2 CONGRUENCES AMONG MODULAR FORMS ON U(2; 2) to 0 is reducible and non-semisimple, thus giving rise to some non-split extension of Galois modules whic

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1 CONGRUENCES AMONG MODULAR FORMS ON U(2; 2) AND THE BLOCH-KATO CONJECTURE KRZYSZTOF KLOSIN Abstract. Let k be a positive integer divisible by 4, ` > k a prime, and f an elliptic cuspidal eigenform (ordinary at `) of weight k, level 4, and non-trivial character. Let f be the `-adic Galois representation attached to f. In this paper we provide evidence for the Bloch-Kato conjecture for the motives ad 0 M 0 ( ) and ad 0 M 0 (2), where M 0 is the motif attached to f. More precisely, let L(Symm 2 f; s) denote the symmetric square L-function of f. We prove that (under certain conditions) ord`(l alg (Symm 2 f; k)) ord`(#s), where S is the (Pontryagin dual of the) Selmer group attached to the Galois module ad 0 f j GK ( ), and K = Q( p ). Our method uses an idea of Ribet [45] in that we introduce an intermediate step and produce congruences between CAP and non-cap modular forms on the unitary group U(2; 2).. Introduction Let ` > 2 be a prime and let be a prime of Q lying over `. The idea of linking up -divisibility of an L-value with the existence of congruences among modular forms and using these congruences to construct elements in a Selmer group goes back to Ribet and his proof of the converse to Herbrand's theorem [45]. In that paper a special value L(; ) of an even Dirichlet character is realized as a constant term of an Eisenstein series E. If L(; ) 0 (mod ), one shows there exists a cuspidal Hecke eigenform f whose Hecke eigenvalues are congruent to those of E (mod ) and as a result of that the mod Galois representation f attached to f is reducible (a consequence of the congruence) but can be chosen to be nonsemisimple (a consequence of the irreducibility of the -adic Galois representation f ), thus giving rise to a non-split extension of one-dimensional Galois modules over F`. This extension can be interpreted as a non-zero element in a certain piece (determined by ) of the class group of Q(`). This strategy can be phrased in the language of automorphic representations suggesting ways to generalize it to other situations. Let be an automorphic representation of an algebraic group M over Q. Realize M as a Levi subgroup in a maximal parabolic subgroup of a larger algebraic group G over Q and lift (e.g., by inducing) to an automorphic representation of G(A). Assuming one knows how to attach -adic Galois representations to automorphic representations of G(A), the one attached to will be reducible and semisimple. If divides a certain L-value L(), construct a representation 0 of G(A), whose Hecke eigenvalues are congruent to those of mod and whose -adic Galois representation is irreducible. These two conditions (respectively) ensure that the mod Galois representation attached Cornell University, Department of Mathematics, 30 Malott Hall, Ithaca, NY , USA Date: April 24, 2008.

2 2 CONGRUENCES AMONG MODULAR FORMS ON U(2; 2) to 0 is reducible and non-semisimple, thus giving rise to some non-split extension of Galois modules which one interprets as lying in an appropriate Selmer group related to. In Ribet's case, M = GL GL, G = GL 2, =, (resp. 0 ) is the automorphic representation of GL 2 (A) attached to E (resp. f). Some versions of this approach have been applied by several authors: Mazur, Wiles, Bellache-Chenevier, Skinner-Urban, Brown, Berger ([4, 60,, 9, 3])) to give lower bounds in terms of L-values on the orders of Selmer groups. The dicult point is the construction of 0 and dierent cases require dierent methods to tackle that point. Let K = Q(i), f 2 S k 4; 4, a normalized cuspidal Hecke eigenform, ` > k a prime such that f is ordinary at `. Write L int (Symm 2 f; k) for the value at k of the symmetric square L-function of f divided by a suitable \integral" period. In this article we implement the above strategy with M = Res K=Q (GL 2=K ), G = U(2; 2) - a quasi-split unitary group associated with the extension K=Q, = base change to K of the automorphic representation associated to f, and L() = L int (Symm 2 f; k). This will allow us to construct elements in the Selmer group of V := ad 0 f j GK ( ), where f is the -adic Galois representation attached to f, G K = Gal(K=K), and denotes a Tate twist. We will now describe the construction of = the lift of to G(A) and of the representation 0. The representation is obtained by lifting f to a Hecke eigenform F f 2 S k, where S k is the space of (weight k and level ) hermitian modular forms as dened by Braun [6, 7, 8], using the Maass lifting constructed by Kojima, Gritsenko and Krieg [36, 23, 37]. Denote by Sk M S k the image of the Maass lift. It is known that the eigenvalues of eigenforms in S k lie in a number eld. In fact we always choose a suciently large nite extension E of Q` and x embeddings Q,! Q`,! C, so we can view all the algebraic numbers of our interest as lying inside the same eld E. From now on will denote a uniformizer of E and O its valuation ring. Assuming L int (Symm 2 f; k) 0 (mod ), we need to construct a hermitian modular eigenform F 0 2 S k orthogonal to the Maass space Sk M whose Hecke eigenvalues are congruent to those of F f (mod ). The form F 0 will give rise to the representation 0 as above. Indeed, the -adic Galois representation attached to F f is reducible and semisimple of the form f ( f ), where is the `-adic cyclotomic character, while it is conjectured that the -adic Galois representation attached to an eigenform orthogonal to the Maass space is irreducible. In proving a bound on the Selmer group we will need to assume this conjecture (see Theorem.2). The construction of F 0 is carried out in several steps. Note that, unlike in Ribet's case, our lift F f is a cusp form, so there is no \constant term" which would \naturally" contain the L-value L int (Symm 2 f; k). Let be a Hecke character of K of innity type (z=jzj) t with k t 6. We rst dene a \nice" hermitian modular form which is essentially a product of a hermitian Siegel Eisenstein series D and a hermitian theta series depending on the character. Using some results of Shimura [49, 50, 5] on algebraicity of Fourier coecients of hermitian Eisenstein series we show that has -integral Fourier coecients. We write (.) = C Ff F f + F with C Ff = () 3 hd ; F f i hf f ; F f i ; where () is a -adic unit, F is orthogonal to F f, i.e., hf f ; Fi = 0, and the inner products are the Petersson inner products on S k. Using results of Shimura [5] and

3 CONGRUENCES AMONG MODULAR FORMS ON U(2; 2) 3 some formulae due to Raghavan and Sengupta [44] we are able to express the inner products by L-functions related to f. More precisely, we get (.2) hd ; F f i hf f ; F f i = () Lint (BC(f); + t+k 2 ;!)Lint (BC(f); 2 + t+k 2 ;!) L int (Symm 2 ; f; k) where () 2 E with ord (()) 0, is the Hida congruence ideal of f,! is the unramied Hecke character of K of innity type (z=jzj) k, and L int (BC(f); s; ) is the L-function of the base change of f to K twisted by the Hecke character. Note that (ignoring the factor for the moment) if one can nd which makes the numerator of the right-hand side of (.2) a -adic unit, then using (.) and -integrality of the Fourier coecients of and of F f we see that if n j L int (Symm 2 f; k) then we can write C Ff = a n with a a -adic unit and hence F f a n F (mod n ). However F 00 := a n F need not be orthogonal to the Maass space. To achieve this last property we modify F 00 appropriately (to obtain F 0 ) using some results of Hida as well as deformation theory of Galois representations. This way we obtain the rst main result of the paper. To simplify the exposition here we omit a certain technical hypothesis on the character. For the full statement see Theorem 7.2. Theorem.. With notation as before, assume that k 2 Z +, 4 j k, ` > k, f is ordinary at ` and the mod representation f restricted to G K is absolutely irreducible. Assume there exists a Hecke character of K of conductor prime to `, innity type (z=jzj) t, k t 6, such that the numerator of the right-hand side of (.2) is a -adic unit. If ord (L int Symm 2 f; k) = n > 0, then there exists F 0 2 S k, orthogonal to the Maass space, such that F 0 F f (mod n ). To be precise, the form F 0 in Theorem. need not be a Hecke eigenform. To measure congruences between F f and eigenforms orthogonal to the Maass space we introduce the notion of a CAP ideal, which is a simple modication of the Eisenstein ideal introduced by Mazur [39]. Theorem. implies that (.3) ord`(i f ) ord`(#o= n ); where I f is the index of the CAP ideal of F f inside the hermitian Hecke O-algebra acting on the orthogonal complement of Sk M localized at the maximal ideal corresponding to F f. We emphasize that the ordinarity assumption on f is essential to our method. It is used to ensure that F 0 is orthogonal to the Maass space (see section 8). For a given f it is unknown for how many primes f is ordinary, although one conjectures that for a non-cm form (which is the case here) this set of primes has Dirichlet density one. An analogous statement for elliptic curves is due to Serre [48]. Let V Q be the E[G Q ]-module ad 0 f ( ). Fix a G Q -stable O-lattice T Q V Q and set W Q := V Q =T Q. For A = V; T; W, write A K for A Q regarded as an E[G K ]- module. Write ` for the set of primes of K lying over `. Let Sel `(W K ) be the Selmer group of W K in the sense of Bloch-Kato (for denition see section 9) and write S `(W K ) for the Pontryagin dual of Sel `(W K ). Then our second main theorem (which uses a result of Urban [55]) is the following (again we omit some mild hypotheses on f - see section 9):

4 4 CONGRUENCES AMONG MODULAR FORMS ON U(2; 2) Theorem.2. Assume that for every eigenform F 2 S k orthogonal to S M k -adic Galois attached to F is absolutely irreducible. Then ord`(#s `(W K )) ord`(i f ): The existence of -adic Galois representations attached to general automorphic forms on G is at this point only conjectural, so Theorem.2 is conditional upon that conjecture. In fact once this conjecture is known in full strength, one should be able to remove the irreducibility assumption from Theorem.2, as it is expected that Galois representations attached to non-maass cuspidal eigenforms are irreducible. Combining Theorem.2 with (.3) which is a consequence of Theorem., we obtain the following corollary. Corollary.3. With the same assumptions as in Theorems. and.2 one has ord`(#s `(W K )) ord`(#o=l int (Symm 2 f; k)): Let us briey explain the relation of Corollary.3 to the Bloch-Kato conjecture for M = ad 0 M 0 ( ), where M 0 is the motif attached to f. Let = f2; `g and write S (W Q ) for the Pontryagin dual of Sel (W Q ) (where we require Q that the classes be unramied away from ` and crystalline at `). Let L(M; s) = p L p(s ) be the L-function of M dened by ( ( p; p;2 (.4) L p (s) := p s ) ( p s ) ( p; p;2p s ) p 6= 2 ( p s ) p = 2; where p;, p;2 are the p-satake parameters of f. Then the Bloch-Kato conjecture for M can be phrased in the following way: Conjecture.4 (Bloch-Kato). One has (.5) #S (W Q ) Tam! (T Q ) = L(M; 0)! (T Q ) O as fractional ideals of E, where Tam! (T Q ) is the `-Tamagawa factor and! (V Q ) is a certain period dened with respect to the \integral structures" T Q and! (for precise denitions see section 9.3). Given our assumptions (which in particular include the ordinarity assumption on f) Corollary.3 falls short of proving that the left-hand side of (.5) contains the right-hand side of (.5), but provides some evidence for it. In fact (for an appropriate choice of T Q and!) the right-hand side of (.5) equals L int (Symm 2 f; k) O. However, the Selmer group on the left-hand side of (.5) could potentially be smaller than S `(W K ) and we do not know if Tam(T Q ) = O. For a more detailed discussion see section 9.3. In that section we also explain the relation of Corollary.3 to the Bloch-Kato conjecture for the motif ad 0 M 0 (2) which is \dual" to ad 0 M 0 ( ). The Bloch-Kato conjecture is currently known only for a few motives - see [33] for a survey of known cases. The most recent result is due to Diamond, Flach and Guo [3] and it concerns the motives ad 0 M 0 and ad 0 M 0 (), while our result concerns the motives ad 0 M 0 ( ) and ad 0 M 0 (2). The method used in [3] is related to the method employed by Wiles and Taylor to prove the Taniyama-Shimura conjecture [6, 54] and so is dierent from ours. Let us briey discuss the organization of the paper. In section 2 we introduce notation which is used throughout the paper. In section 4 we summarize the basic the

5 CONGRUENCES AMONG MODULAR FORMS ON U(2; 2) 5 facts concerning the Maass lifting f 7! F f and compute the Petersson inner product hf f ; F f i in terms of L(Symm 2 f; k). To carry out the calculations we need to rst compute the residue of the hermitian Klingen Eisenstein series and this is done in section 3. The inner product hd ; F f i is computed in section 6. In section 5 we gather the necessary facts concerning the hermitian Hecke algebra which are later used in section 7, where the rst main theorem (Theorem.) is proved assuming existence of a certain Hecke operator which allows one to \kill" the \Maass part" of the form F 00 as above, i.e., obtain a form F 0 that would be orthogonal to Sk M. The existence of such a Hecke operator is proved in section 8 using methods of deformation theory of Galois representations. Finally in section 9 we prove Theorem.2 and Corollary.3 and discuss the relation of the latter to the Bloch-Kato conjecture. We also want to mention that it seems possible to extend our result to an arbitrary imaginary quadratic eld K provided one knows how to construct a Heckeequivariant Maass lifting in that setting. Such a construction has recently been carried out by the author [35] for K with prime discriminant (see also [30]). We hope to use that construction to prove Theorems. and.2 for such a K in a subsequent paper. The author would like to thank Joel Bellache, Tobias Berger, Jim Brown, and Chris Skinner for many useful and inspiring conversations. We would also like to thank the anonymous referee for suggesting various improvements to the Introduction and section Notation and Terminology In this section we introduce some basic concepts and establish notation which will be used throughout this paper unless explicitly indicated otherwise. 2.. Number elds and Hecke characters. Throughout this paper ` will always denote an odd prime. Let i = p, K = Q(i) and let O K be the ring of integers of K. For 2 K, denote by the image of under the non-trivial automorphism of K. Set N := N() :=, and for an ideal n of O K, set Nn := #(O K =n). As remarked below we will always view K as a subeld of C. For 2 C, will denote the complex conjugate of and we set jj := p. Let L be a number eld with ring of integers O L. For a place v of L, denote by L v the completion of L at v and by O L;v the valuation ring of L v. If p is a place of Q, we set L p := Q p Q L and O L;p := Z p Z O L. The letter v will be used to denote places of number elds (including Q and K), while the letter p will be reserved for a (nite or innite) place of Q. For a nite p, let ord p denote the p-adic valuation on Q p. For notational convenience we also dene ord p () :=. If 2 Q p, then jj Qp := p ordp() denotes the p-adic norm of. For p =, j j Q = j j R = j j is the usual absolute value on Q = R. In this paper we x once and for all an algebraic closure Q of the rationals and algebraic closures Q p of Q p, as well as compatible embeddings Q,! Q p,! C for all nite places p of Q. We extend ord p to a function from Q p into Q. Let L be a number eld. We write G L for Gal(L=L). If p is a prime of L, we also write D p G L for the decomposition group of p and I p D p for the inertia group of p. The chosen embeddings allow us to identify D p with Gal(L p =L p ).

6 6 CONGRUENCES AMONG MODULAR FORMS ON U(2; 2) For a number eld L let A L denote the ring of adeles of L and put A := A Q. Write A L; and A L;f for the innite part and the nite part of A L respectively. For = ( p ) 2 A set jj Q A := p jj Qp. By a Hecke character of A L (or of L, for short) we mean a continuous homomorphism : L n A L! C whose image is contained inside fz 2 C j jzj = g: The trivial Hecke character will be denoted by. The character factors into a product of local characters = Q v v, where v runs over all places of L. If n is the ideal of the ring of integers O L of L such that v (x v ) = if v is a nite place of L, x v 2 O L;v and x 2 no L;v no ideal m strictly containing n has the above property, then n will be called the conductor of. If m is an ideal of O L, then we set Q m := v, where the product runs over all the nite places of L such that v j m. For a Hecke character of A L, denote by the associated ideal character. Let be a Hecke character of A K. We will sometimes think of as a character of (Res K=Q GL )(A). We have a factorization = Q p p into local characters p : Res K=Q GL (Qp )! C. For M 2 Z, we set M := Q p6=; pjm p. If is a Hecke character of A K, we set Q = j A The unitary group. To the imaginary quadratic extension K=Q one associates the unitary similitude group where J = GU(n; n) = fa 2 Res K=Q GL n j AJ A t = (A)Jg; I n I n, with I n denoting the n n identity matrix, the bar over A standing for the action of the non-trivial automorphism of K=Q and (A) 2 GL. For a matrix (or scalar) A with entries in a ring aording an action of Gal(K=Q), we will sometimes write A for A t and ^A for (A ). We will also make use of the groups U(n; n) = fa 2 GU(n; n) j (A) = g; and SU(n; n) = fa 2 U(n; n) j det A = g: Since the case n = 2 will be of particular interest to us we set G = U(2; 2), G = SU(2; 2) and G = GU(2; 2). For a Q-subgroup H of G write H for H \ G. Denote by G a the additive group. In G we choose a maximal torus T = 82 >< 6 4 >: a b ^a ^b j a; b 2 Res K=Q GL and a Borel subgroup B = T U B with unipotent radical U B = 82 >< 6 4 >: >= >; ; 5 j ; ; 2 Res K=Q G a ; 2 G a ; + 2 G a 9 >= >; :

7 Let CONGRUENCES AMONG MODULAR FORMS ON U(2; 2) 7 T Q = 82 >< 6 4 >: a b a b j a; b 2 GL denote the maximal Q-split torus contained in T. Let R(G) be the set of roots of T Q, and denote by e j, j = ; 2, the root dened by e j : a a 2 a a ! a j: The choice of B determines a subset R + (G) R(G) of positive roots. We have R + (G) = fe + e 2 ; e e 2 ; 2e ; 2e 2 g: We x a set (G) R + (G) of simple roots (G) := fe e 2 ; 2e 2 g: If (G), denote the parabolic subgroup corresponding to by P. We have P (G) = G and P ; = B. The other two possible subsets of (G) correspond to maximal Q-parabolics of G: the Siegel parabolic P := P fe e 2g = M P U P with Levi subgroup A M P = ^A j A 2 Res K=Q GL 2 ; and (abelian) unipotent radical U P = 82 >< 6 4 >: b b 2 b 2 b >= >; 5 j b ; b 4 2 G a ; b 2 2 Res K=Q G a the Klingen parabolic Q := P f2e2g = M Q U Q with Levi subgroup M Q = 82 >< 6 4 >: x a c ^x b d j x 2 Res K=Q GL ; and (non-abelian) unipotent radical U Q = 82 >< 6 4 >: 3 7 a c b d 9 >= >; 2 U(; ) 5 j ; ; 2 Res K=Q G a ; + 2 G a For an associative ring R with identity and an R-module N we write Nm n to denote the R-module of n m matrices with entries in N. We also set N n := N n, and M n (N) := Nn n. Let x = [ C A D B ] 2 M 2n(N) with A; B; C; D 2 M n (N). Dene a x = A, b x = B, c x = C, d x = D. 9 >= >; ; 9 >= >;

8 8 CONGRUENCES AMONG MODULAR FORMS ON U(2; 2) For M 2 Q, N 2 Z such that MN 2 Z we will denote by D(M; N) the group G(R) Q p- K 0;p(M; N) G(A), where (2.) K 0;p (M; N) = fx 2 G(Q p ) j a x ; d x 2 M 2 (O K;p ) ; b x 2 M 2 (M O K;p ); c x 2 M 2 (MNO K;p ) : If M =, denote D(M; N) simply by D(N) and K 0;p (M; N) by K 0;p (N). For any nite p, the group K 0;p := K 0;p () = G(Z p ) is a maximal (open) compact Qsubgroup of G(Q p ). Note that if p - N, then K 0;p = K 0;p (N). We write K 0;f (N) := p- K 0;p(N) and K 0;f := K 0;f (). Note that K 0;f is a maximal (open) compact subgroup of G(A f ). Set A B K 0; := B A 2 G(R) j A; B 2 GL 2 (C); AA + BB = I 2 ; AB = BA Then K 0; is a maximal compact subgroup of G(R). Let We have U(m) := fa 2 GL m (C) j AA = I m g : K 0; = G(R) \ U(4)! U(2) U(2); where the last isomorphism is given by A B 7! (A + ib; A ib) 2 U(2) U(2): B A Finally, set K 0 (N) := K 0; K 0;f (N) and K 0 := K 0 (). The last group is a maximal compact subgroup of G(A). Let M 2 Q, N 2 Z be such that MN 2 Z. We dene the following congruence subgroups of G(Q): : (2.2) h 0(M; N) := G(Q) \ D(M; N); h (M; N) := f 2 h 0(M; N) j a 2 M 2 (NO K )g; h (M; N) := f 2 h (M; N) j b 2 M 2 (M NO K )g ; and set h 0(N) := h 0(; N), h (N) := h (; N) and h (N) := h (; N). Because we will frequently use the group h 0() = fa 2 GL 4 (O K ) j AJA = Jg, we reserve a special notation for it and denote it by Z. Note that the groups h 0(N), h (N) and h (N) are U(2; 2)-analogues of the standard congruence subgroups 0 (N), (N) and (N) of SL 2 (Z). In general the superscript `h' will indicate that an object is in some way related to the group U(2; 2). The letter `h' stands for `hermitian', as this is the standard name of modular forms on U(2; 2) Modular forms. In this paper we will make use of the theory of modular forms on congruence subgroups of two dierent groups: SL 2 (Z) and Z. We will use both the classical and the adelic formulation of the theories. In the adelic framework one usually speaks of automorphic forms rather than modular forms and in this case SL 2 is usually replaced with GL 2. For more details see e.g. [22], chapter 3. In the classical setting the modular forms on congruence subgroups of SL 2 (Z) will be referred to as elliptic modular forms, and those on congruence subgroups of Z as hermitian modular forms.

9 CONGRUENCES AMONG MODULAR FORMS ON U(2; 2) Elliptic modular forms. The theory of elliptic modular forms is well-known, so we omit most of the denitions and refer the reader to standard sources, e.g. [42]. Let H := fz 2 C j Im (z) > 0g denote the complex upper half-plane. In the case of elliptic modular forms we will denote by 0 (N) the subgroup of SL 2 (Z) consisting of matrices whose lowerleft entries are divisible by N, and by (N) the subgroup of 0 (N) consisting of matrices whose upper left entries are congruent to modulo N. Let SL 2 (Z) be a congruence subgroup. Set M m ( ) (resp. S m ( )) to denote the C-space of elliptic modular forms (resp. cusp forms) of weight m and level. We also denote by M m (N; ) (resp. S m (N; )) the space of elliptic modular forms (resp. cusp forms) of weight m, level N and character. For f; g 2 M m ( ) with either f or g a cusp form, and 0 a nite index subgroup, we dene the Petersson inner product and set hf; gi 0 := Z hf; gi := 0 nh f(z)g(z)(im z) m 2 dx dy; [SL 2 (Z) : 0 ] hf; gi 0 ; where SL 2 (Z) := SL 2 (Z)= h I 2 i and 0 is the image of 0 in SL 2 (Z). The value hf; gi is independent of 0. Every elliptic modular form f 2 M m (N; ) possesses a Fourier expansion f(z) = P n=0 a(n)qn, where throughout this paper in such series q will denote e(z) := e 2iz. For = a b c d 2 GL + 2 (R), set j(; z) = cz + d. In this paper we will be particularly interested in the space S m 4; 4 4, where is the non-trivial character of (Z=4Z). Regarded as a function Z! f; g, it assigns the value to all prime numbers p such that (p) splits in K and the value to all prime numbers p such that (p) is inert in K. Note that since 4 the character is primitive, the space Sm 4; 4 has a basis consisting of primitive normalized eigenforms. We will denote this (unique) basis by N. For f = P n= a(n)qn 2 N, set f := P n= a(n)qn 2 N. Fact 2.. ([42], section 4.6) One has a(p) = 4 p a(p) for any rational prime p - 2. This implies that a(p) = a(p) if (p) splits in K and a(p) = a(p) if (p) is inert in K. For f 2 N and E a nite extension of Q` containing the eigenvalues of T n, n = ; 2; : : : we will denote by f : G Q! GL 2 (E) the Galois representation attached to f by Deligne (cf. e.g., [], section 3.). We will write f for the reduction of f modulo a uniformizer of E with respect to some lattice in E 2. In general f depends on the lattice, however the isomorphism class of its semisimplication ss f is independent of. Thus, if f is irreducible (which we will assume), it is well-dened Hermitian modular forms. For a systematic treatment of the theory of hermitian modular forms see [6, 7, 8] as well as [23, 37, 36]. We begin by dening the hermitian upper half-plane H = fz 2 M 2 (C) j i(z Z ) > 0g;

10 0 CONGRUENCES AMONG MODULAR FORMS ON U(2; 2) where i = [ i i ]. Set Re Z = 2 (Z + Z ) and Im Z = 2 i(z Z ). Let G + (R) := fg 2 G (R) j (g) > 0g: The group G + (R) acts on H by Z = (a Z + b )(c Z + d ), with 2 G + (R). For a holomorphic function F on H, an integer m and 2 G + (R) put Fj m = () 2m 4 j(; Z) m F (Z); with the automorphy factor j(; Z) = det(c Z + d ). Let h be a congruence subgroup of Z. We say that a holomorphic function F on H is a hermitian modular form of weight m and level h if Fj m = F for all 2 h : The group h is called the level of F. If h = h 0(N) for some N 2 Z, then we say that F is of level N. Forms of level will sometimes be referred to as forms of full level. One can also dene hermitian modular forms with a character. Let h = h 0(N) and let : A K! C be a Hecke character such that for all nite p, p(a) = for every a 2 O K;p with a 2 NO K;p. We say that F is of level N and character if Fj m = N (det a )F for every 2 h 0(N): A hermitian modular form of level h (M; N) possesses a Fourier expansion F (Z) = 2S(M) c()e(tr Z); where S(M) = fx 2 S j tr xl(m) Zg with S = fh 2 M 2 (K) j h = hg and L(M) = S \ M 2 (MO K ). As we will be particularly interested in the case when M =, we set S := S() = t t 2 t 2 t 3 2 M 2 (K) j t ; t 3 2 Z; t O K Denote by M m ( h ) the C-space of hermitian modular forms of weight m and level h, and by M m (N; ) the C-space of hermitian modular forms of weight m, level N and character. For F 2 M m ( h ) and 2 G + (R) one has Fj m 2 M m ( h ) and there is an expansion Fj m = 2S c ()e(tr Z): We call F a cusp form if for all 2 G + (R), c () = 0 for every such that det = 0. Denote by S m ( h ) (resp. S m (N; )) the subspace of cusp forms inside M m ( h ) (resp. M m (N; )). If =, set M m (N) := M m (N; ) and S m (N) := S m (N; ). Theorem 2.2 (q-expansion principle, [28], section 8.4). Let ` be a rational prime and N a positive integer with ` - N. Suppose all Fourier coecients of F 2 M m (N; ) lie inside the valuation ring O of a nite extension E of Q`. If 2 Z, then all Fourier coecients of Fj m also lie in O. If F and F 0 are two hermitian modular forms of weight m, level h and character, and either F or F 0 is a cusp form, we dene for any nite index subgroup h 0 of h, the Petersson inner product hf; F 0 i h 0 := Z h 0nH F (Z)F 0 (Z)(det Y ) m 4 ddy; :

11 CONGRUENCES AMONG MODULAR FORMS ON U(2; 2) where = Re Z and Y = Im Z, and hf; F 0 i = [ Z : h 0] hf; F 0 i h 0 ; where Z := Z = hii and h 0 is the image of h 0 in Z. The value hf; F 0 i is independent of h 0. There exist adelic analogues of hermitian modular forms. For F 2 M m (N; ), the function ' F : G(A)! C dened by ' F (g) = j(g; i) m F (g; i) (det d k ); where g = g Q gk 2 G(Q)G(R)K 0;f (N), is an automorphic form on G(A). 3. Eisenstein series The goal of this section is to compute the residue of the hermitian Klingen Eisenstein series (cf. Denition 3. and Theorem 3.0). This computation will be used in the next section. 3.. Siegel, Klingen and Borel Eisenstein series. Siegel and Klingen Eisenstein series are induced from the maximal parabolic subgroups P and Q of G = U(2; 2) respectively. (For the denitions of P and Q see section 2.2.) Let be the modulus character of P (A), (3.) P A ^A P : P (A)! R + with A 2 Res K=Q GL 2 (A), u 2 U P (A), and the modulus character of Q(A), 02 x (3.2) Q 4 u = j det A det Aj 2 A; Q : Q(A)! R + a c ^x b d u C A = jxxj3 A; with x 2 Res K=Q GL (A), a b c d 2 U(; )(A) and u 2 UQ (A). As before, K 0 = K 0; K 0;f will denote the maximal compact subgroup of G(A). Using the Iwasawa decomposition G(A) = P (A)K 0 we extend both characters P and Q to functions on G(A) and denote these extensions again by P and Q. Denition 3.. For g 2 G(A), the series E P (g; s) := P (Q)nG(Q) P (g) s is called the (hermitian) Siegel Eisenstein series, while the series E Q (g; s) := Q(Q)nG(Q) is called the (hermitian) Klingen Eisenstein series. Q (g) s Properties of E P (g; s) were investigated by Shimura in [50]. We summarize them in the following proposition.

12 2 CONGRUENCES AMONG MODULAR FORMS ON U(2; 2) Proposition 3.2 (Shimura). The series E P (g; s) is absolutely convergent for Re (s) > and can be meromorphically continued to the entire s-plane with only a simple pole at s =. One has 4; 4L 3; (3.3) res s= E P (g; s) = 45L 2; 4 where L(; ) denotes the Dirichlet L-function. Properties of the Klingen Eisenstein series were investigated by Raghavan and Sengupta in [44]. The only dierence is that instead of E Q (g; s), [44] uses an Eisenstein series that we will denote by E s (Z). The connection between E Q (g; s) and E s (Z) is provided by Lemma 4.6. After the connection has been established the following proposition follows from Lemma in [44]. Proposition 3.3 (Raghavan-Sengupta). The series E Q (g; s) converges absolutely for Re (s) > and can be meromorphically continued to the entire s-plane. The possible poles of E Q (g; s) are at most simple and are contained in the set f0; =3; 2=3; g: In section 3.4 we will show that E Q (g; s) has a simple pole at s = and calculate the residue. Both E P (g; s) and E Q (g; s) have their classical analogues, i.e., series in which g is replaced by a variable Z in the hermitian upper half-plane H. Let g 2 G(R) be such that Z = gi and set g = (g; ) 2 G(R) G(A f ). Dene E P (Z; s) := E P (g; s) and E Q (Z; s) = E Q (g; s): We will show in Lemma 4.6 that E Q (Z; s) = det Im (Z) 3s (Im (Z)) 2;2 2Q(Z)n Z where for any matrix M we denote its (i; j)-th entry by M i;j. Remark 3.4. Note that we use the same symbols E P (; s) and E Q (; s) to denote both the adelic and the classical Eisenstein series. We distinguish them by inserting g 2 G(A) or Z 2 H in the place of the dot. We will continue this abuse of notation for other Eisenstein series we study. We now turn to the Eisenstein series which is induced from the Borel subgroup B of G, which we call the Borel Eisenstein series. It is a function of two complex variables s and z, dened by E B (g; s; z) := 2B(Q)nG(Q) Q (g) s P (g) z : Note that as the Levi subgroup of B is abelian (it is the torus T ), the character Q s z P is a cuspidal automorphic form on T (A). Thus the following proposition follows from [43], Proposition II..5. Proposition 3.5. The series E B (g; s; z) is absolutely convergent for (s; z) 2 f(s 0 ; z 0 ) 2 C C j Re (s 0 ) > 2=3; Re (z 0 ) > =2g: It can be meromorphically continued to all of C C. ;

13 CONGRUENCES AMONG MODULAR FORMS ON U(2; 2) 3 Remark 3.6. It follows from the general theory (cf. [38], chapter 7) that by taking iterated residues of Eisenstein series induced from minimal parabolics one obtains Eisenstein series on other parabolics. These series are usually referred to as residual Eisenstein series. In fact E P and E Q are residues of E B taken with respect to the variable s and z respectively. We will prove this fact in section 3.5, but see also [32], Remark Siegel Eisenstein series with positive weight. In this section we dene an Eisenstein series induced from the Siegel parabolic, having positive weight, level and non-trivial character. For notation refer to section 2. Let m; N be integers with m 0 and N > 0. Note that K 0; is the stabilizer of i in G(R). Let : K n A K! C be a Hecke character of A K with local decomposition = Q p p, where p runs over all the places of Q. Assume that and (x) = x jxj m p(x p ) = if p 6= ; x p 2 O K;p ; and x p 2 NO K;p : As before we set N = Q pjn p. Let P denote the modulus character of P. We dene P : M P (Q)U P (A) n G(A)! C by setting P (g) = ( 0 g 62 P (A)K 0 (N) (det d q ) N (det d ) j(; i) m g = q 2 P (A)K 0 (N): Note that P has a local decomposition P = Q p P;p, where (3.4) P;p (q p p ) = 8 >< >: p(det d qp ) if p - N; p(det d qp ) p (det d p ) if p j N; p 6= ; (det d q ) j(; i) m if p = and P has a local decomposition P = Q p P;p, where (3.5) P;p A ^A Denition 3.7. The series E(g; s; N; m; ) := u = j det A det Aj Qp : 2P (Q)nG(Q) P (g) P (g) s=2 is called the (hermitian) Siegel Eisenstein series of weight m, level N and character. The series E(g; s; N; m; ) converges for Re (s) suciently large, and can be continued to a meromorphic function on all of C (cf. [50], Proposition 9.). It also has a complex analogue E(Z; s; m; ; N) dened by E(Z; s; m; ; N) := j(g; i) m E(g; s; N; m; )

14 4 CONGRUENCES AMONG MODULAR FORMS ON U(2; 2) for Z = gi, g = g Q g f 2 G(Q)G(R)K 0;f (N). It follows from Lemma 8.7(3) of [50] and formulas (6.40) and (6.48) of [5], together with the fact that K has class number one that (3.6) E(Z; s; m; ; N) = = 2(P (Q)\ h 0 (N))n h 0 (N) N(det d ) (det Im Z) s m=2 j m = 2(P (Q)\ h 0 (N))n h 0 (N) N(det d ) det(c Z + d ) m j det(c Z + d )j 2s+m (det Im Z) s m=2 : 3.3. The Eisenstein series on U(; ). Let B denote the upper-triangular Borel subgroup of U(; ) with Levi decomposition B = T U, where a T := j a 2 Res ^a K=Q GL and U = x j x 2 G a : Let : B (A)! R + be the modulus character given by for u 2 U (A). U(; )(A) with a ^a u = jaaj A Let K = K ; K ;f denote the maximal compact subgroup of K ; = 2 GL 2 (C) j jj 2 + jj 2 = ; 2 R Q being the maximal compact subgroup of U(; )(R) and K ;f = p6= U(; )(Z p). As usually we extend to a map on U(; )(A) using the Iwasawa decomposition. For g 2 U(; )(A), set (3.7) E U(;) (g; s) = 2B (Q)nU(;)(Q) (g) s : The following proposition follows from [50], Theorem 9.7. Proposition 3.8. The series E U(;) (g; s) converges absolutely for Re (s) > and continues meromorphically to all of C. It has a simple pole at s = with residue 3=. We now dene a complex analogue of E U(;) (g; s). As SL 2 (R) acts transitively on H, so does U(; )(R) SL 2 (R). Hence for every z 2 H there exists g 2 U(; )(R) such that z = gi. Set g = (g; ) 2 U(; )(R) U(; )(A f ). An easy calculation shows that (3.8) (g) = Im (z ): For z and g as above, dene the complex Eisenstein series corresponding to E U(;) (g; s) by (3.9) E U(;) (z ; s) := E U(;) (g; s):

15 CONGRUENCES AMONG MODULAR FORMS ON U(2; 2) 5 It is easy to see that (3.0) E U(;) (z ; s) = 2B (Z)nU(;)(Z) (Im (z )) s : The series E U(;) (z ; s) possesses a Fourier expansion of the form where x := Re (z ) and y := Im (z ). E U(;) (z ; s) = n2z c n (y ; s)e 2inx ; Lemma 3.9. Let z and g be as before, i.e., z = gi. Then c 0 (s; y ) = y s + (2s ) (2s) where (s) denotes the Riemann zeta function. s 2 (s) p y s ; Proof. This is a standard argument. See, e.g., [0], the proof of Theorem Residue of the Klingen Eisenstein series. Let E Q (g; s) be the Klingen Eisenstein series dened in section 3.. This section and section 3.5 are devoted to proving the following theorem. Theorem 3.0. The series E Q (g; s) has a simple pole at s = and one has (3.) res s= E Q (g; s) = 52 4 L 2; 4; 4 K (2)L 3; where K (s) denotes the Dedekind zeta function of K. Theorem 3.0 is a consequence of the following proposition. Proposition 3.. The following statements hold: (i) For any xed s 2 C with Re (s) > 2=3 the function E B (g; s; z) has a simple pole at z = =2 and (3.2) res z==2 E B (g; s; z) = 3 2 E Q(g; s + =3): (ii) For any xed z 2 C with Re (z) > =2 the function E B (g; s; z) has a simple pole at s = 2=3 and (3.3) res s= 2 3 E B (g; s; z) = 2 6 K (2) E P (g; z + =2) : Indeed, using Proposition 3. and interchanging the order of taking residues we obtain: res s= 2 3 E Q g; s + = K (2) res z= E P g; 2 + z : 2 By Proposition 3.2, and thus we nally get res z= 2 E P g; 45L z 2; = 4; 4L 3; res s= E Q (g; s) = 52 L 2; 4 4 K (2)L 3; 4 ;

16 6 CONGRUENCES AMONG MODULAR FORMS ON U(2; 2) which proves Theorem 3.0. We now prepare for the proof of Proposition 3., which will be completed in section 3.5. Let x a b ^x c d 2 M Q (A). Since a b c d 2 U(; )(A), we can use the Iwasawa decomposition for U(; )(A) with respect to the upper-triangular Borel to write a b with 2 2 K, where K is as in section 3.3. Note that if = [ 3 4 ], c d = ^ then by Q B 2 x and a character by: a c 2 K 0. Dene a character ^x b d C A (3.4) P = ^A 02 x P 6 4 A = Q y Q : M Q (A)! R x ^x P : M P (A)! R + ^x ^y C ^ C A = jj A; A = jxy (xy )j A ; where we used the Iwasawa decomposition for GL 2 (A K ) = Res K=Q GL 2 (A) with respect to its upper-triangular Borel B R, and its maximal compact subgroup K R = U(2) Q v- GL 2(O K;v ) to write A 2 GL 2 (A K ) as We again have " x A = 2 y # K B R (A)K R : Extend Q and P as well as Q and P to functions on G(A) using the Iwasawa decompositions (3.5) G(A) = B(A) K 0 = P (A) K 0 = Q(A) K 0 : A simple calculation shows that (3.6) Q s P z = s+ 3 2 z Q 2z Q = 3 4 s+z for any complex numbers s and z. Let E B (g; s; z) be the Borel Eisenstein series dened in section 3.. By Proposition 3.5 the series is absolutely convergent if Re (s) > 2=3 and Re (z) > =2 and admits meromorphic continuation to all of C 2. Using identity (3.6) and rearranging terms we get: E B (g; s; z) := 2Q(Q)nG(Q) Q (g) s+ 2 3 z P 3 2 s P 2B(Q)nQ(Q) Q (g) 2z =

17 (3.7) = CONGRUENCES AMONG MODULAR FORMS ON U(2; 2) 7 2P (Q)nG(Q) P (g) 3 4 s+z 2B(Q)nP (Q) P (g) 3 2 s : Let E U(;) (g; s) be the Eisenstein series dened by formula (3.7). We also dene an Eisenstein series on Res K=Q GL 2 (A) by: (3.8) E ResK=Q GL 2 (g; s) = 2B R(Q)nRes K=Q GL 2(Q) where R denotes the modulus character on B R dened by: (3.9) R a b The following maps (3.20) and (3.2) x a c R : B R! R + = jaab b j =2 A : Q : M Q U Q! U(; ) ^x b d ; u C A 7! a P : P! Res K=Q GL 2 A ^A 7! A c b d ; R (g) s ; give bijections B(Q) n Q(Q) = B (Q) n U(; )(Q) and B(Q) n P (Q) = BR (Q) n Res K=Q GL 2 (Q); respectively. On the A-points we can extend Q to a map G(A)! U(; )(A)=K and P to a map G(A)! Res K=Q GL 2 (A)=K R by declaring them to be trivial on K 0. Hence we can rewrite (3.7) as (3.22) E B (g; s; z) := 2Q(Q)nG(Q) = Q (g) s+ 2 3 z E U(;) ( Q (g); 2z) = 2P (Q)nG(Q) P (g) 3 4 s+z E ResK=Q GL 2 ( P (g); 3 2 s): 3.5. E Q (g; s) as a residual Eisenstein series. In this section we complete the proof of Proposition 3.. We will only present a proof of part (i) of the proposition as the proof of (ii) is completely analogous. (In part (ii) the role of E U(;) (see below) is played by E ResK=Q GL 2 for which an easy computation shows that res s= E ResK=Q GL 2 (g; s) = 2 =(4 K (2)).) In what follows Z will denote a variable in the hermitian upper half-plane H, and z a variable in the complex upper half-plane H. Otherwise we use notation from sections Write g = g Q g 2 G(A) with g Q 2 G(Q), g 2 G(R) and 2 K 0;f. We have E B (g; s; z) = E B (g; s; z) and E Q (g; s) = E Q (g; s), hence it is enough to prove (3.2) for g = (g; ) 2 G(R) G(A f ). Let K denote the maximal compact subgroup of U(; )(A) and

18 8 CONGRUENCES AMONG MODULAR FORMS ON U(2; 2) let Q : G(A)! U(; )(A)=K be as in formula (3.20). Lemmas 3.2 and 3.4 are easy. Lemma 3.2. If g = (g; ) 2 G(A), then Im ( Q (g)i) = Im (gi) 2;2. Remark 3.3. Note that for any 2 2 matrix M with entries in C one has Im (M 2;2 ) = (Im (M)) 2;2. Hence the conclusion of Lemma 3.2 can also be written as Im ( Q (g)i) = Im ((gi) 2;2 ). Lemma 3.4. For any Z 2 H, there exists 2 Q(Z) such that (Im Z) 2;2 > 2. The next lemma is just a simple adaptation to the case of hermitian modular forms of the proof of Hilfsatz 2.0 of [2]. Lemma 3.5. For every Z 2 H, we have sup 2 Z det Im (Z) < : Proposition 3.6. Let > 0 and g = (g; ) 2 G(R) G(A f ). For every s 2 C with Re (s) > + and every z 2 C with jz 2j <, the series (3.23) D := jz =2j converges. 2Q(Q)nG(Q) Q (g) s+2z=3 E U(;) ( Q (g); 2z) Proof. Using the same arguments as in the proof of Lemma 4.6 (cf. section 4.2) one shows that det Im (Z) 3s+2z D = (Im (Z)) jz =2j je U(;)( Q (g)i; 2z)j: 2;2 2Q(Z)n Z (Note that z 0 := Q (g)i is a complex variable.) As g = (g; ) and 2 Z K 0;f, we have Q (g) = Q ((g; )). By Lemmas 3.2 and 3.4 we can nd a set S of representatives of Q(Z) n Z such that for every 2 S we have (3.24) Im ( Q (g)i) = Im ((gi) 2;2 ) > 2 : The series E U(;) (z ; 2z) has a Fourier expansion of the form E U(;) (z ; 2z) = n2z c n (2z; Im (z ))e 2inRe (z) ; and E U(;) (z ; 2z) c 0 (2z; Im (z )) for every xed z continues to a holomorphic function on the entire z-plane and for every xed z is rapidly decreasing as Im (z )!. It follows that for any given N > 0 there exists a constant M(N) (independent of z and independent of z as long as jz =2j < ) such that je U(;) (z ; 2z) c 0 (2z; Im (z ))j < M(N) as long as Im (z ) > N. Set x := Re ( Q (g)i) and y := Im ( Q (g)i) = Im ((gi) 2;2 ) : Taking N = =2, we see by formula (3.24) that there exists a constant M (independent of ) such that je U(;) (x + iy ; 2z)j M + jc 0 (2z; y )j. Using (3.8) and Lemma 3.9 one sees that there exists a positive constant C independent of z and of such that jz =2jjc 0 (2z; y )j < C + jy j +2 :

19 CONGRUENCES AMONG MODULAR FORMS ON U(2; 2) 9 Thus we conclude that there exists a positive constant A (independent of z and ) such that (3.25) z 2 E U(;) ( Q (g)i; 2z) A( + Im ( Q(g)i) +2 ) = = A( + Im (gi) +2 2;2 ): For s 0 2 C lying inside the region of absolute convergence of E s 0(Z) let det Im (Z) jej s 0(Z) := 2Q(Z)n Z (Im (Z)) 2;2 s0 denote the majorant of E s (Z). By formula (3.25) we have det Im (Z) 3s+2z (Im (Z)) 2;2 (3.26) D AjEj 3s+2z (Z) + A 2S (Im (Z))+2 2;2 : Note that jej 3s+2z (Z) is well-dened (i.e., 3s + 2z is in the region of absolute convergence of E s 0(Z)) by our assumption on s and z. Denote the second term of the right-hand side of formula (3.26) by D 2. Then D 2 = A 2S det Im (Z) 3s+2z (+2) (Im (Z)) (det Im (Z))+2 : 2;2 By Lemma 3.5 there exists a constant M(Z) such that det Im (Z) M(Z) for every 2 S and hence D 2 AM(Z) +2 jej 3s+2z (+2) < as Re (3s + 2z ( + 2)) > 3 by our assumptions on z and s. This nishes the proof. Proof of Proposition 3.. We need to show that for a xed s 2 C with Re (s) > 2=3 and for every > 0 there exists > 0 such that jz =2j < implies (3.27) D(z) := z 2 2Q(Q)nG(Q) Q (g) s+2z=3 E U(;) ( Q (g); 2z) 3 2 2Q(Q)nG(Q) Q (g) s+=3 < : As remarked at the beginning of the section we can assume without loss of generality that g = (g; ) 2 G(R) G(A f ). We rst show that (3.27) holds for s with Re (s) >. Fix s 2 C with Re (s) > and 0 > 0 such that 0 < 0 < Re (s). From now on assume jz =2j < 0. Fix a set S of representatives of Q(Q)nG(Q). By Proposition 3.6 and the fact that E Q (g; s 0 ) converges absolutely for s 0 with Re (s 0 ) >, there exists a nite subset S of S such that the following two inequalities: (3.28) (3.29) 2S 2 ( Q (g)) s+=3 < 6 ; 2S 2 z 2 Q (g) s+2z=3 E U(;) ( Q (g); 2z) < 4

20 20 CONGRUENCES AMONG MODULAR FORMS ON U(2; 2) are simultaneously satised. Here S 2 denotes the complement of S in S. We have D(z) D (z) + D 2 (z), where D j (z) := z 2 2S j Q (g) s+2z=3 E U(;) ( Q (g); 2z) 3 2 Q (g) s+=3 2S j : Note that if we replace 0 with a smaller 00 > 0, then estimates (3.28) and (3.29) remain true as long as jz =2j < 00 for the same choice of S. Hence we nd > 0 with < 0 such that D (z) < 2. This is clearly possible as D (z) is a nite sum and it follows from Proposition 3.8 that 3=2 is the residue of E U(;) ( Q (g); 2z) at z = =2. On the other hand D 2 (z) D 3 (z) + D 4 (z), where D 3 (z) := 2S 2 z 2 Q (g) s+2z=3 E U(;) ( Q (g); 2z) and D 4 (z) := 3 ( Q (g)) s+=3 : 2 2S 2 Formulas (3.28) and (3.29) imply now that D 3 (z) < =4 and D 4 (z) < =4. Hence D(z) D (z) + D 2 (z) D (z) + D 3 (z) + D 4 (z) < as desired. We have thus established the equality res z==2 E B (g; s; z) = 2 3 E Q(g; s + =3) for s with Re (s) >. However, both sides are meromorphic functions in s and since the right-hand side is holomorphic for Re (s) > 2=3, so must be the left-hand side. Hence they agree for Re (s) > 2=3. 4. The Petersson norm of a Maass lift The goal of this section is to express the denominator of C Ff in formula (.) by a special value of the symmetric square L-function of f. 4.. Maass lifts. Let H, as before, denote the complex upper half-plane. The space HCC aords an action of the Jacobi modular group J := SL 2 (Z)nO K 2, under which a b c d. ; ; takes (; z; w) 2 H C C to a+b c+d ; Denition 4.. A holomorphic function : H C C! C z c+d ; w c+d is called a Jacobi form of weight k and index m if for every a b c d 2 SL2 (Z) and ; 2 O K, and = j k;m a c b d := (c + d) k e m czw a + b m c + d c + d ; z c + d ; w c + d = j m [; ] := e(mt + z + w) m (; z + + ; w + t + ): Let k be a positive integer divisible by 4 and F a hermitian P cusp form of weight k and full level. By rearranging the Fourier expansion F (Z) = B2S c(b)e(tr BZ) of F we obtain (4.) F (Z) = m (; z; w)e(m 0 ) m2z >0

21 where Z = [ w z 0 ] 2 H and CONGRUENCES AMONG MODULAR FORMS ON U(2; 2) 2 m (; z; w) = l2z0;t2 2 OK ttlm c l t t m e(l + tz + tw) is a Jacobi form of weight k and index m. The expansion (4.) is called the Fourier- Jacobi expansion of F. Denition 4.2. The Maass space denoted by Sk M( Z) is the C-linear subspace of S k ( Z ) consisting of those F 2 S k ( Z ) which satisfy the following condition: there exists a function c F : Z 0! C such that c F (B) = d2z >0;dj(B) for all B 2 S, where (B) := max Maass form or a CAP form. d k c F (4 det B=d 2 ) n q 2 Z >0 j q B 2 So : We call F 2 S M k ( Z) a Theorem 4.3 (Raghavan-Sengupta [44]). There exists a C-linear isomorphism between the Maass space and the space 4 (4.2) S k + 4; := = ( 2 S k 4; 4 j = n= b(n)q n ; b(n) = 0 if 4 ) = : n We will describe this isomorphism in more detail. Any Jacobi form of weight k and index can be written as a nite linear combination: (4.3) (; z; w) = t2a f t () t (; z; w); P where A = 0; 2 ; i 2 ; i+ 2, t (; z; w) := 2t+O K e( + lz + w) and f t () = l0;l 4nt (mod 4) c F (l)e(l=4): The map (; z; w) 7! f 0 () gives an injection of J k;, the space of Jacobi forms of weight k and index, into S k 4; 4. If we put = and dene by j k = f0, the composite F 7! (; z; w) 7! f 0 () 7! gives the 4 isomorphism alluded to in Theorem 4.3. Denoting this isomorphism by, we can map any normalized Hecke eigenform f = P n b(n)qn 2 S k 4; 4 to the element F f := (f f ) 2 S P k M( Z). Here f = n b(n)qn. This lifting is Hecke equivariant in a sense, which will be explained in section 5.4. Note that F f = F f and F f 6= 0 if and only if f 6= f. Denition 4.4. If f 6= f, then F f is called the Maass lift of f or the CAP lift of f. P Proposition 4.5. If f = n= b(n)qn 2 S k 4; 4 is a normalized eigenform, then (4.4) c F f (n) = ( 2i u(n) (b(n) b(n)) if n 6 (mod 4) 0 if n (mod 4) ;

22 22 CONGRUENCES AMONG MODULAR FORMS ON U(2; 2) where u(n) := #ft 2 A j 4N(t) n (mod 4)g. Proof. This follows from formula (4) on page 670 in [37] The Petersson norm of F f. To express hf f ; F f i by an L-value we will use an identity proved in [44] that involves a variant E s (Z) (dened below) of the Klingen Eisenstein series E Q (g; s) (which was dened in section 3.). For a matrix M, denote by M i;j the (i; j)-th entry of M. Let C be the subgroup of Z consisting of all matrices whose last row is Set det Im Z s E s (Z) = : (Im Z) ; 2Cn Z The series converges for Re (s) > 3 ([44], Lemma ). Lemma 4.6. Let g = (g; ) 2 G(A) and Z = gi. Then (4.5) E Q (g; s) = 4 E 3s(Z): Proof. First note that E s (Z) = 4 det Im Z s ; (Im Z) 2C 0 ; n Z where C 0 is the subgroup of Z consisting of matrices whose last row is ofthe form with 2 O K. Moreover we have C 0 = wq(z)w with w = This gives 2C 0 n Z det Im Z (Im Z) ; s = det Im s ww Z (Im ww = Z) ; 2Q(Z)n Z det Im Z s ; (Im Z) 2;2 2Q(Z)n Z as w 2 Z. Now for 2 Z we have Q (g) = Q (q), where q = (q; ) and g = q with q 2 Q(R), 2 K 0;. If q = um with m = u 2 U Q (R), then Moreover x a b ^x c d Q (g) = Q (um) = Q (m(m um)) = Q (m) = jxxj 3 A : Im Z = Im gi = Im qi = Im um i: 2 M Q (R) and A direct calculation shows that det Im u(mi) = det Im m i and that (Im u(mi)) 2;2 = (Im mi) 2;2. On the other hand Im m i = " xx (ci+d)(ci+d) hence we have det Im Z = Q (g) =3 : (Im Z) 2;2 The lemma now follows from the fact that the natural injection Q(Z) n Z! Q(Q) n G(Q) # ;

23 CONGRUENCES AMONG MODULAR FORMS ON U(2; 2) 23 Q is a bijection. This is a consequence of the identity Q(A) = Q(Q) Q(R) Q( p- Z p), which follows from Lemma 8.4 of [50]. Set (4.6) E s (Z) := 2s (s) (s )(2s 2) K (s)e s (Z): In [44] Raghavan and Sengupta prove that E s (Z) can be analytically continued in s to the entire complex plane except for possible simple poles at s = 0; ; 2 ; 3. Using Lemma 4.6 and Theorem 3.0 we conclude that E s (Z) has a simple pole at s = 3 and (4.7) res s=3 E s (Z) = 2 2 (3): Combining results of section 3 of [44] with a formula on page 200 in [loc. cit.] we get (4.8) Ff ; Es k+3f f = 4 3s 3s+2k 6 (s) (s k + 2) (s k + 3) Y 3 j= (s k + j) A L(Symm 2 f; s) h ; i : Here we dene L(Symm 2 f; s) for a normalized eigenform f = P n= a(n)qn as an Euler product: (4.9) L(Symm 2 f; s) = ( a(2) 2 2 s ) ( a(2) 2 2 s ) Y p6=2 ( 2 p; p s )( p; p;2 p s )( 2 p;2 p s ) where the complex numbers p; and p;2 are the p-satake parameters of f dened by the equation a(p)x + Combining formulas (4.7) and (4.8) we obtain: 4 p k 2 x 2 = ( p; x)( p;2 x): p (4.0) hf f ; F f i = 2 2k 3 (k) k 2 h ; i L(Symm 2 f; k): Finally, to relate h ; i to hf; fi, in the next subsection we will prove the following lemma. Lemma 4.7. The following identity holds: (4.) h ; i = 2 hf; fi (N) = 24 hf; fi : Combining Lemma 4.7 with formula (4.0) we nally obtain: Theorem 4.8. The following identity holds: (4.2) hf f ; F f i = 2 2k+2 3 (k) k 2 hf; fi L(Symm 2 f; k):

24 24 CONGRUENCES AMONG MODULAR FORMS ON U(2; 2) 4.3. Inner product formula for Jacobi forms. This section is devoted to proving Lemma 4.7. Proof of Lemma 4.7. Let and 2 denote two Jacobi forms of weight k and index m. It is easy to show that (4.3) h ; 2 i = ZF vk ZF (; z; w) 2 (; z; w)e jz wj2 v dz 0 dz dw 0 dw du dv; where F is the standard fundamental domain for the action of SL 2 (Z) on the complex upper half-plane and F fg C C is a fundamental domain for the action of the matrices 0 0 and (; 2 O K ) on C C. After performing a change of variables on C C (keeping xed) z 0 = z + w w 0 = z w; and denoting by F 0 the fundamental domain F in the new variables, the integral over F in (4.3) becomes 8 Z F 0 (; z 0 ; w 0 ) 2 (; z 0 ; w 0 )v k e j z0 +w0 2 z 0 w0 2 j 2 v dz 0 0 dz 0 dw 0 0 dw 0 : Set = 2 =, where is the rst Fourier-Jacobi coecient of the CAP form F f. Using formula (4.3) we can write: (4.4) h ; i = 8 with (4.5) I(t; t 0 ; ) = Z F 0 Z t2a t 0 2A F f t()f t 0()v k 4 I(t; t 0 ; ) du dv e(n(a) + az + aw) e(n(b) + bz + bw) a2t+o K b2t 0 +O K e v ((Im (z0 )) 2 +(Re (w 0 )) 2 ) dz 0 0 dz 0 dw 0 0 dw 0 : Changing variables again we get (4.6) I(t; t 0 ; ) = e(n(a) N(b))I I 2 a2t+o K b2t 0 +O K with Z I = 4 e(2x 0 Re (a) 2x 0 Re (b)) e 2 (2x0 ) dx 0 0 dx0 ; where is the parallelogram in C spanned by the two R-linearly independent complex numbers and, and x 0 = x ix 0 2 C, with x 0 0; x 0 2 R. Before we dene I 2 we note that I can be written as (4.7) I = 4 Z Im () 0 e v 4(x0 )2 Z 0 e(2x 0 Re (a) 2x 0 Re (b)) dx 0 0 Now the integral inside the parantheses in (4.7) equals e 8Re (a) x0 if Re (a) = Re (b) and 0 otherwise. Hence (4.8) I = 4 R Im () 0 e 4 v (x0 )2 e 8Re (a) x0 dx 0 if Re (a) = Re (b) 0 if Re (a) 6= Re (b) dx 0 :