ESI The Erwin S hr�odinger International Boltzmanngasse 9Institute for Mathemati al Physi s A-1090 Wien, AustriaOn the Global Gross{Prasad onje turefor Yoshida LiftingsSiegfried B�o hererMasaaki FurusawaRainer S hulze-Pillot

Vienna, Preprint ESI 1264 (2003) January 8, 2003Supported by the Austrian Federal Ministry of Edu ation, S ien e and CultureAvailable via http://www.esi.a .at

ON THE GLOBAL GROSS-PRASAD CONJECTUREFOR YOSHIDA LIFTINGSSIEGFRIED B�OCHERER, MASAAKI FURUSAWA, AND RAINERSCHULZE-PILLOTProf. J. Shalika to his 60th birthdayIntrodu tionIn two arti les in the Canadian Journal [16, 17℄, B. Gross and D. Prasadpro laimed a global onje ture on erning the de omposition of an au-tomorphi representation of an adeli spe ial orthogonal group G1 uponrestri tion to an embedded orthogonal group G2 of a quadrati spa ein smaller dimension and also its lo al ounterpart. In the lo al situa-tion, one an summarize the onje ture by saying that the o urren eof �2 in the restri tion of �1 depends on the �-fa tor atta hed to therepresentation �1 �2; in the global situation, assuming the existen eof the lo al nontrivial invariant fun tional at all pla es and its non-vanishing on the spheri al ve tor at almost all unrami�ed pla es, one onsiders a spe i� linear fun tional given by a period integral. Thisperiod integral is then onje tured to give a nontrivial fun tional if andonly if the entral riti al value of the L-fun tion atta hed to �1 �2is nonzero. In parti ular in the ase when G1 is the group of an n-dimensional nondegenerate quadrati spa e V and G2 is the group ofan (n � 1)-dimensional subspa e W of V , they showed that in low di-mensions (n � 4) known results an be interpreted as eviden e for this onje ture, using the well known isomorphisms for orthogonal groupsin low dimensions.The ase n = 5 has been treated in the lo al situation by Prasad [27℄;it an also be reinterpreted using these isomorphisms: The split spe ialorthogonal group in dimension 5 is isomorphi to the proje tive sym-ple ti similitude group PGSp2; and the spin group of the 4-dimensionalPart of this work was done during a stay of all three authors at the IAS, Prin e-ton, supported by von Neumann Fund, Bell Fund and Bankers Trust Fund. Fu-rusawa was also supported by Sumitomo Foundation. We thank the IAS for itshospitality.S. B�o herer and R. S hulze-Pillot thank D. Prasad and the Harish ChandraResear h Institute, Allahabad, India for their hospitality. S hulze-Pillot's visit toHCRI was also supported by DFG. 1

2 S. B�OCHERER, M. FURUSAWA, AND R. SCHULZE-PILLOTsplit orthogonal group is SL2�SL2: Prasad then showed that for formson PGSp2 that are lifts from the orthogonal group of a 4-dimensionalspa e, the situation an be understood in terms of the seesaw dualredu tive pair (in Kudla's sense)GSp2RRRRRRRRRRRRRRRR

GO(4) �GO(4)G(SL2 � SL2) llllllllllllll GO(4) :In lassi al terms, the analogous global question leads to the problemto determine those pairs of uspidal ellipti modular forms (eigenformsof almost all He ke operators) that an o ur as summands if one de- omposes the restri tion of a uspidal Siegel modular form F of degree2 (that is an eigenform of almost all He ke operators) to the diagonallyembedded produ t of two upper half planes into a sum of (produ tsof) eigenforms of almost all He ke operators (for a dis ussion of theproblems that arise in translating the representation theoreti state-ment into a lassi al statement see below and Remark 2.13. One analso rephrase this as the problem of al ulating the period integralZ(�nH)�(�nH) F ��z1 00 z2�� f1(z1)f2(z2)d�z1d�z2for two ellipti He ke eigenforms f1; f2: The L-fun tion that should o - ur then a ording to the onje ture of Gross and Prasad is the degree16 L-fun tion asso iated to the tensor produ t of the 4-dimensional rep-resentation of the L-group Spin(4) of SO(5) with the two 2-dimensionalrepresentation asso iated to two opies of SO(3) (due to the de ompo-sition of SO(4) or rather its overing group mentioned above). Wedenote this L-fun tion as L(Spin(F ); f1; f2; s):In this reformulation it is natural to go beyond the original questionof nonvanishing and to try and get an expli it formula onne ting theL-value in question with the period integral. In general, it seems ratherdiÆ ult to al ulate the period integral as above, sin e little is knownabout the restri tion of Siegel modular forms to the diagonally embed-ded produ t of two upper half planes. One should also point out thatan integral representation for the degree 16 L-fun tion in question isnot known yet. However, for theta series of quadrati forms the re-stri tion to the diagonal of a degree two theta series be omes simplythe produ t of the degree one theta series in the variables z1; z2; whi hallows one to get a al ulation started. Thus here we only onsider

GLOBAL GROSS-PRASAD CONJECTURE FOR YOSHIDA LIFTINGS 3Siegel modular forms (of trivial hara ter) that arise as linear om-binations of theta series of quaternary quadrati forms. Su h Siegelmodular forms, if they are eigenforms, are alled Yoshida liftings, at-ta hed to a pair of ellipti usp forms or, equivalently, to a pair ofautomorphi forms on the multipli ative group of an adeli quaternionalgebra. These liftings have been investigated in [32, 4, 8℄, and the onne tion between the trilinear forms on the spa es of automorphi forms on the multipli ative group of an adeli quaternion algebra andthe triple produ t L-fun tion has been investigated in [20, 15, 7℄. Ifone ombines these results and applies them to the present situation,it turns out that the period integral in question an indeed be expli -itly al ulated in terms of the entral riti al value of the L-fun tionmentioned; in the ase of a Yoshida-lifting atta hed to the pair h1; h2of ellipti usp forms, this L-fun tion is seen to split into the produ tL(h1; f1; f2; s)L(h2; f1; f2; s); so that the entral riti al value be omesthe produ t of the entral riti al values of these two triple produ t L-fun tions. We prove a formula that expresses the square of the periodintegral expli itly as the produ t of these two entral riti al values,multiplied by an expli itly known non-zero fa tor. We reformulate theobtained identity in a way whi h makes sense as well for an arbitrarySiegel modular form F in terms of the original L(Spin(F ); f1; f2; s),in pla e of L(h1; f1; f2; s)L(h2; f1; f2; s), hoping that su h an identityindeed holds for any Siegel modular form F . At present we annotprove it ex ept for the ase when F is a Siegel or Klingen Eisensteinseries of level 1: In the ase when F is the Saito-Kurokawa lifting of anellipti He ke eigenform h, one sees easily that the period integral iszero unless one has f1 = f2 = f ; in this ase the period integral an betransformed into the Petersson inner produ t of the restri tion of the�rst Fourier Ja obi oeÆ ient of F to the upper half plane with f andthen leads us to a onje tural identity for the square of this Peterssoninner produ t with the entral riti al value of L(h; f; f ; s):Our al ulation leaves the question open whether it an happen thatthe period integral vanishes for the lassi al modular forms onsideredbut is non-zero for other fun tions in the same adeli representationspa e. Viewed lo ally, this amounts to the question whether an in-variant nontrivial linear fun tional on the lo al representation spa e isne essarily non-zero at the given ve tor. At the in�nite pla e we anexhibit su h a ve tor (depending on the weights given) by applying asuitable di�erential operator to the Siegel modular form onsidered. Atthe �nite pla es not dividing the level, it omes down to the questionwhether (for an unrami�ed representation) a nontrivial invariant lin-ear fun tional is ne essarily non-zero at the spheri al (or lass 1) ve tor

4 S. B�OCHERER, M. FURUSAWA, AND R. SCHULZE-PILLOTinvariant under the maximal ompa t subgroup. This is generally ex-pe ted, at least for generi representations. We intend to ome ba k tothis question in future work.We also investigate the situation where the pair f1; f2 and the prod-u t H �H are repla ed by a Hilbert modular form and the modularembedding of a Hilbert modular surfa e; in terms of the Gross-Prasad onje ture this amounts to repla ing the split orthogonal group of a 4-dimensional spa e from above by a non split (but quasisplit) orthogonalgroup that is split at in�nity. It turns out that one gets an analogousresult; we prove this only in the simplest ase when all modular formsinvolved have weight 2; the lass number of the quadrati �eld involvedis 1 and the order in a quaternion algebra belonging to the situation isa maximal order. The proof for the general ase should be possible inan analogous manner.1. Yoshida liftings and their restri tion to the diagonalFor generalities on Siegel modular forms we refer to [Fre1℄. For asymple ti matrix M = � A BC D � 2 GSpn(R) (with n � n-blo ksA;B;C;D) we denote by (M;Z) 7!M < Z >= (AZ +B)(CZ+D)�1the usual a tion of the group G+Sp(n;R) of proper symple ti simili-tudes on Siegel's upper half spa e Hn.We shall mainly be on erned with Siegel modular forms for ongruen esubgroups of type�(n)0 (N) = �� A BC D � 2 Sp(n;Z) j C � 0 mod N� :The spa e of Siegel modular forms (and usp forms respe tively) ofdegree n and weight k for �(n)0 (N) will be denoted byMkn (N) (Skn(N));for a ve tor valued modular form transforming a ording to the repre-sentation � the weight k above should be repla ed by �: By < ; > wedenote the Petersson s alar produ t.We re all from [4, 8, 5℄ some notations on erning the Yoshida-liftingswhose restri tions we are going to study in this arti le. For details werefer to the ited arti les. We onsider a de�nite quaternion algebraD over Q and an Ei hler order R of square free level N in it andde ompose N as N = N1N2 where N1 is the produ t of the primesthat are rami�ed in D. On D we have the involution x 7! x, the(redu ed) tra e tr(x) = x+ x and the (redu ed) norm n(x) = xx.

GLOBAL GROSS-PRASAD CONJECTURE FOR YOSHIDA LIFTINGS 5The group of proper similitudes of the quadrati form q(x) = n(x) onD is isomorphi to (D� �D�)=Z(D�) (as algebrai group) via(x1; x2) 7! �x1;x2 with �x1;x2(y) = x1yx�12 ;the spe ial orthogonal group is then the image off(x1; x2) 2 D� �D� j n(x1) = n(x2)g:We denote by H the orthogonal group of (D;n) and by H+ the spe ialorthogonal group.For � 2 N let U (0)� be the spa e of homogeneous harmoni polynomialsof degree � on R3 and view P 2 U (0)� as a polynomial onD(0)1 = fx 2 D1jtr(x) = 0gby putting P ( 3Xi=1 xiei) = P (x1; x2; x3)for an orthonormal basis feig of D(0)1 with respe t to the norm form n.The spa e U (0)� is known to have a basis of rational polynomials (i.e.,polynomials that take rational values on ve tors in D(0) = D(0)1 \D).The group D�1=R� a ts on U (0)� through the representation �� (of high-est weight (�)) given by(��(y))(P )(x) = P (y�1xy):Changing the orthonormal basis above amounts to repla ing P by(��(y))(P ) for some y 2 D�1:By hh ; ii0 we denote the suitably normalized invariant s alar prod-u t in the representation spa e U (0)� .For �1 � �2 the H+(R)-spa eU (0)�1 U (0)�2(irredu ible of highest weight (�1 + �2; �1 � �2)) is isomorphi to theH+(R)-spa e U�1;�2 of C[X1;X2℄-valued harmoni forms on D21 trans-forming a ording to the representation of GL2(R) of highest weight(�1 + �2; �1 � �2):An intertwining map has been given expli itly in [5, Se tion 3℄; forx = (x1; x2) 2 D21 the polynomial (Q)(x) 2 C[X1;X2℄ is homoge-neous of degree 2�2: We write now for Q 2 U (0)�1 U (0)�2(1.1) (Q)(x) = X�1+�2=2�2 �1�2(x; Q)X�11 X�22 :

6 S. B�OCHERER, M. FURUSAWA, AND R. SCHULZE-PILLOTThe map x 7! �1�2(x; Q) is (for �xed Q) a polynomial in x1; x2 thatis harmoni of degree �01 = �1 + �1 � �2 in x1 and harmoni of degree�02 = �2 + �1 � �2 in x2; and for h 2 H+(R) we have �1�2(hx; Q) = �1�2(x; h�1Q):The irredu ibility of the spa e U (0)�1 U (0)�2 implies that this map isnonzero for some Q: We denote by U� the spa e of harmoni polyno-mials of degree � on D1 with invariant s alar produ t hh ; ii. If �is even, the H(R)-spa es U� and U (0)�=2 U (0)�=2 are isomorphi and willbe identi�ed.The map(1.2) (Q;R1; R2) 7! hh �1�2(�; Q); R1 R2iifor Q 2 U (0)�1 U (0)�2 ; R1 2 U�01; R2 2 U�02 de�nes then a nontrivial invari-ant trilinear form for the triple of H(R)-spa es ((U (0)�1 U (0)�2 ); U�01; U�02):Lemma 1.1. Let integers �1 � �2 and �1; �2 be given for whi h� 01 = �1 + �1 � �2; � 02 = �2 + �1 � �2are even. Then there exists a nontrivial H(R)- invariant trilinear formT on the spa e U�1;�2 U�01 U�02 if and only if there exist integers�1; �2; su h that �i = �i + and �1 + �2 = 2�2 holds. This form isunique up to s alar multiples and an be de omposed asT = T (0)1 T (0)2with (up to s alars) unique nontrivial invariant trilinear formsT (0)i = T (0)i;�01;�02on U (0)�i U (0)�01=2 U (0)�02=2:In parti ular, for = 0 and T �xed, the trilinear form given in (1.2)is proportional to T (with a nonzero fa tor ~ (�1; �2; �1; �2)).Proof. De omposingU�01 = U (0)�01=2 U (0)�01=2; U�02 = U (0)�02=2 U (0)�02=2as a D�1=R� �D�1=R�-spa e one sees that T as asserted exists if andonly if there are nontrivial invariant trilinear formsT (0)i = T (0)i;�01;�02on U (0)�i U (0)�01=2 U (0)�02=2

GLOBAL GROSS-PRASAD CONJECTURE FOR YOSHIDA LIFTINGS 7for i = 1; 2: In this ase T de omposes asT = T (0)1 T (0)2 :The T (0)i are known to exist if and only if the triples(� 01=2; � 02=2; �1); (� 01=2; � 02=2; �2)are balan ed, i.e, the numbers in either triple are the lengths of thesides of a triangle (and then the form is unique up to s alars); theyare unique up to s alar multipli ation. It is then easily he ked thatthe numeri al ondition given above is equivalent to the existen e ofnonnegative integers �1; �2; satisfying �i = �i+ and �1+�2 = 2�2:Consider now the Gegenbauer polynomial G(�)(x; x0) = obtained fromG(�)1 (t) = 2� [�2 ℄Xj=0 (�1)j 1j!(�� 2j)! (� � j)!22j t��2jby ~G(�)(x; x0) = 2�(n(x)n(x0))�=2G(�)1 ( tr(xx0)2pn(x)n(x0))and normalize the s alar produ t on U� su h that G(�) is a reprodu ingkernel, i. e. hhG(�)(x; x0); Q(x)ii� = Q(x0)for all Q 2 U�. Then for �1; �2; �01; �02 as above and some �xed Q 2U�1;�2 the map (x1; x2) 7! T (Q;G(�01)(x1; �); G(�02)(x2; �))de�nes a polynomial RQ(x1; x2) in x1; x2 that is harmoni of degree�01 = �1 + �1 � �2 in x1 and harmoni of degree �02 = �2 + �1 � �2 inx2; and for h 2 H+(R) we have RQ(hx) = Rh�1Q(x):As above we an therefore on lude that one has(1.3) �1�2(x; Q) = ~ (�1; �2; �1; �2)T (Q;G(�01)(x1; �); G(�02)(x2; �);where the fa tor of proportionality ~ (�1; �2; �1; �2) is not zero.We denote by A(D�A; R�A; �)the spa e of fun tions ' : D�A ! U (0)� satisfying '( xu) = ��(u�11 )'(x)for 2 D�Q and u = u1uf 2 R�A, whereR�A = D�1 �Yp R�p

8 S. B�OCHERER, M. FURUSAWA, AND R. SCHULZE-PILLOTis the adeli group of units of R. These fun tions are determined bytheir values on the representatives yi of a double oset de ompositionD�A = [ri=1D�yiR�A(where we hoose the yi to satisfy yi;1 = 1 and n(yi) = 1):The natural inner produ t on the spa e A(D�A; R�A; �) is given byh'; i = rXi=1 hh'(yi); (yi)ii0ei ;where ei =j (yiRy�1i )� j is the number of units of the order Ri = yiRy�1iof D:On the spa e A(D�A; R�A; �) we have for p 6 jN (hermitian) He ke oper-ators ~T (p) (given expli itly by the End(U (0)� )-valued Brandt matri es(Bij(p))) and for p j N involutions fwp ommuting with the He ke op-erators and with ea h other.For i = 1; 2 and �1 � �2 with �1 � �2 even we onsider now fun tions'i in A(D�A; R�A; �i):The Yoshida lifting (of degree 2) of the pair ('1; '2) is then given as(1.4) Y (2)('1; '2)(Z)(X1;X2) == rXi;j=1 1eiej X(x1;x2)2(yiRy�1j )2('1(yi) '2(yj))(x1; x2)(X1;X2)�� exp(2�itr�� n(x1) tr(x1x2)tr(x1x2) n(x2) �Z� :This is a ve tor valued holomorphi Siegel modular form for the group�(2)0 (N) with trivial hara ter and with respe t to the representation�2�2 det�1��2+2 (where �2�2 denotes the 2�2-th symmetri power rep-resentation of GL2).If we onsider the restri tion of su h a modular form to the diagonal� z1 00 z2 � ; the oeÆ ient of X�11 X�22 be omes a fun tion F (�1;�2)(z1; z2)whi h is in both variables a s alar valued modular form for the group�0(N) with trivial hara ter of weight�1 + �1 � �2 + 2 in z1; �2 + �1 � �2 + 2 in z2:In parti ular the weights in the variables z1; z2 add up to 2�1 + 4 forea h pair (�1; �2) with �1 + �2 = 2�2 and the oeÆ ient of X�11 X�22vanishes unless �01 = �1 + �1 � �2; �02 = �2 + �1 � �2

GLOBAL GROSS-PRASAD CONJECTURE FOR YOSHIDA LIFTINGS 9are even.If f1; f2 are ellipti modular forms of weights k1; k2 we de�ne thenhF �� z1 00 z2 �� ; f1(z1)f2(z2)ik1;k2to be the double Petersson produ thhF (�1;�2)(z1; z2); f1(z1)ik1 ; f2(z2)ik2if k1 = �1 + �1 � �2 + 2; k2 = �2 + �1 � �2 + 2for some �1; �2 with �1 + �2 = 2�2 and to be zero otherwise (thisde�nition oin ides with the Petersson produ t of the orrespondingautomorphi forms on the groups Sp2(A) or Sp2(R) (restri ted to thenaturally embedded SL2(A)�SL2(A)) and SL2(A) (or the respe tivereal groups)).We will mainly onsider Yoshida liftings for pairs of forms that areeigenforms of all He ke operators and of all the involutions. It is theneasy to see that Y (2)('1; '2)(Z) is identi ally zero unless '1; '2 havethe same eigenvalue under the involution fwp for all p j N . The pre ise onditions under whi h the lifting is nonzero have been stated in [8℄.We will �nally need some fa ts about the orresponden e studied e.g.in [10, 22, 29, 24℄ between modular forms for �0(N) (with trivial har-a ter) and automorphi forms on the adeli quaternion algebra D�A:We onsider the essential partAess(D�A; R�A; �) onsisting of fun tions ' that are orthogonal to all 2 A(D�A; (R0A)�; �)for orders R0 stri tly ontaining R; this spa e is invariant under the~T (p) for p 6 jN and the ~wp for p j N and hen e has a basis of ommoneigenfun tions of all the ~T (p) for p 6 jN and all the involutions ~wp forp j N . Being the omponents of eigenve tors of a rational matrix withreal eigenvalues the values of these eigenfun tions are real, (i.e., poly-nomials with real oeÆ ients in the ve tor values ase) when suitablynormalized.Moreover the eigenfun tions are in one to one orresponden e with thenewforms in the spa e S2+2�(N)of ellipti usp forms of weight 2 + 2� for the group �0(N) that areeigenfun tions of all He ke operators (if � is the trivial representationand R is a maximal order one has to restri t here to fun tions or-thogonal to the onstant fun tion 1 on the quaternion side in order toobtain usp forms on the modular forms side). This orresponden e

10 S. B�OCHERER, M. FURUSAWA, AND R. SCHULZE-PILLOT(Ei hler's orresponden e) preserves He ke eigenvalues for p 6 jN , andif ' orresponds to f 2 S2+2�(N) then the eigenvalue of f under theAtkin-Lehner involution wp is equal to that of ' under ~wp if D splits atp and equal to minus that of ' under ~wp ifDp is a skew �eld. The orre-sponden e an be expli itly des ribed by asso iating to ' the modularform h(z) = rXi;j=1 1eiej Xx2(yiRy�1j )('(yi) '(yj))(x) exp(2�in(x)z)(where as above '(yi)'(yj) denotes the harmoni polynomial in U2�obtained by identifying U (0)� U (0)� with U2�:)An extension of Ei hler's orresponden e to forms ' as above that arenot essential but eigenfun tions of all the involutionsfwp has been givenin [21, 7℄. 2. Computation of periodsOur goal is the omputation of the periodshY (2)('1; '2) �� z1 00 z2 �� ; f1(z1)f2(z2)ik1;k2de�ned above for ellipti modular forms f1; f2 for the group �0(N): Forthis we study �rst how the vanishing of this period integral depends onthe eigenvalues of the fun tions involved under the Atkin-Lehner invo-lutions or their quaternioni and Siegel modular forms ounterparts.For a Siegel modular form F for the group �(2)0 (N) we let the Atkin-Lehner involutions with respe t to the variables z1; z2 a t on the restri -tion of F to the diagonal matri es � z1 00 z2 � and denote by F �� z1 00 z2 �� jfWpthe result of this a tion.Lemma 2.1. Let N be squarefree and let F be a ve tor valued Siegelmodular form of degree 2 for the representation � of GL2(C) of highestweight (�1; �2) in the spa e C[X1;X2℄�1��2 of homogeneous polynomialsof degree �1 � �2 in X1;X2 with respe t to �(2)0 (N) and assume forp j N that the restri tion of F to the diagonal is an eigenform of fWpwith eigenvalue ~�(0)p . Let f1; f2 be ellipti usp forms of weights k1; k2for �0(N) that are eigenforms of the Atkin-Lehner involution wp witheigenvalues �(1)p ; �(2)p . Then the period integral(2.1) hF �� z1 00 z2 �� ; f1(z1)f2(z2)ik1;k2is zero unless one has ~�(0)p �(1)p �(2)p = 1:

GLOBAL GROSS-PRASAD CONJECTURE FOR YOSHIDA LIFTINGS 11Proof. Applying the Atkin-Lehner involution wp to both variables z1; z2one sees that this is obvious.We an view the ondition of Lemma 2.1 as a (ne essary) lo al on-dition for the nonvanishing of the period integral at the �nite primesdividing the level, with a similar role being played at the in�nite primesby the ondition that modular forms of the weights k1; k2 of f1; f2 ap-pear in the de omposition of the restri tion of the ve tor valued mod-ular form F to the diagonal (or of a suitable form in the representationspa e of F , see below).Lemma 2.2. Let f1; f2; h1; h2 be modular forms for �0(N) that areeigenfun tions of all Atkin-Lehner involutions for the p j N with f1; f2 uspidal. Let h1; h2 have the same eigenvalue �0p for all the wp for thep j N and denote by �(1)p ; �(2)p the Atkin-Lehner eigenvalues at p j N off1; f2:For a fa torization N = N1N2 where N1 has an odd number of primefa tors let DN1 be the quaternion algebra over Q that is rami�ed pre- isely at 1 and the primes p j N1 and RN1 an Ei hler order of level Nin DN1:Let '(N1)1 ; '(N1)2 be the forms in A((DN1)�A; (RN1)�A; �i)(i = 1; 2) orre-sponding to h1; h2 under Ei hler's orresponden e.Then the period integral(2.2) hY (2)('(N1)1 ; '(N1)2 )��z1 00 z2�� ; f1(z1)f2(z2)ik1 ;k2is zero unless �0p�(1)p �(2)p = �1 holds for pre isely those p that divide N1;in parti ular it is always zero unless QpjN �0p�(1)p �(2)p = �1 holds.Proof. For ea h fa torization of N as above we denote by e�p(N1) theeigenvalue under fwp of '(N1)1 ; '(N1)2 ; we have e�p(N1) = ��0p for the pdividing N1 and e�p(N1) = �0p for the p dividing N2. Hen e the produ te�p(N1)�(1)p �(2)p is 1 for all p dividing N if N1 is the produ t of the primesp j N su h that �0p�(1)p �(2)p = �1 and is �1 for at least one p j Notherwise; in parti ular a de omposition for whi h e�p(N1)�(1)p �(2)p = 1for all p j N holds and N1 has an odd number of prime fa tors exists ifand only if we have QpjN e�p(N1)�(1)p �(2)p = �1:The fWp-eigenvalue of the restri tion of Y (2)('(N1)1 ; '(N1)2 ) to the diag-onal is e�p(N1) by the result of Lemma 9.1 of [4℄ on the eigenvalue ofY (2)('(N1)1 ; '(N1)2 ) under the analogue for Siegel modular forms of the

12 S. B�OCHERER, M. FURUSAWA, AND R. SCHULZE-PILLOTAtkin-Lehner involution. The assertion then follows from the previouslemma.For simpli ity we will in the sequel assume that h1; h2; f1; f2 are allnewforms of (square free) level N ; essentially the same results an beobtained for more general quadruples of forms of square free level usingthe methods of [7℄.Lemma 2.3. Let N 6= 1 be squarefree, D;R as des ribed in Se tion 1,let f1; f2 be normalized newforms of weights k1; k2 for the group �0(N).Let '1; '2 2 A(D�A; R�A; �i) be as above. Assume that the (even) weightsk1; k2 of f1; f2 an be written as ki = �i + �1 � �2 + 2 = �0i + 2 withnonnegative integers �i satisfying �1+�2 = 2�2 and denote for i = 1; 2by i the U�0i=2-valued form in A(D�A; R�A; ��0i) orresponding to fi underEi hler's orresponden e.Then the period integral(2.3) hY (2)('1; '2)��z1 00 z2�� ; f1(z1)f2(z2)ik1 ;k2has the (real) value(2.4) hf1; f1ihf2; f2i( rXj=1 T (0)�1;�01;�02('1(yi) 1(yi) 2(yi)))� ( rXj=1 T (0)�2;�01;�02('2(yi) 1(yi) 2(yi)));with a nonzero onstant depending only on �1; �2; k1; k2:Proof. The oeÆ ient of X�11 X�22 of the (i; j)-term in F (z1; z2) =Y (2)('1; '2) � z1 00 z2 � is (with Qij := '1(yi) '2(yj)) equal to(2.5) ~ (�1; �2; �1; �2) X(x1;x2)2Iij T (Qij; G(�01)(x1; �); G(�02)(x2; �))� exp(2�in(x1)z1) exp(2�in(x2)z2)by (1.3). We write(2.6) �(�0)ij (z)(x0) = Xx2IijG(�0)(x; x0) exp(2�in(x)z)for the U�0-valued theta series atta hed to Iij and the Gegenbauerpolynomial G(�)(x; x0) and rewrite (2.5) as(2.7) ~ (�1; �2; �1; �2)T (Qij;�(�01)ij (z1);�(�02)ij (z2)):

GLOBAL GROSS-PRASAD CONJECTURE FOR YOSHIDA LIFTINGS 13The ij-term of the period integral (2.3) be omes then(2.8) ~ (�1; �2; �1; �2)T (Qij; h�(�01)ij (z1); f1(z1)i; h�(�02)ij (z2); f2(z2)i;whi h by (3.13) of [6℄ and the fa torization(2.9) T = T�1;�2;�01;�02 = T (0)�1;�01;�02 T (0)�2;�01;�02 = T (0)1 T (0)2is equal to(2.10) ~ (�1; �2; �1; �2)hf1; f1ihf2; f2iT (0)1 ('1(yi); 1(yi); 2(yi))� T (0)2 ('2(yj); 1(yj); 2(yj)):Summation over i; j proves the assertion. The value omputed is realsin e the values of the 'i; i are so and sin e T0 is known to be real.Theorem 2.4. Let h1; h2; f1; f2; 1; 2 be as in Lemma 2.2, Lemma2.3 with Atkin-Lehner eigenvalues �0p for h1; h2 and �(1)p ; �(2)p for f1; f2;assume QpjN �0p�(1)p �(2)p = �1. Let D be the quaternion algebra over Qwhi h is rami�ed pre isely at the primes p j N for whi h �0p�(1)p �(2)p = �1holds and R an Ei hler order of level N in D; let '1; '2 be the formsin A(D�A; R�A; �1;2) orresponding to h1; h2 under Ei hler's orrespon-den e.Then the square of the period integral(2.11) hY (2)('1; '2)��z1 00 z2�� ; f1(z1)f2(z2)ik1 ;k2is equal to(2.12) hh1; h1ihh2; h2iL(h1; f1; f2; 12)L(h2; f1; f2; 12);where is an expli itly omputable nonzero number depending only on�1; �2; k1; k2; N and the triple produ t L-fun tion L(h; g; f ; s) is nor-malized to have its fun tional equation under s 7! 1� s:In parti ular the period integral is nonzero if and only if the entral riti al value of L(h1; f1; f2; s)L(h2; f1; f2; s) is nonzero.Proof. The hoi e of the de omposition N = N1N2 made above impliesthat we an use Theorem 5.7 of [7℄ to express the right hand side of(2.4) by the produ t of entral riti al values of the triple produ t L-fun tions asso iated to (h1; f1; f2); (h2; f1; f2): The Petersson norms off1; f2 appearing in Theorem 5.7 of [7℄ an el against those appearingin the proof of Lemma 2.3.

14 S. B�OCHERER, M. FURUSAWA, AND R. SCHULZE-PILLOTRemark 2.5. a) If the produ tQpjN �0p�(1)p �(2)p is +1 we know from[7℄ that the sign in the fun tional equation of the triple produ tL-fun tions L(h1; f1; f2; s); L(h2; f1; f2; s) is �1 and hen e the entral riti al values are zero; from Lemma 2.2 we know thatfor any Yoshida lifting F asso iated to h1; h2 as in Lemma 2.2the Petersson produ t of the restri tion of F to the diagonal andf1(z1); f2(z2) is zero as well.b) It should be noti ed that given h1; h2 there are 2!(N)�1 possible hoi es of the quaternion algebra with respe t to whi h one on-siders the Yoshida lifting asso iated to h1; h2: All these Yoshidaliftings are di�erent, but have the same Satake parameters forall p - N: Given f1; f2 with QpjN �0p�(1)p �(2)p = �1 there is thenpre isely one hoi e of quaternion algebra that leads to a non-trivial result for the period integral, all the others give automat-i ally zero by Lemma 2.2. The hoi e of this quaternion algebrashould be seen as variation of the Vogan L-pa ket of the p-adi omponent of the adeli representation generated by the Siegelmodular form for the p j N in su h a way that the resultingL-pa ket satis�es the lo al Gross-Prasad ondition for the split5-dimensional and 4-dimensional orthogonal groups.We want to rephrase the result of Theorem 2.4 in order to repla e thefa tor of omparison hh1; h1ihh2; h2i o urring by a fa tor dependingonly on F = Y (2)('1; '2) instead of h1; h2: Con erning the symmetri square L-fun tion of F o urring in the following orollary we remindthe reader that we view F as as an automorphi form on the adeli or-thogonal group of the 5-dimensional quadrati spa e V of dis riminant1 over Q that ontains a 2-dimensional totally isotropi subspa e.Corollary 2.6. Under the assumptions of Theorem 2.4 and the addi-tional assumption that h1; h2 are not proportional, the value of (2.12)is equal to:(2.13) hF;F iL(N)(F;Sym2; 1)L(h1; f1; f2; 12)L(h2; f1; f2; 12);where again is an expli itly omputable nonzero onstant dependingonly on the levels and weights involved and L(N)(F;Sym2; s) is the N-free part of the L fun tion of F with respe t to the symmetri squareof the 4 dimensional representation of the L-group of the group SO(V )(V as above).Proof. Sin e h1; h2 are not proportional, the Siegel modular form Fis uspidal and the Petersson produ t hF;F i is well de�ned. From [4,

GLOBAL GROSS-PRASAD CONJECTURE FOR YOSHIDA LIFTINGS 15Proposition 10.2℄ we re all that hF;F i is (up to a nonzero onstant)equal to the the residue at s = 1 of theN -free partD(N)F (s) of the degree5 L-fun tion asso iated to F (normalizing the 'i to h'i; 'ii = 1; theformulas given in [4℄ generalize easily to the situation where the 'i takevalues in harmoni polynomials). It is also well known that hhi; hii isequal (up to a nonzero onstant depending only on weights and levels)toD(N)hi (1) whereD(N)hi (s) is the symmetri square L-fun tion asso iatedto hi. Comparing the parameters of the L-fun tions L(N)(F;Sym2; s)and D(N)F (s)D(N)h1 (s)D(N)h2 (s) we see that the value of L(N)(F;Sym2; s)at s = 1 is equal to the residue at s = 1 of D(N)F (s)D(N)h1 (s)D(N)h2 (s);whi h gives the assertion.Remark. We an as well view L(N)(F;Sym2; s) as the exterior squareof the degree 5 L-fun tion asso iated to F .Let us dis uss now two degenerate ases:Corollary 2.7. a) Under the assumptions of Theorem 2.4 repla e'2 by the onstant fun tion (Pi 1ei )�1 (and hen e h2(z) by theEisenstein series E(z) = (Pi;j 1eiej )�1Pi;j 1eiej�(0)ij (z)): Thenthe period integral(2.14) hY (2)('1; '2)��z1 00 z2�� ; f1(z1)f2(z2)ik1 ;k2is zero unless f1 = f2 =: f; in whi h ase its square is equal tothe value at s = 1 of(2.15) hF;F iL(N)(F;Sym2; s)L(h1; f; f ; s� 12)L(E; f; f ; s� 12);where again is an expli itly omputable nonzero onstant de-pending only on the levels and weights involved and L(E; f1; f2; s)is de�ned in the same way as the triple produ t L-fun tion fora triple of usp forms, setting the p-parameters of E equal top1=2; p�1=2 for p - N:b) Under the assumptions of Theorem 2.4 let h = h1 = h2: Thenthe period integral (2.11) is equal to(2.16) hh; hiL(h; f1; f2; 12);where is a nonzero onstant depending only on the levels andweights involved.

16 S. B�OCHERER, M. FURUSAWA, AND R. SCHULZE-PILLOTProof. Both assertions are obtained in the same way as Theorem 2.4and Corollary 2.6; noti e that in ase a) (with !(N) denoting the num-ber of prime fa tors of N) both L(E; f; f ; s� 12) and L(N)(F;Sym2; s)are of order !(N)� 1 at s = 1:

GLOBAL GROSS-PRASAD CONJECTURE FOR YOSHIDA LIFTINGS 17Remarka) The form of our result given in Corollary 2.6 ould in prin iplebe true for any Siegel modular form F instead of a Yoshidalifting if one repla es L(h1; f1; f2; 12)L(h2; f1; f2; 12) by the valueL(Spin(F ); f1; f2; 12) of the spin L-fun tion mentioned in the in-trodu tion. There is, however, not mu h known about the an-alyti properties of L(N)(F;Sym2; s); in parti ular this L- fun -tion might have a zero or a pole at s = 1:b) In the degenerate ase of Corollary 2.7 a) the Yoshida liftingF is the Saito-Kurokawa lifting asso iated to h1: The result ofthat ase ould also be true in the ase that h1; f1; f2 are oflevel 1 and F is the Saito-Kurokawa lifting of h1; but we annot prove this at present (ex ept for the vanishing of the periodintegral in the ase that f1 6= f2, whi h is easily proved).We noti e that in that last ase (as well as in the related aseof a Yoshida lifting of Saito-Kurokawa type) the period integralis seen to be equal to the Petersson produ t h�1(�; 0); f(� )i;where �1(�; z) is the �rst Fourier Ja obi oeÆ ient of F:In the ase of Corollary 2.7 b) the Yoshida lift F an beviewed as an Eisenstein series of Klingen type asso iated to h;in parti ular its image under Siegel's �-operator is equal to h(more pre isely, the Klingen Eisenstein series in question is asum of Yoshida liftings asso iated to various quaternion alge-bras of level dividing N (see [6℄), where the other ontributionsyield a vanishing period integral). One he ks that the result ofCorollary 2.7 b) is valid also for F denoting the Klingen Eisen-stein series atta hed to a uspidal normalized He ke eigenformh of level 1 and f1; f2 two uspidal normalized He ke eigenformsof level 1:We an obtain a result similar to that of Theorem 2.4 for more generalweights k1; k2 of the modular forms f1; f2: For this, remember thata ording to Lemma 1.1 the value given in (2.4) for the period integralin question also makes sense if one repla es throughout �01; �02 by � 01 =�01 + ; � 02 = �02 + for some �xed > 0; the forms f1; f2 then havingweights ki = �0i + 2 + for i = 1; 2: As noti ed above, our periodintegral be omes 0 in this situation. We an, however, modify thefun tion Y (2)('1; '2)(Z) by a di�erential operator ~D 2;�1;�2 in su h away that ~D 2;�1;�2Y (2)('1; '2) is a fun tion on H�H that is a modularform of weights k1; k2 of z1; z2 as des ribed above and yields a value forthe period integral of the same form as the one given in form (2.4).More pre isely, we have:

18 S. B�OCHERER, M. FURUSAWA, AND R. SCHULZE-PILLOTProposition 2.8. For nonnegative integers k, r and l with k � 2 andany partition l = a+ b, there exists a (non-zero) holomorphi di�eren-tial operator Drk;a;b (polynomial in X21 ��z1 ;X1X2 ��z12 ;X22 ��z2 , evaluated inz12 = 0) mapping C[X1;X2℄l-valued fun tions on H2 to C �Xa+r1 Xb+r2 -valued fun tions on H�H and satisfyingDrk;a;b �F jk;l M"1M#2� = (Drk;a;bF ) jz1k+a+r M1 jz2k+b+r M2for all M1; M2 2 SL(2;R); here the upper indi es z1 and z2 at theslash-operator indi ate the variable, with respe t to whi h one has toapply the elements of SL(2;R) and "# denote the standard embeddingof SL(2) � SL(2) into Sp(2) given by� a b d �" �� A BC D �# = 0BB� a 0 b 00 A 0 B 0 d 00 C 0 D 1CCAOf ourse one an onsider Drk;a;bF as a C-valued fun tion.Remark 2.9. One may indeed show, that there exists (up to multipli- ation by a onstant) pre isely one su h nontrivial holomorphi di�er-ential operator.Corollary 2.10. The di�erential operator Drk;a;b de�ned above givesrise to a map~Drk;a;b :M2k;l(�20(N)) �!M1k+a+r(�10(N))M1k+b+r(�10(N))of spa es of modular forms. (It is easy to see that for r > 0 this mapa tually goes into spa es of usp foms).Corollary 2.11. Denoting by Sym2(C) the spa e of omplex symmet-ri matri es of size 2 we de�ne a polynomial fun tionQ : Sym2(C) �! C[X1;X2℄2rby Drk;a;betr(TZ) = Q(T )et1z1+t2z2where Z = � z1 z3z3 z2 � 2 H2.Furthermore we assume (with k = m2 +�) that P : C(m;2) �! C[X1;X2℄lis a polynomial fun tion satisfyinga) P is pluriharmoni

GLOBAL GROSS-PRASAD CONJECTURE FOR YOSHIDA LIFTINGS 19b) P ((X1;X2)A) = ��;l(A)P (X1;X2) for all A 2 GL(2;C).Then, for Y1;Y2 2 Cm(Y1;Y2) 7�! �P (Y1;Y2) �Q(� Yt1Y1 Yt1Y2Yt2Y1 Yt2Y2 �)�a+r+�;b+r+�de�nes an element of Ha+r+� (m) Hb+r+� (m), where H�(m) is thespa e of harmoni polynomials in m variables (for the standard qua-drati form), homogeneous of degree � and for any R 2 C[X1;X2℄l+2rwe denote by fRg�;� the oeÆ ient of X�1X�2 in R, �+ � = l + 2r.The proof of Corollary 2.11 is a ve tor-valued variant of similar state-ments in [2℄ and [9, p.200℄, using the proposition above and the har-a terization of harmoni polynomials by the Gau�-transform; we leavethe details of proof to the reader.Proof of Proposition 2.8.We start from a Maa�-type di�erential operator Æk+l whi h mapsC[X1;X2℄l-valued fun tions on H2 to C[X1;X2℄l+2-valued ones and satis�es(Æk+lF ) jk;l+2 M = Æk+l (F jk;l M)for all M 2 Sp(2;R).It is well known, how su h operators arise from elements of the universalenveloping algebra of the omplexi�ed Lie algebra of Sp(2;R), see e.g.[19℄ . In our ase (we refer to [3℄ for details) we an des ribe theseoperators quite expli itly in terms of the simple operatorsDF := � 12�i ��Z� [X℄and NF := �� 14� (ImZ)�1F� [X℄Here X stands for the olumn ve tor � X1X2 �. Then we de�neÆkF = kNF +DFIt is remarkable (and already in orporated in our notation!) that Æk+ldepends only on k + l.The iteration Ærk+l := Æk+l+2r�2 Æ � � � Æ Æk+l+2 Æ Æk+l an also be des ribed expli itly byÆrk+l = rXi=0 �(k + l + r)�(k + l + r � i)�ri�N iDr�i

20 S. B�OCHERER, M. FURUSAWA, AND R. SCHULZE-PILLOTFor a fun tion F : H2 �! C[X1;X2℄l and a de omposition l = a + bwe putrrk+l(a; b)F =: Xa+r1 Xb+r2 � oeÆ ient of �Ær2;k+lF �jH�HThenr has already the transfomation properties required in the propo-sition, i.e.rrk+l(a; b)�F jk;l M"1M#2� = �rrk+l(a; b)F � jz1k+a+r M1 jz2k+b+r M2for M1; M2 2 SL(2;R).Moreover, if F is in addition a holomorphi fun tion on H2, thenrrk+l(a; b)F is a nearly holomorphi fun tion in the sense of Shimura(with respe t to both variables z1 and z2), as polynomials in 1y1 and 1y2they are of degree � r. Shimura's stru ture theorem on nearly holo-morphi fun tions [30℄ says that all nearly holomorphi fun tions onH an be obtained from holomorphi fun tions by applying Maa� typeoperators Æk := k2iy + ��zand their iterates. This however is only true if the weight (i.e. k+a+ror k + r + b) is bigger than 2r, whi h is not ne essarily true in oursituation. We therefore use a weaker version of Shimura' theorem (see[31, Theorem 3.3℄), valid under the assumption "w > 1 + r", where wis the weight at hand and r is the degree of the nearly holomorphi fun tion: Every su h fun tion f on H of degree � r has an expressionf = fhol + Lw( ~f)where fhol is holomorphi and ~f is again nearly holomorphi of degree� r; in this expressionLw := Æw�2�y2 ���z� = w2iy ���z + y2 �2�z��zis a "Lapla ian" of weight w ommuting with the jw-a tion of SL(2;R).We also point out that fhol is uniquely determined by f (in parti ular,f = fhol, if f is holomorphi ) and we have (f jw M)hol = (fhol) jw Mfor all M 2 SL(2;R).If we apply this statement to rrk+l(a; b)F , onsidered as fun tion of z1and z2, we get an expression of typerrk+l(a; b)F = f + Lz1k+a+rg1 + Lz2k+r+bg2 + Lz1k+a+rLz2k+r+bhwhere f; g1; g2; h are nearly holomorphi fun tions on H�H, f beingholomorphi in both variables, g1 holomorphi in z2, g2 holomorphi in z1. Note that (due to our assumption k � 2 ) Shimura's theorem

GLOBAL GROSS-PRASAD CONJECTURE FOR YOSHIDA LIFTINGS 21is appli able here. An inspe tion of Shimura's proof (whi h is quiteelementary for our ase) shows that f is indeed of the form f = DF ,where D is a polynomial p in ��z1 ; ��z12 ; ��z2 , evaluated in z12 = 0. Thispolynomial does not depend on F at all and it has the required trans-formation properties.It remains however to show that D is not zero:For this purpose, we onsider the spe ial fun tionzr12 : � H2 �! C[X1;X2℄lZ 7�! zr12Xa1Xb2It is easy to see that rrk+l(a; b)(zr12) is then equal to the onstant fun -tion r!, therefore rrk+l(a; b)(zr12) = D(zr12) = r!;in parti ular, D is non-zero and we may putDrk;a;b = p(X21 ��z1 ;X1X2 ��z12 ;X22 ��z2 );evaluated at z12 = 0We an then prove in the same way as above:Corollary 2.12. The assertions of Lemma 2.3 and Theorem 2.4 re-main true if f1; f2 have weights ki = �0i + 2 + (i = 1; 2) with some > 0; if one repla es Y (2)('1; '2)��z1 00 z2��by ~D 2;�1;�2Y (2)('1; '2) (z1; z2)Remark 2.13. Appli ation of the di�erential operator toY (2)('1; '2) (Z)) before restri tion to the diagonal does not hange theSp2(R)-representation spa e of that fun tion, i.e., we have found adi�erent fun tion in the same representation spa e whose period inte-gral assumes the value that is predi ted by the onje ture of Gross andPrasad. More pre isely, (using remark 2.9 and some additional on-siderations) one an show that the vanishing of this predi ted value isalready suÆ ient for the vanishing of the period integral for all triplesF 0; f 01; f 02 of fun tions in the Harish-Chandra modules generated by theoriginal fun tions F; f1; f2. To obtain a similar statement for the lo- al representations at the �nite pla es not dividing the level one wouldhave to show that a nonvanishing invariant linear fun tional on thetensor produ t of the representations is not zero on the produ t of the

22 S. B�OCHERER, M. FURUSAWA, AND R. SCHULZE-PILLOTspheri al (or lass 1) ve tors invariant under the maximal ompa t sub-group. This is expe ted to be true as well; we plan to ome ba k to theseproblems in future work.3. Restri tion to an embedded Hilbert modular surfa eTo avoid te hni al diÆ ulties we deal here only with the simplest ase:The quaternion algebra D is rami�ed at all primes p dividing the levelN and we have �1 = �2 = 0, i.e., the Yoshida lifting is a s alar valuedSiegel modular form of weight 2 and the order R we are onsideringis a maximal order. We put F = Q(pN) and assume that N is su hthat the lass number of F is 1. We denote by � the dis riminant ofF; by a 7! a� its nontrivial automorphism and onsider the basis 1; wwith w = �+p�2 of the ring oF of F . Denoting by C the matrixC := �1 1w �w�we have the usual modular embedding(3.1) � : (z1; z2) 7! C �z1 00 z2� tCof H�H into the Siegel upper half plane H2 and(3.2) ~� : �a b d� 7! �C 00 tC�1�0BB�a 0 b 00 a� 0 b� 0 d 00 � 0 d�1CCA�C�1 00 tC�from SL2(F ) into Sp2(Q):We have then for 2 SL2(F ) :(3.3) ~�( )(�((z1; z2)) = ~�( ((z1; z2)))with the usual a tions of the groups SL2(F ) on H�H and of Sp2(Q)on the Siegel upper half plane.We put now J = 0BB�1 0 0 00 0 0 10 0 1 00 1 0 01CCAand onsider for a Siegel modular form f of weight k for the group� � Sp2(Q) the fun tion(3.4) ~f(�1; �2) := f jkJ(�(�1; �2)):

GLOBAL GROSS-PRASAD CONJECTURE FOR YOSHIDA LIFTINGS 23Writing �0 = J Æ �; e�0( ) := J�( )J�1 we see that ~f is a Hilbert modularform for the group e�0�1(�):By al ulating e�0( ) expli itly for 2 SL2(F ) one he ks that e�0( )is in �(2)0 (N) if and only if is in SL2(oF � d), the group of matri es( a b d ) with a; d 2 oF ; 2 d; b 2 d�1, where d is the di�erent of F:If L is a Z-latti e of (even) rank m = 2k with quadrati form q andasso iated bilinear form B(x; y) := q(x + y) � q(x) � q(y) satisfyingq(L) � Z and Nq(L#)Z = Z (N is the level of (L; q)) then it is shownin [4℄ that(3.5)#(2)(L; q; Z)jkJ = 1 Xx12L;x22L# exp�2�itr�� q(x1) B(x1; x2)=2B(x1; x2)=2 q(x2) �Z��;where 1 is a nonzero onstant depending only on the genus of (L; q)and where #(2)(L; q; Z) is the usual theta series of degree 2 of (L; q):One he ks therefore that, writing K = fx1 + x2w 2 L F j x1 2L; x2 2 L#g; we have(3.6) #(2)(L; q; e�0(z1; z2)) = #(K; q; (z1; z2));where we denote by #(K; q; (z1; z2)) =Py2K exp(2�i(q(y)z1+q(y)�z2))the theta series of the oF -latti e K with the extended form q on it.It is again easily he ked that L# � N�1L implies that K is an integralunimodular oF -latti e, and it is well known that then the theta series#(K; q; (z1; z2)) is a modular form of weight k for the group SL2(oF�d):Lemma 3.1. Let ~D be a quaternion algebra over F rami�ed at bothin�nite primes and let ~R be a maximal order in ~D: LetA( ~D�A; ~R�A; 0) =: A( ~D�A; ~R�A)be de�ned in the same way as in Se tion 1 for D and let A( ~D�A; ~R�A) beequipped with the natural a tion of He ke operators T (p) for the p notdividing N des ribed by Brandt matri es as explained in [11℄.Then byasso iating to a He ke eigenform 2 A( ~D�A; ~R�A) the Hilbert modularformf(z1; z2) = Z( ~D�n ~D�A)�( ~D�n ~D�A) (y) (y0)#(y0 ~Ry�1; (z1; z2))dy dy0one gets a bije tive orresponden e between the as above and theHe ke eigenforms of weight 2 and trivial hara ter for the group �0(n; d)of pre ise level n giving an expli it realization of the orresponden e ofShimizu und Ja quet/Langlands [29, 24℄. Here n denotes the produ t of

24 S. B�OCHERER, M. FURUSAWA, AND R. SCHULZE-PILLOTthe prime ideals rami�ed in ~D and �0(n; d) is the subgroup of SL2(oF �d) whose lower left entries are in nd:The fun tion �(y; y0) on ~D�A � ~D�A given by setting �(y; y0) equal tothe Petersson produ t of f with #(y0 ~Ry�1; (z1; z2)) is proportional to(y; y0) 7! (y) (y0):Proof. The �rst part of this Lemma is due to Shimizu [29℄ (taking intoa ount that by [11℄ the group �0(n; d) is the orre t transformationgroup for the theta series in question). Let �̂ be the fun tion on theadeli orthogonal group of ~D indu ed by � and let ̂ be the fun tionon the adeli orthogonal group of ~D indu ed by (y; y0) 7! (y) (y0):The fun tion ̂ generates an irredu ible representation spa e of ~D�Awhose theta lifting to SL2(FA) is generated by f; and �̂ is a ve tor inthe theta lifting of this latter representation of SL2(FA); whi h by [25℄ oin ides with the original representation spa e generated by ̂: Sin eboth �̂; ̂ are invariant under the same maximal ompa t subgroup of~D�A; the uniqueness of su h a ve tor implies that they must oin ideup to proportionality. That �̂ is not zero follows from the obvious fa tthat f by its onstru tion an not be orthogonal to all the theta series.Lemma 3.2. With the above notations let '1; '2 in A(D�A; R�A; 0) beHe ke eigenforms with the same eigenvalue under the involutions fwpfor the p j N with asso iated newforms h1; h2 of weight 2 and level N:Let f be a Hilbert modular form of weight 2 for the group SL2(oF � d)that orresponds in the way des ribed in Lemma 3.1 to the fun tion 2 A( ~D�A; ~R�A) for ~D = D F and ~R being the maximal order in ~R ontaining R: Then the value of the period integral(3.7) ZSL2(oF�d)nH�H (Y (2)('1; '2)j2J(�((z1; z2)))f((z1; z2))dz1dz2is equal to(3.8) 2hf; fi(Xi '1(yi) (yi))(Xi '2(yi) (yi));where we identify yi with yi 1 2 ~D and where 2 is some onstantdepending only on N:In order to interpret the value obtained in (3.8) in the same way asin Se tion 2 as the entral riti al value of an L-fun tion, we reviewbrie y the integral representation of the L-fun tion that one obtainswhen one repla es in a triple (h; f1; f2) of ellipti usp forms the pair(f1; f2) by one Hilbert usp form f for a real quadrati �eld.

GLOBAL GROSS-PRASAD CONJECTURE FOR YOSHIDA LIFTINGS 25For the moment, both the Hilbert usp form f and the ellipti uspform h an be of arbitrary even weight k: Now we onsider the Siegeltype Eisenstein series of weight k, de�ned on H3 byEk3 (W; s) = X =0� � �C D 1A2�30(N)1n�30(N) det(CW+D)�kdet(=( < W >)sHere and in the sequel we denote by G1 the subgroup of G de�ned by"C = 0", where G is any group of symple ti matri es.We restri t this Eisenstein series to W = � � 00 Z � with � 2 H,Z 2 H2 and furthermore we onsider then the modular embeddingwith respe t to Z.In this way we get a fun tion E(�; z1; z2; s), whi h behaves like a mod-ular form for � and like a Hilbert modular form for (z1; z2) of weight k.We want to ompute the twofold integralI(f; h; s) := ZSL2(oF�d)nH2 Z�0(N)nH h(� )f(z1; z2)E(�; z1; z2; s)d� �dz�1dz�2where dz� = yk�2dxdy for z = x+ iy 2 H.This an be done in several ways: One an relate this integral to similarones in [26℄ or in [14℄ (both these works are in an adeli setting) or one an try to do it along lassi al lines as in [13, 28, 9℄. We sket h the latterapproa h here (for lass number one, h being a normalized newform oflevel N .)The inner integration over � (whi h an be done with Z 2 H2 in-stead of the embedded (z1; z2)) is the same as in the papers mentionedabove, produ ing an L-fa tor L2(h; 2s + 2k � 2) (with L2(; ) denot-ing the symmetri square L-fun tion) times a Klingen type Eisensteinseries E2;1(h; s), whi h is de�ned as follows: We denote by C2;1 themaximal paraboli subgroup of Sp(2) for whi h the last line is of theform (0; 0; 0; �) and we put C2;1(N) = C2;1(Q) \ �20(N). Furthermorewe de�ne a fun tion hs(Z) on H2 byhs(Z) = h(z1)�det(Y )y1 �s ;where z1 = x1 + iy1 denotes the entry in the upper left orner ofZ = X + iY 2 H2. Then we putE2;1(h; s)(Z) = X 2C2;1n�20(N)hs(Z) jk

26 S. B�OCHERER, M. FURUSAWA, AND R. SCHULZE-PILLOTTo do the se ond integration, one needs information on ertain osets:This is the only new ingredient entering the pi ture:Lemma 3.3. A omplete set of representatives for C2;1(N)n�20(N) isgiven by fd(M)J�1~�0( )gwith running over SL2(oF � d)1nSL2(oF � d), and M = � � �u v �running over those elements of SL(2;Z)1nSL(2;Z) with v � 0(N),where d denotes the standard embedding of GL(2) in Sp(2) given byd(M) = � (M�1)t 00 M �.This lemma is related to the double oset de ompositionC2;1(N)n�20(N)=~�0(SL2(oF � d))and somewhat analogous to the oset de omposition in [28, p.692℄; weomit the proof.We may now do the usual unfolding to get(3.9) ZSL2(oF�d)nH2 h(z1; z2) ^E2;1(h; �; s)(z1; z2)dz�1dz�2= ZSL2(oF�d)1nH2 XM=0� � �v u 1Ah((v; u)C� z1 00 z2 �C t� vu �)� f(z1; z2)� Dy1y4(v + u!)2y1 + (v + u�!2y2�s dz�1dz�2Using the Fourier expansions of f and F ,h(z) = 1Xn=1 a(n)e2�inz; f(z1; z2) = X�20F ;��0A(�)e2�itr(��z)one an (after some standard al ulations) write the integral above as (s)Xn X�n(u;v) a(n)A(n(v + u!)2)n�s�2k+2N(v + u!)�2s�2k+2where we use the following equivalen e relation: two pairs (u; v) and(u0; v0) are alled equivalent i� v + u! and v0 + u0! are equal up to aunit from oF as a fa tor.Assume now in addition that h is a normalized eigenfun tion of all

GLOBAL GROSS-PRASAD CONJECTURE FOR YOSHIDA LIFTINGS 27He ke operators; then we de�ne the L-fun tion L(h f; s) as an Eulerprodu t over all primes p with Euler fa tors (at least for p oprime toN) Lp(h f; s) := LAsaip (f; �pp�s)LAsaip (f; �pp�s)where we use the Euler fa tors LAsaip (f; s) of the Asai-L-fun tion at-ta hed to f (see [1℄) and �p and �p are the Satake-p-parameters at-ta hed to the eigenform h (normalized to have absolute values p k�12 ).We will write later L(h; f ; s) to denote the shift of this L-fun tion thatis normalized to have fun tional equation under s 7! 1� s.By standard al ulation, we see that the integral above is, after multi-pli ation by L2(h; 2s+2k�2), equal to the L-fun tion L(hf; s+2k�2)(up to elementary fa tors; the ondition v � 0(N) also reates someextra ontribution for p-Euler fa tors with p j N). This al ulation of ourse requires some formal al ulations similar to those given e.g. in[13℄.Remark 3.4. If the lass number H of F is di�erent from one, then theorbit stru ture is more ompli ated. One gets H di�erent sets of rep-resentatives of the type des ribed in the lemma above (ea h one twistedby a matrix in SL2(F ) mapping a usp into 1). After unfolding, onegets then a Diri hlet series also involving Fourier oeÆ ients of f atall the H di�erent usps. If we assume that f is the �rst omponent(i.e. the one orresponding to the prin ipal ideal lass) in a tuple of HHilbert modular forms su h that the orresponding adeli modular formis an eigenform of all He ke operators, then it is possible (but quiteunpleasant) to transfer that Diri hlet series into the Euler produ t inquestion.Now we return to the ase of weight 2. We an ompute the integralI(f; h; s) at s = 0 not only by unfolding as above but also by using theSiegel-Weil formula for the Eisenstein series in the integrand. Thenone gets in the same way as in [7℄ that the square of the right handside of (3.8) is (up to an expli it onstant) the produ t of the entral riti al values of the L-fun tions atta hed to the pairs h1; f and h2; fas above:Theorem 3.5. Let '1; '2; h1; h2; f; be as in Lemma 3.2. Then thesquare of the period integral(3.10) ZSL2(oF�d)nH�H (Y (2)('1; '2)j2J(�((z1; z2)))f((z1; z2))dz1dz2

28 S. B�OCHERER, M. FURUSAWA, AND R. SCHULZE-PILLOTis equal to(3.11) 3hh1; h1ihh2; h2iL(h1; f ; 1=2)L(h2; f ; 1=2);where 3 is an expli itly omputable nonzero number depending onlyon N and the produ t L-fun tion L(h; f ; s) is normalized to have itsfun tional equation under s 7! 1� s:In parti ular the period integral is nonzero if and only if the entral riti al value of L(h1; f ; s)L(h2; f; s) is nonzero.

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30 S. B�OCHERER, M. FURUSAWA, AND R. SCHULZE-PILLOT[22℄ H. Hijikata, H. Saito: On the representability of modular forms by theta series,p. 13-21 in Number Theory, Algebrai Geometry and Commutative Algebra,in honor of Y. Akizuki, Tokyo 1973[23℄ T. Ibukiyama: On di�erential operators on automorphi forms and invariantpluriharmoni polynomials.Comm.Math.Univ.S.Pauli 48, 103-118(1999)[24℄ H. Ja quet, R. Langlands: Automorphi forms on GL(2), Le t. Notes in Math.114, Berlin-Heidelberg-New York 1970[25℄ C. Moeglin: Quelques propri�etes de base des series theta, J. of Lie Theory 7(1997), 231-238[26℄ I.Piatetski-Shapiro, S.Rallis: Rankin triple L-fun tions. Compos.Math.64, 31-115(1987)[27℄ D. Prasad: Some appli ations of seesaw duality to bran hing laws, Math. An-nalen 304, 1-20 (1996)[28℄ T. Satoh: Some remarks on triple L-fun tions. Math.Ann.276, 687-698(1987)[29℄ H. Shimizu: Theta series and automorphi forms on GL2. J. of the Math. So .of Japan 24 (1972), 638-683[30℄ G. Shimura, The spe ial values of the zeta fun tions asso iated with uspforms, Comm. pure appl. Math. 29, 783-804 (1976)[31℄ G. Shimura,Di�erential operators, holomorphi proje tion, and singular forms,Duke Math.J.76, 141-173 (1994)[32℄ H. Yoshida: Siegel's modular forms and the arithmeti of quadrati forms.Invent. Math. 60 (1980), no. 3, 193{248Siegfried B�o hererKunzenhof 4B79117 FreiburgGermanyboe h�siegel.math.uni-mannheim.deMasaaki FurusawaDepartment of Mathemati sGraduate S hool of S ien eOsaka City UniversitySugimoto 3{3{138, Sumiyoshi-kuOsaka 558-8585, Japanfurusawa�s i.osaka- u.a .jpRainer S hulze-PillotFa hri htung 6.1 MathematikUniversit�at des Saarlandes (Geb. 27)Postfa h 15115066041 Saarbr�u kenGermanys hulzep�math.uni-sb.de