N ) × F k ( N ) : L fin (3 k − , f × g × h ) = 0 } ≫ N,ǫ k δ − ǫ . At last we study the nonvanishing modulo suitable primes p of the algebraic partof triple product L -values. Given f, g, h ∈ F k ( N ), we define L alg (3 k − , f × g × h ) := Γ(3 k − k ) ω ( N ) N k − π k − L fin (3 k − , f × g × h )( f, f )( g, g )( h, h ) . According to [BSP96] Theorem 5.7 (a revised version of this theorem can be foundin [BSSP03], in the proof of Proposition 2.1), L alg (3 k − , f × g × h ) lies in the subfieldof C generated by the Fourier coefficients of f , g and h and hence is algebraic. Corollary 6.8.
Let p be a prime such that p ≥ k − and p = 2 , and p be a placein Q above p . Let N be a square-free integer with an odd number of prime factorswhich has a prime factor ≡ and one ≡ . Then, when k > ,for any h ∈ F k ( N ) , there exist f, g ∈ F k ( N ) such that L alg (3 k − , f × g × h ) p ) . This holds for k = 2 too if in addition p ∤ ϕ ( N ) − . VERAGES AND NONVANISHING OF TRIPLE PRODUCT CENTRAL L -VALUES 63 Proof.
The corollary is obvious, noting from (3) that, for any h ∈ F k ( N ), X f,g ∈F k ( N ) ε p = − , ∀ p | N L alg (3 k − , f × g × h ) = (1 − ϕ ( N ) δ ( k ))2 k − Γ(3 k − k ) Γ(2 k − Γ(2 k )= ( ϕ ( N ) − ϕ ( N ) , k = 2;2 k − k − k ) Γ(2 k − Γ(2 k ) , k > . (cid:3) Appendix A. Hilbert Symbol
For a local field F , the Hilbert symbol ( · , · ) F : F × / ( F × ) × F × / ( F × ) → {± } can be defined by that ( a, b ) F = 1 if ( a,bF ) ∼ = M (2 , F ) is split, and = − a,bF ) isa division algebra. It can be calculated by the following algorithm (see [Ser73]): • For F = R , ( a, b ) R = 1 if a or b is >
0, and = − a and b are < • For F = Q p , if we write a, b in the form p α u, p β v where u, v ∈ Z × p , we have( a, b ) Q p = ( ( − αβε ( p ) ( up ) β ( vp ) α if p = 2 , ( − ε ( u ) ε ( v )+ αω ( v )+ βω ( u ) if p = 2 . Here ( up ) denotes the Legendre symbol, and ε , ω are defined by ε ( z ) ≡ z −
12 (mod 2) = ( z ≡ , z ≡ ω ( z ) ≡ z −
18 (mod 2) = ( z ≡ ± , z ≡ ± . A.1.
An application to quaternion algebras.
We can use the Hilbert symbolto prove Lemma 2.4.
Proof of Lemma 2.4. (1) When p ∤ N ,( − N, − Q p = ( − (cid:16) − Np (cid:17) (cid:16) − p (cid:17) = 1 . When p | N but p = 2 (then p ≡ − N, − Q p = ( − (cid:16) − N/pp (cid:17) (cid:16) − p (cid:17) = − . When p = 2, if 2 ∤ N , N is the product of an odd number of primes of the form4 n + 3, and therefore − N ≡ − N, − Q = ( − ε ( − N ) ε ( − = 1;if 2 | N , N is the product of an even number of primes of the form 4 n + 3, and then − N ≡ − − N, − Q = ( − ε ( − N/ ε ( − ω ( − = − . (2) First we notice that, for a prime p = 2 ,
3, by the Quadratic Reciprocity Law,(38) (cid:16) − p (cid:17) = (cid:16) p (cid:17)(cid:16) − p (cid:17) = ( − ε ( p ) (cid:16) p (cid:17)(cid:16) − p (cid:17) = (cid:16) p (cid:17) = ( p ≡ − p ≡ . When p ∤ N , ( − N, − Q p = 1. When p | N but p ∤ p ≡ − N, − Q p = ( − (cid:16) − N/pp (cid:17) (cid:16) − p (cid:17) = − . When p = 2, if 2 ∤ N , ( − N, − Q = ( − ε ( − N ) ε ( − = 1;if 2 | N , ( − N, − Q = ( − ε ( − N/ ε ( − ω ( − = − . When p = 3, if 3 ∤ N , N is the product of an odd number of primes of the form3 n + 2 (2 might be included), and therefore − N ≡ − N, − Q = ( − (cid:16) − N (cid:17) (cid:16) − / (cid:17) = 1;if 3 | N , N is the product of an even number of primes of the form 3 n + 2, and then − N ≡ − − N, − Q = ( − ε (3) (cid:16) − N/ (cid:17) (cid:16) − / (cid:17) = ( − − −
1) = − . (cid:3) A.2.
An application to quadratic forms.
Another application of the Hilbertsymbol is to study what numbers can be represented in a given quadratic form.
Lemma A.1 ([Ser73]) . Let f ∼ a X + · · · + a n X n be a quadratic form of rank n and a ∈ Q × p / ( Q × p ) . When n = 3 , in order that f represents a in Q p , it is necessaryand sufficient that a = − a a a in Q × p / ( Q × p ) , or a = − a a a and ( − , a ) Q p = ( a , a ) Q p ( a , a ) Q p ( a , a ) Q p . With this lemma we give a proof of Proposition 4.7.
Proof of Proposition 4.7.
To prove the necessity, by the definition of the Hilbertsymbol, we write D ( Q ) = ( a,b Q ) with ( a, b ) Q p = − p | N . Recall that N D ( X + X i + X j + X k ) = X − aX − bX + abX . (1) When p ≡ N , we claim that − aX − bX + abX cannot represent 1 in Q p . Then it cannot represent 1 in Q , i.e. D ( Q ) has no elementwith trace 0 and norm 1.To prove the claim, we only need to show1 = − ( − a )( − b )( ab ) in Q × p / ( Q × p ) , and ( − , = ( − a, − b )( − b, ab )( − a, ab )(we omit the subscript Q p when there is no confusion). Notice that when p ≡ − Z /p Z ) × and therefore a square in Q × p by Hensel’s VERAGES AND NONVANISHING OF TRIPLE PRODUCT CENTRAL L -VALUES 65 Lemma. This proves the first statement. For the second one, by the properties ofHilbert symbols (see [Voi20] Lemma 5.6.3 and Lemma 12.4.1),( − a, − b ) = ( − a, − ( − a )( − b )) = ( − a, − ab );( − b, ab )( − a, ab ) = ( ab, ab ) = ( − , ab ) = ( − , − − , − ab );( − a, − b )( − b, ab )( − a, ab ) = ( − a, − ab )( − , − − , − ab ) = ( − , − a, − ab ) . With the assumption ( a, b ) = − a, − ab ) = −
1) we have( − a, − b )( − b, ab )( − a, ab ) = − ( − , − . (This result does not depend on the congruence condition of p .) When p ≡ − Q × p , and then ( − , −
1) = 1 = ( − , D ( x ) = 1 , N D ( x ) = 1 ⇔ Tr D (2 x −
1) = 0 , N D (2 x −
1) = 3 . So for this case we will show that, when p ≡ N , − aX − bX + abX cannot represent 3 in Q p ; i.e. − ∈ ( Q × p ) , and ( − , = ( − a, − b )( − b, ab )( − a, ab ) . When p ≡ − Z /p Z ) × according to (38), and thereforea square in Q × p by Hensel’s Lemma. By the deduction of the previous case we stillhave ( − a, − b )( − b, ab )( − a, ab ) = − ( − , − − ∈ ( Q × p ) implies( − ,
3) = ( − , − − , −
1) = 1 · ( − , − . This completes the proof of the proposition. (cid:3)
Acknowledgements
This paper contains the main result of the author’s doctoral dissertation atCUNY the Graduate Center. The author would like to thank his advisor, ProfessorBrooke Feigon for her patient guidance and encouragement, without whom thecompletion of this work would not have been possible. The author is also thankfulto Shih–Yu Chen, Bingrong Huang, Peter Humphries, Yiannis Sakellaridis, andLiyang Yang for many helpful discussions and comments regarding this work.
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