Local and global Maass relations (expanded version)

Transcription

1 Contents Local and gloal Maass relations (expanded version) Ameya Pitale, Ahishek Saha and Ralf Schmidt Astract. We characterize the irreducile, admissile, spherical representations of GSp 4 (F ) (where F is a p-adic field) that occur in certain CAP representations in terms of relations satisfied y their spherical vector in a special Bessel model. These local relations are analogous to the Maass relations satisfied y the Fourier coefficients of Siegel modular forms of degree 2 in the image of the Saito-Kurokawa lifting. We show how the classical Maass relations can e deduced from the local relations in a representation theoretic way, without recourse to the construction of Saito-Kurokawa lifts in terms of Fourier coefficients of half-integral weight modular forms or Jacoi forms. As an additional application of our methods, we give a new characterization of Saito-Kurokawa lifts involving a certain average of Fourier coefficients. Introduction 2 Spherical Bessel functions 4 3 Adelization and Fourier coefficients 0 4 Special automorphic forms 2 5 A proof of the classical Maass relations 4 6 Normalization of the Bessel functions 8 7 Explicit formulas for Bessel functions 20 8 Formulas for Fourier coefficients 23 9 Another proof of the classical Maass relations 24 0 A new characterization of Saito-Kurokawa lifts 26 Introduction Let F e a holomorphic Siegel modular form of degree 2 and weight k with respect to the full Siegel modular group Sp 4 (Z). Then F has a Fourier expansion of the form F (Z) = S a(s)e 2πi Tr(SZ), () The first and third authors are supported y NSF grant DMS-0054.

2 INTRODUCTION 2 where Z is a point in the Siegel upper half space H 2, and where the sum is taken over the set P 2 of semi-integral, symmetric and positive semidefinite 2 2-matrices S. We say that F satisfies a /2 the Maass relations if, for all S =, /2 c a a( 2 2 c ) = r k a( r gcd(a,,c) ac r 2 2r 2r ). (2) Such a relation was first known to e satisfied y Eisenstein series; see 20. Maass, in, started a systematic investigation of the space of modular forms satisfying these relations, calling this space the Spezialschar. Within a few years it was proven, through the efforts of Maass, Andrianov and Zagier, that the Spezialschar is precisely the space of modular forms spanned y Saito-Kurokawa liftings. Recall that a Saito-Kurokawa lifting is a Siegel modular form of weight k constructed from an elliptic modular form of weight 2k 2 with k even. The ook 4 gives a streamlined account of the construction of these liftings, and of the proof that they span the Spezialschar. In addition to this classical approach, it is possile to construct Saito-Kurokawa liftings using automorphic representations theory. For simplicity, we only consider cuspforms. The procedure may e illustrated as follows: π PGL 2 (A) Wald SL 2 (A) θ PGSp 4 (A) Π (3) f F Here, f is an elliptic cuspform of weight 2k 2 with k even and with respect to SL 2 (Z). Assuming that f is an eigenform, it can e translated into an adelic function which generates a cuspidal automorphic representation π of GL 2 (A). This representation has trivial central character, so really is a representation of PGL 2 (A). Since PGL 2 = SO3, there is a theta correspondence etween this group and SL 2 (A), the doule cover of SL 2 (A). More precisely, one considers the Waldspurger lifting from SL 2 (A) to PGL 2 (A), which is a variant of the theta correspondence. Let τ e any pre-image of π under this lifting 2. Since PGSp 4 = SO5, there is another theta correspondence etween SL 2 (A) and PGSp 4 (A). We can use it to forward τ to an automorphic representation Π of PGSp 4 (A). From this Π one can extract a Siegel modular form F of weight k. It turns out that Π is cuspidal, so that F is a cuspform. This F coincides with the classical Saito-Kurokawa lifting of f. For the details of this construction, see 6 and 26. There is a marked difference etween the classical and the representation theoretic constructions. The classical construction directly provides the Fourier coefficients of the modular form F (in terms of the Fourier coefficients of the half-integral weight modular form corresponding to f via the Shimura correspondence). In contrast, the representation theoretic Saito-Kurokawa lifting consists of the following statement: For each cuspidal elliptic eigenform f of weight 2k 2 Recall that a matrix is semi-integral if its off-diagonal entries are in 2 Z, and its diagonal entries are in Z. 2 In this classical situation, there is precisely one pre-image. For modular forms with level, there can e several possiilities for τ, leading to the phenomenon that one elliptic cusp form f may have several Saito-Kurokawa liftings. We refer to 27 for more explanation.

3 INTRODUCTION 3 with even k there exists a cuspidal Siegel eigenform F of weight k such that its spin L-function is given y L(s, F ) = L(s, f)ζ(s k + )ζ(s k + 2). In this case the Fourier coefficients of F are not readily availale. At the very least, one would like to know that the Fourier coefficients of the modular form F constructed in the representation theoretic way satisfy the Maass relations. One quick argument consists in referring to either 5 or 3. In these papers it is proven that the Fourier coefficients of F satisfy the Maass relations if and only if L(s, F ) has a pole. The pole condition is satisfied ecause of the appearance of the zeta factors aove. However, it would e desirale to deduce the Maass relations directly from the representation theoretic construction. One reason is that this construction opens the way to generalizations in various directions, and for these more general situations results similar to 5 or 3 are not availale. For example, what happens if we replace the aove condition k even y k odd? In this case it turns out that one can still do the representation theoretic construction, the difference eing that the archimedean component of the automorphic representation Π is no longer in the holomorphic discrete series. Hence, one will otain a certain type of non-holomorphic Siegel modular form whose adelization generates a gloal CAP representation. As far as we know, the full details of this construction have yet to e written out (though see 2). But since the non-archimedean situation is no different from the case for even k, we expect this new type of Siegel modular form to admit a Fourier expansion for which the Maass relations hold as well. One could prove such a statement if one had a direct representation-theoretic way of deducing the Maass relations. Similarly, we expect that a representation-theoretic proof of the Maass relations would easily generalize to the case of Saito-Kurokawa lifts with respect to congruence sugroups. It was shown in 4 that a representation theoretic method for proving the Maass relations exists. In the present paper we take a similar, ut slightly different approach. Common to oth approaches is the fact that certain local Jacquet modules are one-dimensional. This may e interpreted as saying that the local representations in question admit a unique Bessel model, and this Bessel model is special (see Sect. 2 for precise definitions). While 4 makes use of certain Siegel series to derive an explicit formula for local p-adic Bessel functions, we use Sugano s formula, to e found in 28. Our main local result, Theorem 2. elow, asserts the equivalence of five conditions on a given irreducile, admissile, spherical representation π of GSp 4 (F ) with a special Bessel model (where F is a p-adic field). The first condition is that one of the Satake parameters of π is q /2 (where q is the cardinality of the residue class field); in particular, such representations are non-tempered. The second condition is that π is a certain kind of degenerate principal series representation; these representations occur in gloal CAP representations with respect to the Borel or Siegel paraolic sugroup. The third and fourth conditions are formulas relating certain values of the spherical Bessel function; the third formula is a local analogue of the Maass relations. The fifth condition is an explicit formula for some of the values of the spherical Bessel function; this formula is very similar to one appearing in 9 and 4 for the values of a Siegel series. In the gloal part of this paper, we will explain how the classical Maass relations follow from this local result. It is known that the local components of the automorphic representation Π in the diagram (3) are of the kind covered in Theorem 2.. Hence, the corresponding spherical Bessel functions satisfy the local Maass relations (this implication is all that is needed from

4 2 SPHERICAL BESSEL FUNCTIONS 4 Theorem 2.). Since Bessel models are closely related to Fourier coefficients, one can deduce the gloal (classical) Maass relations from the local relations. To make this work one has to relate the classical notions with the representation theoretic concepts. While this is standard, some care has to e taken, which is why we carry these arguments out in some detail. In fact, we give two different proofs of the classical Maass relations; one uses a result proved y the second author in collaoration with Kowalski and Tsimerman 0, while the other relies on some explicit computations with Bessel functions which may e of independent interest. As explained aove, what we have in mind are future applications to more general situations. Finally, in the last section, we prove a result (Theorem 0.) that gives a new characterization of Saito-Kurokawa lifts involving a certain average of Fourier coefficients. Notation Let G = GSp 4 e the group of symplectic similitudes of semisimple rank 2, defined y GSp 4 = {g GL 4 : t 0 2 gjg = µ(g)j, µ(g) GL }, where J =. 2 0 Here, µ is called the similitude character. Let Sp 4 = {g GSp 4 : µ(g) = }. The Siegel paraolic sugroup P of GSp 4 consists of matrices whose lower left 2 2-lock is zero. Its X unipotent radical U consists of all elements of the lock form, where X is symmetric. A The standard Levi component M of P consists of all elements u t A with u GL and A GL 2. Over the real numers, we have the identity component G(R) + := {g GSp 4 (R) : µ(g) > 0}. Let H 2 e the Siegel upper half space of degree 2. Hence, an element of H 2 is a symmetric, complex 2 2-matrix with positive definite imaginary part. The group G(R) + acts on H 2 via A B g Z = (AZ + B)(CZ + D) for g =. C D Given any commutative ring R, we denote y Sym 2 (R) the set of symmetric 2 2-matrices with coefficients in R. The symol P 2 denotes the set of positive definite, half-integral symmetric 2 2-matrices. 2 Spherical Bessel functions In this section only, F is a local non-archimedean field with ring of integers o, prime ideal p, uniformizer ϖ and order of residue field q. An irreducile, admissile representation of GSp 4 (F ) is called spherical if it admits a spherical vector, i.e., a non-zero GSp 4 (o)-invariant vector. Let (π, V ) e such a representation. Then π is a constituent of a representation paraolically induced from a character γ of the standard Borel sugroup of GSp 4 (F ). The numers ϖ γ () = γ( ϖ ϖ ), γ(2) = γ( ϖ ),

5 2 SPHERICAL BESSEL FUNCTIONS 5 γ (3) = γ( ϖ ϖ ), γ(4) = γ( ϖ ϖ ), (4) are called the Satake parameters of π. The conjugacy class of diag(γ (), γ (2), γ (3), γ (4) ) in GSp 4 (C) determines the isomorphism class of π. Note that γ () γ (3) = γ (2) γ (4) = ω π (ϖ), where ω π is the central character of π. Hence, in the case of trivial central character, the Satake parameters are {α ±, β ± } for some α, β C. In this case we allow ourselves a statement like one of the Satake parameters of π is α ±. In this work we will employ the notation of 29 and the classification of 2 for constituents of paraolically induced representations of GSp 4 (F ). According to Tale A.0 of 2, the spherical representations are of type I, II, III, IVd, Vd or VId, and a representation of one these types is spherical if and only if the inducing data is unramified. Note that type IVd is comprised of one-dimensional representations, which are irrelevant for our purposes. Representations of type I are irreducile principal series representations, and they are the only generic spherical representations. By 26, representations of type II occur as local components of the automorphic representations attached to classical Saito-Kurokawa liftings. Recall that these automorphic representations are CAP (cuspidal associated to paraolic) with respect to the Siegel paraolic sugroup; this property has een defined on p. 35 of 6, where it was called strongly associated to P. One can show that representations of type Vd and VId occur as local components of automorphic representations which are CAP with respect to B, the Borel sugroup. By Theorem 2.2 of 6, P -CAP and B-CAP representations with trivial central character have a common characterization as eing theta liftings from the metaplectic cover of SL 2. We will see elow that spherical representations of type II, Vd and VId with trivial central character have a common characterization in terms of their spherical Bessel functions. We will riefly recall the notion of Bessel model; for more details see 22 (and 7 for the a /2 archimedean case). Let ψ e a fixed character of F. Let S = with a,, c F. Such a /2 c matrix defines a character θ = θ S of U, the unipotent radical of the Siegel paraolic sugroup, y X θ( ) = ψ(tr(sx)), X Sym 2 (F ). Every character of U is of this form for a uniquely determined S. From now on we will assume that θ is non-degenerate, y which we mean that S is invertile. If det(s) F 2 we set L = F F and say that we are in the split case. Otherwise we set L = F ( det(s)) and say that we are in the non-split case. Below we will use the Legendre symol ( L ) if L/F is an unramified field extension, = 0 if L/F is a ramified field extension, (5) F if L = F F. Let T = T S e the torus defined y T = {g GL 2 : t gsg = det(g)s}. Then T (F ) = L. g We think of T (F ) emedded into GSp 4 (F ) via g det(g) t g. Then T (F ) is the identity

6 2 SPHERICAL BESSEL FUNCTIONS 6 component of the stailizer of θ in the Levi component of the Siegel paraolic sugroup. We call the semidirect product R = T U the Bessel sugroup defined y S. Given a character Λ of T (F ), the map tu Λ(t)θ(u) (t T (F ), u U(F )) defines a character Λ θ of R(F ). Now let (π, V ) e an irreducile, admissile representation of GSp 4 (F ). A (Λ, θ)-bessel functional for π is a non-zero element of Hom R (V, C Λ θ ). Equivalently, a (Λ, θ)-bessel functional for π is a non-zero functional β on V satisfying β(π(r)v) = (Λ θ)(r)v for all r R(F ), v V. Given such a functional β, the corresponding Bessel model for π consists of the functions B(g) = β(π(g)v), where v V. By Theorem 6..4 of 22 every infinite-dimensional π admits a Bessel functional for some choice of θ and Λ. The question of uniqueness is discussed in Sect. 6.3 of 22. Bessel models with Λ = are called special. In the case of spherical representations, one may ask aout an explicit formula for the spherical vector in a (Λ, θ)-bessel model. Such a formula was given y Sugano in 28 (at the same time proving that such models are unique for spherical representations). In the case that Λ is unramified, Sugano s formula is conveniently summarized in Sect. (3.6) of 8. We recall the result. To egin, we assume that the elements a,, c defining the matrix S satisfy the following standard assumptions: a, o and c o. In the non-split case, 2 4ac is a generator of the discriminant of L/F. In the split case, 2 4ac o. This is not a restriction of generality; using some algeraic numer theory, one can show that, after a suitale transformation S λ t ASA with λ F and A GL 2 (F ), the standard assumptions are always satisfied. One consequence of (6) is the decomposition GL 2 (F ) = ϖ m T (F ) GL 2 (o); (7) m 0 see Lemma 2-4 of 28. In conjunction with the Iwasawa decomposition, this implies GSp 4 (F ) = R(F )h(l, m)gsp 4 (o), (8) where h(l, m) = l,m Z m 0 ϖ l+2m ϖ l+m Hence, a spherical Bessel function B is determined y the values B(h(l, m)). It is easy to see that B(h(l, m)) = 0 if l < 0 (see Lemma (3.4.4) of 8). Sugano s formula now says that B(h(l, m))x m y l H(x, y) = (9) P (x)q(y) l,m 0 where P (x), Q(y) and H(x, y) are polynomials whose coefficients depend on the Satake parameters, on the value of ( ) L F, and on Λ; see p. 205 of 8 for details. The formula implies in particular that B() 0, so that we may always normalize B() to e. With these preparations, we may now formulate our main local theorem. ϖ m. (6)

7 2 SPHERICAL BESSEL FUNCTIONS 7 2. Theorem. Let π e an irreducile, admissile, spherical representation of GSp(4, F ) with trivial central character. Assume that π admits a special Bessel model with respect to the matrix S. Let B e the spherical vector in such a Bessel model for π, normalized such that B() =. Then the following are equivalent. i) One of the Satake parameters of π is q ±/2. ii) π is one of the representations in the following list: χ GL(2) χ for an unramified character χ of F (type II). L(νξ, ξ ν /2 ), where ξ is the non-trivial, unramified, quadratic character of F (type Vd). L(ν, F ν /2 ) (type VId). iii) For all l, m 0, B(h(l, m)) = l q i B(h(0, l + m i)). (0) i=0 iv) The following relation is satisfied: v) The following two conditions are satisfied: For all m 0, α m α m α α B(h(, 0)) = B(h(0, )) + q. () q 3m/2 B(h(0, m)) = αm+ α m ( L α α )q /2 αm α m, (2) F α α where α ± is one of the Satake parameters of π. Here, if α = α, we understand = m i= αm+ 2i. We are not in the following exceptional situation: ( L F ) = (the inert case) and π = χ GL(2) χ ξ (type II), where χ, ξ are the unramified characters with χ(ϖ) = ±i and ξ(ϖ) =. Proof. i) ii) follows y inspecting the list of Satake parameters of all spherical representations; see Tale A.7 of 2. iii) iv) is trivial. iv) i) iii) Oserve that (0) is equivalent to the following identity etween generating series, B(h(l, m))x m y l = l q i B(h(0, l + m i))x m y l. (3) l,m 0 l,m 0 i=0 By Sugano s formula, the left hand side equals LHS = H(x, y) P (x)q(y),

8 2 SPHERICAL BESSEL FUNCTIONS 8 with H, P, Q as in Proposition 2-5 of 28. For the right hand side of (3), we calculate RHS = = = = = m=0 m=0 m=0 l=0 i=0 i=0 q y q y l q i B(h(0, l + m i))x m y l i=0 q i B(h(0, l + m i))x m y l l=i q i B(h(0, l + m))x m y l+i l=0 m=0 j=0 B(h(0, l + m))x m y l l=0 l+m=j B(h(0, j))x m y l = q B(h(0, j)) xj+ y j+ y x y j=0 ( ) = ( q x B(h(0, j))x j y B(h(0, j))y j y)(x y) j=0 j=0 ( ) H(x, 0) H(y, 0) = ( q x y y)(x y) P (x)q(0) P (y)q(0) ( ) H(x, 0) H(y, 0) = ( q x y. y)(x y) P (x) P (y) Hence, (3) is equivalent to ( ) ( q y)(x y)h(x, y)p (y) Q(y) xp (y)h(x, 0) yp (x)h(y, 0) = 0. (4) If one of the Satake parameters is q ±/2, then one can verify that (4) is satisfied. This shows that i) iii). Conversely, assume that iv) is satisfied. Let F (x, y) e the polynomial on the left hand side of (4). Then iv) is equivalent to saying that the coefficient of y of the power series T (y) = H(0, y) Q(y) H(y, 0) ( q y)p (y), which has no constant term, vanishes. In particular, this means that the y 2 -coefficient of F (0, y) = yq(y)p (y)( q y)t (y) is equal to 0. But it can e easily checked that the y 2 -coefficient of F (0, y) is given y (q /2 α )(q /2 α)(q /2 β )(q /2 β), q δ where α ±, β ± are the Satake parameters of π. It follows that α = q ±/2 or β = q ±/2. The completes the proof of iv) i).

9 2 SPHERICAL BESSEL FUNCTIONS 9 i) v) Let α ±, β ± e the Satake parameters of π. By Sugano s formula, P (x) m 0 B(h(0, m))x m = H(x, 0). (5) Note that H(x, 0) (resp. P (x)) is a degree 3 (resp. degree 4) polynomial. Write P (x) = P 0 + P x + P 2 x 2 + P 3 x 3 + P 4 x 4, H(x, 0) = H 0 + H x + H 2 x 2 + H 3 x 3. Areviating A = α + α and B = β + β, as well as δ = ( L F ), we have and H 0 =, H = q 2 (q δ) H 2 = q 5 (q δ) H 3 = q 7 δ. P 0 =, P = q 2 AB, P 2 = q 4 (A 2 + B 2 2), P 3 = q 6 AB, P 4 = q 8, ( ) q + + δ(δ + ) q /2 (δ + )(A + B) + δab, ( ) q(δ + ) + δ 2 (q + ) q /2 δ(δ + )(A + B) + δqab, Taking the m th derivative of oth sides of (5), setting x = 0 and dividing y m!, we get the following recurrence relation for B(h(0, m)), 4 B(h(0, m i))p i = H m, (6) i=0 where H m = 0 if m > 3. Assume that the Satake parameters are α ± and q ±/2, i.e., B = q /2 +q /2. Then, using (6) and induction on m, it is easy to verify that (2) holds. Conversely, assume that (2) holds for all m 0. Then, from (6) for m =, we find that either B = q /2 + q /2 or A = q /2 (δ + ). Assume that A = q /2 (δ + ). From (6) for m = 2, 3, 4, we conclude, after some calculation, that B = q /2 q /2 and δ =. It follows that A = 0, so that α = ±i. Looking at Satake parameters, this is precisely the excluded exceptional situation. Remarks: i) The second condition in part v) of this theorem cannot e omitted, since in this exceptional situation the formula (2) holds as well. ii) There is a certain analogy of the identity (0) with the classical Maass relations (2). In fact, in the proof of Theorem 9. we will show that the Maass relations are implied y the local relations (0).

10 3 ADELIZATION AND FOURIER COEFFICIENTS 0 iii) Comining the formulas iii) and v) of Theorem 2., we otain q 3m/2 B(h(l, m l)) = l ( α q i/2 m i+ α (m i+) ( L α α )q /2 αm i α (m i) ) F α α (7) i=0 for m l 0. The expression on the right hand side, viewed as a polynomial in α, is related to the value of a certain Siegel series; see Hilfssatz 0 in 9 and Corollary 5. in 4. In fact, formula (7) appears as Lemma 8. of 4. 3 Adelization and Fourier coefficients We turn to classical Siegel modular forms and their adelization. Let Γ = Sp 4 (Z) and S (2) k (Γ) e the space of holomorphic cuspidal Siegel modular forms of degree 2 and weight k with respect to Γ. Hence, if F S (2) k (Γ), then for all γ Γ we have F kγ = F, where (F k g)(z) := µ(g) k j(g, Z) k F (g Z ) (8) for g G(R) + and Z H 2, the Siegel upper half space. Here j(g, Z) = det(cz + D) for A B g = G(R) C D +. The Fourier expansion of F is given y F (Z) = S a(s)e 2πi Tr(SZ), (9) where the sum is taken over the set P 2 of semi-integral, symmetric and positive definite matrices S. Let A e the ring of adeles of Q. It follows from the strong approximation theorem for Sp 4 that G(A) = G(Q)G(R) + K 0 (20) where K 0 := Γ p with Γ p = G(Z p ). Let F S (2) k (Γ). Write g G(A) as g = g Qg g 0 with p< g Q G(Q), g G(R) +, g 0 K 0, and define Φ F : G(A) C y the formula Φ F (g) := (F k g )(i 2 ). (2) Since G(R) + K 0 G(Q) = Γ, the function Φ F is well-defined. From the definition it is clear that for all g G(A), ρ G(Q), k 0 K 0, k K and z Z(A), it satisfies Φ F (zρgk k 0 ) = Φ F (g)j(k, i 2 ) k. (22) Here Z GL is the center of GSp 4 and K U(2) is the standard maximal compact sugroup of Sp 4 (R). Let P = M U e the Siegel paraolic sugroup of G. By the Iwasawa decomposition, G(A) = U(A)M(A)K K 0. Let ψ : Q\A C e the character such that ψ(x) = e 2πix if x R and ψ(x) = for x Z p. Given S Sym 2 (Q), one otains a character Θ S of U(Q)\U(A) y X Θ S ( ) = ψ(tr(sx)).

11 3 ADELIZATION AND FOURIER COEFFICIENTS Note that every character of U(Q)\U(A) is otained in this way. For S Sym 2 (Q) we define the following adelic Fourier coefficient of Φ F, Φ S F (g) := Φ F (ng)θ S (n) dn for g G(A). (23) U(Q)\U(A) The following result, which is standard, provides a formula for Φ S F (g) in terms of the Fourier coefficients of F. 3. Proposition. Let g = n 0 mk k 0, with n 0 U(A), m M(A), k K and k 0 K 0. Let m = m Q m m 0, with m Q M(Q), m M(R) + and m 0 M(A) K 0. Write m Q = A v t A with A GL 2 (Q) and v Q. Let Z 0 = m i 2. Let S Sym 2 (Q) e nondegenerate, and let S = v t ASA. Let F S (2) k (Γ) with Fourier expansion (9) and let ΦS F e as defined in (23). Then { Φ S Θ S (n 0 )µ(m ) k j(g, i 2 ) k a(s )e 2πi Tr(S Z 0 ) if S P 2, F (g) = (24) 0 otherwise, where g = m k. In particular, if S P 2, we have Φ S F () = a(s)e 2π Tr(S). (25) Also, for S Sym 2 (Q) and S = v t ASA, with A GL 2 (Q) and v Q, we have Φ S F (g) = Φ S A F ( v t A g) for all g G(A). (26) Proof. From (22) and the definition (23), we get A change of variale n m Q nm Q Φ S F (n 0 mk k 0 ) = Θ S (n 0 )j(k, i 2 ) k Φ S F (m Q m ). in (23) gives Φ S F (m Q m ) = Φ S F (m ), where S = v t ASA. For every prime p, let a p Z p e the largest ideal such that whenever n U(a p ) then Θ S (n) =. Let t = p rp Z, where a p = p rp Z p. Note that t = if and p< p only if S is semi-integral. Since (Sym 2 (Z)\Sym 2 (R)) Sym 2 (Z p ) p< is a fundamental domain for U(Q)\U(A) (where we identify U = Sym 2 ), so is (t Sym 2 (Z)\Sym 2 (R)) Sym 2 (a p ). p<

12 4 SPECIAL AUTOMORPHIC FORMS 2 Therefore, Φ S F (m ) = ( ( = ( = p< U(a p) p< U(a p) p< U(a p) dn p ) dn p ) dn p )( t Sym 2 (Z)\Sym 2 (R) Φ F (nm )Θ S (n) dn η t Sym 2 (Z)\Sym 2 (Z) Sym 2 (Z)\Sym 2 (R) η t Sym 2 (Z)\Sym 2 (Z) Θ S (η) ) U(Z)\U(R) Φ F (ηnm )Θ S (ηn) dn Φ F (nm )Θ S (n) dn. Since Θ S is a non-trivial character of tu(z)\u(z) if t >, the sum is zero unless t =. Hence, assume that t =, i.e., a p = Z p for all p. By (2), ( X ) Φ S F (m ) = (F k m )(i2 )Θ X S ( ) dx Sym 2 (Z)\Sym 2 (R) = µ(m ) k j(m, i 2 ) k Sym 2 (Z)\Sym 2 (R) Now, sustitute the Fourier expansion of F to get Φ S F (m ) = µ(m ) k j(m, i 2 ) k Sym 2 (Z)\Sym 2 (R) F (Z 0 + X)e 2πi Tr(S X) dx. T P 2 a(t )e 2πi Tr(T Z0) e 2πi Tr(T X) e 2πi Tr(S X) dx. If S P 2, then the integral aove is zero for every T, otherwise it is equal to one exactly for T = S. This proves (24). One can otain (26) y a simple change of variales. 4 Special automorphic forms We now assume that F S (2) k (Γ) is a Hecke eigenform and is a Saito-Kurokawa lift of f S 2k 2 (SL 2 (Z)), with k even, as in 6 of 4. Let Φ F e as defined in (2) and let (π F, V F ) e the irreducile cuspidal automorphic representation of G(A) generated y right translates of Φ F. Then π F is isomorphic to a restricted tensor product p π p with irreducile, admissile representations π p of GSp 4 (Q p ). The following is well-known (see, for example, 26): The archimedean component π is a holomorphic discrete series representation with scalar minimal K-type determined y the weight k. Following the notation of 7, we denote this representation y E(k, k). For a prime numer p, the representation π p is a degenerate principal series representation χ GL(2) χ with an unramified character χ of Q p. Here, we are using the notation of 2. In particular, π p is a representation of type II according to Tale A. of 2. Note that these are non-tempered, non-generic representations.

13 4 SPECIAL AUTOMORPHIC FORMS 3 4. Lemma. Let p e a prime numer or p =. Let S Sym 2 (Q p ) e non-degenerate. In the archimedean case, assume also that S is positive or negative definite. Let T S e the connected component of the stailizer of the character Θ S of U(Q p ). Explicitly, T S = {g GL 2 : t gsg = det(g)s}, (27) g where we emed GL 2 into GSp 4 via g det(g) t g. Let V p e a model for π p, and consider functionals β p : V p C with the property Then: β p (π p (n)v) = Θ S (n)β p (v) for all v V p and n U(Q p ). (28) i) The space of such functionals β p is one-dimensional. ii) If β p satisfies (28), then it automatically satisfies β p (π p (m)v) = β p (v) for all v V p and m T S (Q p ). (29) Proof. This follows from Lemma of 22 in the non-archimedean case, and from Theorem 3.0 of 5 in the archimedean case. In the language of Bessel models, Lemma 4. states that the only such model admitted y π p is special, i.e., with trivial character on T S (Q p ); see Sect. 2 for the definition of Bessel models in the non-archimedean case, and 5, Sect. 2.6, for the definition in the archimedean case. Part i) of Lemma 4. asserts the uniqueness of such models. We remark that property ii) in this lemma is precisely the U-property of 5. We fix a distinguished vector vp 0 V p for each of our local representations π p : If p is finite, we let vp 0 e a spherical (i.e., non-zero G(Z p )-invariant) vector, and if p = we let vp 0 e a vector spanning the one-dimensional K -type determined y k. Note that the construction of the restricted tensor product V p depends on the choice of distinguished vectors almost everywhere, and we use the vectors vp 0 for this purpose. Let S Sym 2 (Q) e positive or negative definite. We will see in a later section that β p (vp) 0 0 for almost all p. For those places where this is the case, we normalize the β p such that β p (vp) 0 =. The following lemma states that the automorphic forms in the space of π are special in the sense of 5, p Lemma. Let the notations e as aove. Then, for any non-degenerate S Sym 2 (Q), and all Φ V F, we have Φ S (mg) = Φ S (g) for all g G(A) and m T S (A). (30) Proof. We fix S. The assertion is trivial if the functional β : V C, β(φ) := Φ S () = U(Q)\U(A) Φ(n)Θ (n) dn S

14 5 A PROOF OF THE CLASSICAL MAASS RELATIONS 4 is zero. Assume that β is non-zero. For each place p, let V p e a model for π p. By a standard argument, β induces a non-zero functional β p : V p C with the property (28). Looking at the archimedean place, Corollary 4.2 of 7 implies that S is positive or negative definite. By the uniqueness asserted in Lemma 4., it follows that there exists a non-zero constant C S such that β(φ) = C S β p (v p ), (3) p whenever Φ V F corresponds to the pure tensor v p via V F = Vp ; note that the product on the right is finite y our normalizations. Using ii) of Lemma 4., it follows that β(π(m)φ) = β(φ) for all m T S (A). Since Φ is aritrary, this implies the assertion of the lemma. 5 A proof of the classical Maass relations a /2 For S = P /2 c 2 we denote c(s) = gcd(a,, c) (resp. disc(s) = 4 det(s)) and call c(s) the content (resp., call disc(s) the discriminant) of S. For S, S 2 P 2, we say that S S 2 if there exists a matrix A SL 2 (Z) such that t AS A = S 2. For any S P 2, let S denote the equivalence class of S under the aove relation; note that all matrices in a given equivalence class have the same content and discriminant. For any discriminant D < 0 (recall that a discriminant is an integer congruent to 0 or modulo 4) and any positive integer L, we let H(D; L) denote the set of equivalence classes of matrices in P 2 whose content is equal to L and whose discriminant is equal to DL 2. In particular, if S P 2, then S H(D; L) where L = c(s) and D = disc(s)/c(s) 2. It is clear that the map S LS gives a ijection of sets H(D; ) H(D; L). Our ojective in this section is to prove the following theorem. 5. Theorem. Let F e a cuspidal Siegel Hecke eigenform of weight k with respect to Sp 4 (Z). For S P 2, let a(s) denote the Fourier coefficient of F at S. Suppose that F is a Saito-Kurokawa lift. Then the following hold. i) Assume that disc(s ) = disc(s 2 ) and c(s ) = c(s 2 ). Then a(s ) = a(s 2 ). ii) For D < 0 a discriminant and L a positive integer, define a(d; L) = a(s) where S is any memer of P 2 satisfying c(s) = L, disc(s) = DL 2 ; this is well-defined y the previous part. Then the following relation holds: a(d; L) = r L r k a(d(l/r) 2 ; ). (32) 5.2 Corollary. (Maass relations) Let F e a cuspidal Siegel Hecke eigenform of weight k with respect to Sp 4 (Z) which is a Saito-Kurokawa lift. For S P 2, let a(s) denote the Fourier coefficient of F at S. Then a a( 2 2 c ) = r k a( r gcd(a,,c) ac r 2 2r 2r ). (33)

15 5 A PROOF OF THE CLASSICAL MAASS RELATIONS 5 Proof. In the notation of the aove Theorem, a a( 2 ) = a(( 2 4ac)/gcd(a,, c) 2 ; gcd(a,, c)), a( 2 c Now the corollary follows immediately from (32). ac r 2 2r 2r ) = a(( 2 4ac)/r 2 ; ). Let us now prove Theorem 5.. As a first step, we recall a very useful characterization of the elements of H(D; L). Let d < 0 e a fundamental discriminant 3. We define S d P 2 as follows. d 4 0 if d 0 (mod 4), 0 S d = (34) d if d (mod 4). For any positive integer M, we let K p (0) (M) e the sugroup of GL 2 (Z p ) consisting of elements that are congruent to 0 (mod M). We define Cl d (M) as follows: where Cl d (M) = T d (A)/T d (Q)T d (R)Π p< (T d (Q p ) K p (0) (M)). T d = {g GL 2 : t gs d g = det(g)s d }. (35) It is easy to see that Cl d (M) is finite. For example, Cl d () is canonically isomorphic to the ideal class group of Q( d). Let c Cl d (M) and let t c p< T d(q p ) e a representative for c. By strong approximation, we can write (non-uniquely) t c = γ c γ c, κ c (36) with γ c GL(2, Q) +, and κ c Π p< K p (0) (M). It is known (see 8, p. 209) that S c := det(γ c ) ( t γ c )S d γ c P 2 and satisfies c(s c ) =, disc(s c ) = d. Also, the (2, 2)-coefficient of S c is modulo M. For any positive integer L, we define L M M φ L,M (c) = S L c. (37) It follows that c(φ L,M (c)) = L, disc(φ L,M (c)) = dl 2 M 2. We remark here that the matrix φ L,M (c) depends on our choice of representative t c as well as on our choice of the matrix γ c involved in strong approximation. However, the equivalence class φ L,M (c) is independent of these choices. In fact, we have the following proposition. 3 Recall that an integer n is called a fundamental discriminant if n is the discriminant of the field Q( n). This is equivalent to saying that either n is a squarefree integer congruent to modulo 4 or n = 4m where m is a squarefree integer congruent to 2 or 3 modulo 4.

16 5 A PROOF OF THE CLASSICAL MAASS RELATIONS Proposition. For each pair of positive integers L, M, the map c φ L,M (c) gives a welldefined ijection etween Cl d (M) and H(dM 2 ; L). Proof. Let us first show that the map is well defined. Let Γ 0 (M) (resp. Γ 0 (M)) e the usual congruence sugroups of SL 2 (Z) consisting of matrices whose upper-right (resp. lower-left) entry is divisile y M. Let c Cl d (M). Suppose that γ c (), γ c (2) are two distinct elements otained in (36) from c and that φ () L,M (c), φ(2) L,M (c) are the matrices otained via (37). Then our definitions imply that there exists t T (Q), k Γ 0 (M) such that γ c (2) = tγ c () k. It follows immediately that φ (2) L,M (c) = t Rφ () L,M (c)r where M M R = k Γ 0 (M) SL 2 (Z). Hence φ (2) L,M (c) = φ() L,M (c). Next, we show that the map c φ L,M (c) is injective. Suppose that φ L,M (c ) = φ L,M (c 2 ). Then there exists A SL 2 (Z) such that t Aφ L,M (c 2 )A = φ L,M (c ). An easy calculation involving M M the entries of the matrices shows that A Γ 0 (M). Then R = A Γ 0 (M) and t RS c2 R = S c. Let t = γ γ, κ and t 2 = γ 2 γ2, κ 2 e our chosen representatives in p< T (Q p) of c and c 2, respectively. Then γ 2 Rγ T d (Q) t 2 GL 2 (R) + p K p (0) (M)t. It follows that t and t 2 represent the same element of Cl d (M), completing the proof. Finally, we show that the map c φ L,M (c) is surjective. Since we have already proved injectivity, it is enough to show that Cl d (M) = H(dM 2 ; L). Now it is a classical fact (see e.g. 2, p. 27) that H(dM 2 ; L) = H(dM 2 ; ) = M u(d) Cl d() ( ( ) d p)p, p M where u( 3) = 3, u( 4) = 2 and u(d) = for other d. On the other hand, a simple argument along the lines of 25, p. 68, shows that Cl d (M) = M u(d) Cl d() ( ( ) d p)p. p M This completes the proof. We now return to the proof of Theorem 5.. In order to prove the first assertion, it is enough to prove that (for some fixed fundamental discriminant d < 0, fixed positive integers L, M and fixed elements c, c 2 Cl d (M)) that a(φ L,M (c )) = a(φ L,M (c 2 )). (38)

17 5 A PROOF OF THE CLASSICAL MAASS RELATIONS 7 Let t p< T d(q p ) e a representative for c and let t 2 p< T d(q p ) e a representative for c 2. By Lemma 4.2, it follows that Φ S d F (t g) = Φ S d F (t 2g) for all g G(A). (39) Define the element (H L,M ) f p< G(Q p) to e the diagonal emedding in p< G(Q p) of the element LM 2 H L,M = LM. M Let us explicate (39) in the special case g = (H L,M ) f. Note that for i =, 2, we have t i (H L,M ) f = γ ci (H L,M ) Q (H L,M ) (γ ci ) (H L,M ) f κ c i (H L,M ) f, where we note that γ ci (H L,M ) Q M(Q) and (H L,M ) (γ ci ) M(R) +. Furthermore, the element (H L,M ) f κ c i (H L,M ) f lies in the group K 0 defined after (20). It follows from Proposition 3. that Φ S d F (t (H L,M ) f ) = e 2π Tr(S d) (LM) k a(φ L,M (c )), Φ S d F (t 2(H L,M ) f ) = e 2π Tr(S d) (LM) k a(φ L,M (c 2 )). Comining (39) and (40), we deduce (38). This completes the proof of the first assertion of Theorem 5.. To prove the second assertion of Theorem 5., we need the following result, which is Theorem 2.0 of Theorem. (Kowalski Saha Tsimerman) Let d < 0 e a fundamental discriminant and Λ = p Λ p e a character of Cl d () (note that Λ induces a character on Cl d (M) for all positive integers M via the natural surjection Cl d (M) Cl d ()). Let F e a cuspidal Siegel Hecke eigenform of weight k with respect to Sp 4 (Z) and let π = p π p e the irreducile cuspidal representation of GSp 4 (A) attached to F. For S P 2, let a(s) denote the Fourier coefficient of F at S. For each prime p, let B p e the spherical vector in the (S d, Λ p, θ p )-Bessel model for π p, normalized so that B p () =. Then for any positive integers L = p plp and M = p pmp the relation ( (LM) k ) Λ(c)a(φ L,M (c)) = Λ(c)a(S c ) B p (h(l p, m p )) Cl d (M) Cl d () holds. c Cl d (M) c Cl d () Let us see what this implies in the setup of Theorem 5.. For F a cuspidal Siegel Hecke eigenform of weight k with respect to Sp 4 (Z) which is a Saito-Kurokawa lift, D < 0 a discriminant and L a positive integer, define a(d; L) = a(s) where S is any memer of P 2 satisfying c(s) = L and disc(s) = DL 2 (this is well-defined y the first assertion of Theorem 5., which we have already proven). Then Theorem 5.4, in the special case Λ =, F as aove, tells us that p LM (40) a(dm 2 ; L) = (LM) k a(d; ) B p (h(l p, m p )). (4) p LM

18 6 NORMALIZATION OF THE BESSEL FUNCTIONS 8 We need to prove a(dm 2 ; L) = r L r k a(d(lm/r) 2 ; ). (42) By (4), the left side is equal to r L (LM) k a(d; ) p LM p LM B p (h(l p, m p )), and the right side is equal to r k (LM/r) k a(d; ) B p (h(0, l p + m p r p )). Hence, we are reduced to showing that p L B p (h(l p, m p )) = r L r p L B p (h(0, l p + m p r p )). This equation would follow provided for each prime p LM we could prove that B p (h(l p, m p )) = l p i=0 p i B p (h(0, l p + m p i)). But this follows from Theorem 2.. Note here that, y our remarks in Sect. 4, the nonarchimedean local components π p associated to F are of the form χ GL(2) χ for an unramified character χ of Q p (type II). This concludes the proof of Theorem Normalization of the Bessel functions We return to the setup of Section 4. We will prove certain explicit formulas for the Bessel functions and their effect under change of models. This will lead to another proof of the classical Maass relations which does not use Theorem 5.4. In the following let a /2 S = P /2 c 2 e fixed. Recall that in Sect. 4 we have fixed distinguished vectors vp 0 in each local representation V p. For any place p let β p e a non-zero functional V p C as in Lemma 4.. Let Bp S e the Bessel function corresponding to vp, 0 i.e., Bp S (g) = β p (π p (g)vp) 0 for g G(Q p ). We are going to normalize the β p, hence the Bp S, in a certain way. Non-archimedean case Assume that p is a prime. It follows from Sugano s formula (9) that Bp S () 0, provided S satisfies the standard assumptions (6). Hence, if S satisfies these conditions, then we can normalize Bp S () =.

19 6 NORMALIZATION OF THE BESSEL FUNCTIONS 9 For aritrary positive definite S in P 2 we proceed as follows. Let disc(s) = N 2 d, where d is a fundamental discriminant. Let S d e as in (34). Then we have S d = a t ASA, where A := 2N a N a 2N a + 2a N a if d 0 (mod 4), if d (mod 4). For revity, we put S = S d. We oserve that the matrix S satisfies the standard assumptions (6) for every prime p <. Consequently, we can normalize our Bessel functions so that Bp S () = A for all primes p. However, for applications we require the values Bp S () = Bp S ( a t ). A Calculating this value requires decomposing the argument of Bp S in the form (8) and then using Sugano s formula (9). We postpone the proof of the following lemma until the next section. A 6. Lemma. Let A, S, S, d, N e as aove. Let L = c(s). Then a t = th(l, m)k, with A t T S (Q p ), k GSp 4 (Z p ) and m = v p (N /L) and l = v p (N ) m = v p (L). Consequently, B S p () = B S p (h(l, m)) with these values of l and m. Since S satisfies the standard assumptions (6), the right hand side can e evaluated using Sugano s formula (9). Archimedean case If p =, let A /(N a) /a / d =, where as efore we write disc(s) = N 2/N 0 2 d, with d a fundamental discriminant. Then S = a t A SA = 2, the identity matrix. We normalize so that B 2 () = e 4π ; this is possile y Theorem 3.0 of 7. What naturally appears when we relate Bessel models to Fourier coefficients is the value B () S = B 2 A ( a t A ). Calculating this value requires decomposing the argument of B 2 as thk, where t T S (R), the matrix h is diagonal, and k is in the standard maximal compact sugroup of Sp 4 (R). Then one may use the explicit formula given in Theorem 3.0 of 7. The result is as follows. 6.2 Lemma. With the aove notations and normalizations, we have We postpone the proof of this lemma until the next section. (43) B S () = det(s) k/2 e 2π Tr(S). (44)

20 7 EXPLICIT FORMULAS FOR BESSEL FUNCTIONS 20 Gloal normalization Recall that our starting point was a Saito-Kurokawa lift F S (2) k (Γ) and its associated adelic function Φ F defined in (20). Having fixed the local vectors vp 0 at each place, we may normalize the isomorphism V = V p such that Φ F corresponds to the pure tensor vp. 0 Given S P 2, let C S e the constant defined y (3). Having fixed the vectors vp, 0 the functionals β p, and the isomorphism V = V p, the constants C S are well-defined. By definition, Φ S F (g) = C S Bp S (g p ), (45) p for all g = (g p ) G(A). The values of the constants C S are unknown, ut the following property will e sufficient to derive the Maass relations. 6.3 Lemma. Let S, S P 2 such that Φ S F (), ΦS F () 0 and S = v t ASA for some A GL 2 (Q) and v Q. Then C S = C S. Proof. We have Φ S F () = Φ S A F ( v t A ) = C S p On the other hand, the left hand side equals C S Bp S A ( v t A ) = C S Bp S (). p Bp S (). The assertion follows. p 7 Explicit formulas for Bessel functions In this section we provide the proofs of Lemmas 6. and 6.2. Non-archimedean case We first consider p <. Recall that S satisfies the standard assumptions (6). In view of (8), h given an element g GSp 4 (Q p ) of the form g = v t h, we want to know which doule coset R(Q p )h(l, m)gsp 4 (Z p ) it elongs to. For this, we first state the following result for GL 2. a /2 7. Lemma. Suppose that S = satisfies the standard assumptions (6). Let h /2 c GL 2 (Q p ), and assume that, according to the decomposition (7), p m h = t k (46) with t T S (Q p ), a non-negative integer m, and k GL 2 (Z p ). Define a,, c, d Q p y a c d = det(h) t hs h. Then m = max(0, v(a ), v(d )).

21 7 EXPLICIT FORMULAS FOR BESSEL FUNCTIONS 2 Proof. Taking determinants on the identity (46), we get A calculation shows that If we let k = w x, it follows that y z det(h) = det(t)p m det(k). (47) det(h) t hs h = det(k) t ap k m /2 /2 cp m k. a = det(k) ( w 2 ap m + wy + y 2 cp m), d = det(k) ( x 2 ap m + xz + z 2 cp m). Note that all quantities except (possily) p m are integral, and that c is a unit. Also, one of y or z is a unit. Hence, a, d o m = 0. Assume a and d are not oth in o. Then m > 0, the valuation of a and d is m, and at least one of a or d has valuation m. It follows that m = min(v(a ), v(d )), or equivalently m = max( v(a ), v(d )). This concludes the proof. Using this lemma, it is straightforward to derive the following result for GSp 4. h 7.2 Lemma. Let g = v t h GSp 4 (Q p ) with h GL 2 (Q p ) and v Q p. Then g R S (Q p )h(l, m)gsp 4 (Z p ), where m is the integer otained from Lemma 7. for h, and the value of l is given y l = v p (det(h)/v) m. We can now give the proof of Lemma 6.: From Lemmas 7. and 7.2, since det(a ) t A S A = as det(a) ( and det(a) = /(N a), we get m = max(0, v N ) ( p a, N ) vp c ). Making use of gcd(a,, c) = gcd(a, c, N ), the result follows. Archimedean case Now consider p =. The values of B S have een computed in 7. In this case S = 2, γ and hence T S (R) = { : γ > 0}SO(2). We have the following disjoint doule coset γ decomposition, GSp 4 (R) = R(R){h(λ, ζ) : λ R, ζ }K, where K is the maximal compact sugroup of Sp 4 (R), R(R) = T 2 (R)U(R) is the Bessel sugroup, and ζ λ h(λ, ζ) = ζ ζ. ζ

22 7 EXPLICIT FORMULAS FOR BESSEL FUNCTIONS 22 If g = uth(λ, ζ)k 0, with u U(R), t T 2 (R) and k 0 K, then, y Theorem 3.0 of 7, B 2 (g) = Θ 2 (u)b 2 (h(λ, ζ)) = Θ 2 (u)λ k e 2πλ(ζ2 +ζ 2). (48) A Now we need to otain the decomposition of g A = a t A as uth(λ, ζ)k 0. Clearly, u =. Also, with A = 7.3 Lemma. Let h = a dn 2 2 = a 2 dn = u 2 dn u = a, x = /2 2 u, and y = a u. y x y, y x ζ y with y 0. Then h = k ζ k 2, with k, k 2 SO(2) and ζ 2 = + x2 y 2 + y 4 + ( + x 2 y 2 + y 4 ) 2 4y 4 2y 2. Proof. We may assume that x 0. By the Cartan decomposition of SL 2 (R), there exist ζ cos(θ) sin(θ) k, k 2 SO(2) and ζ > such that h = k k ζ 2. Write k = for sin(θ) cos(θ) θ 0, 2π. Applying oth sides of h = k ζ ζ k 2 to i as fractional linear transformations, and using that SO(2) stailizes i, we get y 2 i + xy = cos(θ)ζ2 i + sin(θ) sin(θ)ζ 2 i + cos(θ). Simplifying and comparing the coefficients of i and the constant terms, we get ζ 2 xy sin(θ) = cos(θ)(ζ 2 y 2 ), ( ζ 2 y 2 ) sin(θ) = xy cos(θ). Note that, since x, y 0, we have sin(θ), cos(θ) 0 and y ±ζ, ±/ζ. Hence, we can divide the aove two equations and after simplification otain y 2 ζ 4 ( + x 2 y 2 + y 4 )ζ 2 + y 2 = 0, which gives the lemma. Finally, we can give the proof of Lemma 6.2: Apply Lemma 7.3 with x = (/2)/u and y = a/u, where u = a dn 2, to get A = uk ζ ζ k 2,

23 8 FORMULAS FOR FOURIER COEFFICIENTS 23 with ζ as in the statement of the lemma and k, k 2 SO(2). Then u 2 ζ g A = au k a t k ζ ζ ζ as required. Now (44) follows from (48). 8 Formulas for Fourier coefficients We again consider the situation where F S (2) k (Sp 4(Z)), k even, is a Hecke eigenform and is a Saito-Kurokawa lift of f S 2k 2 (SL 2 (Z)). For every prime p < let α p e the Satake parameter of f at p. Let a(s) denote the Fourier coefficient of F at S P 2. a /2 8. Proposition. Let S = P /2 c 2 e such that a(s) 0. Assume that disc(s) = N 2d where d < 0 is a fundamental discriminant. Let L = c(s) = gcd(a,, c). For any prime p <, set a p = v p (N ) and p = v p (L) and let δ p = ( d p) e the Legendre symol (y convention, δp = 0 if p divides d). Let C S e the constant defined y (45). Then k2 t k 2, a(s) = C S (N ) 3/2 det(s) k 2 p p i 2 p N i=0 ( α a p i+ p α ap+i p α p α p δ p p 2 α ap i p α ap+i p α p α p ). (49) Proof. By Proposition 3. and (45), Using Lemma 6.2, it follows that a(s)e 2πTr(S) = Φ S F () = C S a(s) = C S det(s) k/2 p< p B S p (). B S p (). Hence, y Lemma 6., a(s) = C S det(s) ( ( ( k/2 N ))) Bp S h v p (L), v p L p N = C S det(s) k/2 p N B S p (h( p, a p p )). By our remarks in Sect. 4, the non-archimedean local components π p of the automorphic representation π F are of the form χ p GL(2) χ p for an unramified character χ p of Q p (type II). In fact, χ p is the unramified character with χ p (p) = α p. Sustituting the formulas in iii) and v) of Theorem 2. for the local Bessel functions, we otain (49). Proposition 8. has een generalized to Hilert-Siegel modular forms in 4.

24 9 ANOTHER PROOF OF THE CLASSICAL MAASS RELATIONS 24 The formula of Das-Kohnen-Sengupta Let f(τ) = n= c(n)e2πinτ S 2k 2 (SL 2 (Z)) e an elliptic eigenform, with k even, and assume that F is the Saito-Kurokawa lift of f. If α p is the Satake parameter at p, then Hence formula (49) may e rewritten as a(s) = C S N 3/2 det(s) k 2 = C S N k det(s) k 2 c(p µ µ(2k 3)/2 αµ+ p ) = p p p N i=0 p p N i=0 α µ p α p α p. ( ) p i 2 p (a p i)(2k 3)/2 c(p ap i ) δ p p 2 p (2k 3)/2 c(p ap i ) p i(k )( ) c(p ap i ) δ p p k 2 c(p ap i ). Now assume that L = c(s) = N. This is equivalent to saying that S is a multiple of a matrix with fundamental discriminant. More precisely, putting n = L = N, we see that S = nt where T P 2 is such that disc(t ) = d. Hence, a(nt ) = C nt n k det(s) k 2 = C T n k det(s) k 2 = C T det(t ) k 2 p ν n p n p ν n p i=0 ν i=0 p i(k )( ) c(p ap i ) δ p p k 2 c(p ap i ) p i(k )( ) c(p ν i ) δ p p k 2 c(p ν i ) ν p (ν i)(k )( ) c(p i ) δ p p k 2 c(p i ). i=0 This coincides with the formula in Lemma 3. of 3. Comparison with this formula shows that C T det(t ) k 2 is a Fourier coefficient of the modular form of weight k /2 corresponding to f under the Shimura lifting. 9 Another proof of the classical Maass relations In this section we will give another proof of the Maass relations satisfied y the Fourier coefficients of a Saito-Kurokawa lift using our knowledge aout Bessel models for the underlying automorphic representation, without recourse to the classical construction. This proof will not use Theorem Theorem. Let F e a cuspidal Siegel Hecke eigenform of weight k with respect to Sp 4 (Z) which is a Saito-Kurokawa lift. For S P 2, let a(s) denote the Fourier coefficient of F at S. Then a a( 2 2 c ) = r k a( r gcd(a,,c) ac r 2 2r 2r ). (50)

25 9 ANOTHER PROOF OF THE CLASSICAL MAASS RELATIONS 25 Proof. Let S = a 2 2 c Let c(s) = L. For r L, set S (r) = Using Lemma 6.2, we get Hence, y Lemma 6., Analogously, a(s) = C S det(s) k/2. As usual, we write disc(s) = N 2 d where d is a fundamental discriminant.. By Proposition 3. and (45), = C S det(s) k/2 ( p L ac r 2 2r 2r a(s)e 2πTr(S) = Φ S F () = C S a(s) = C S det(s) k/2 p L or p (N /L) B S p a(s (r) ) = C S (r) det(s (r) ) k/2 p N /r ( h = C S (r) det(s (r) ) k/2 ( p L p N /r p< p B S p (). B S p (). ( ( ( N ))) Bp S h v p (L), v p L ( v p (L), v p ( N L ( ( ( N ))) Bp S h 0, v p r ( ( Bp S h 0, v p ( N r ))) )( p L p N ))) )( ( Bp S h(0, vp (N )) )). p L p N ( Bp S h(0, vp (N )) )). If N /r is not divisile y p, then B S p (h(0, v p (N /r))) = y our normalizations. Hence, the second condition under the first product sign can e omitted. Since C S = C S (r) y Lemma 6.3, we conclude from the aove equations that r k a(s (r) ) ( ( ( N ))) Bp S h v p (L), v p L p L = a(s) r p L ( ( Bp S h 0, v p ( N r ))). Applying r L to oth sides gives ( ) r k a(s (r) ) r L p L ( ( Bp S h v p (L), v p ( N L ))) = a(s) r L r p L ( ( Bp S h 0, v p ( N r ))). Since our assertion is a(s) = r L rk a(s (r) ), we are done if we can prove that ( ( ( N ))) Bp S h v p (L), v p = L r L p L r p L ( ( Bp S h 0, v p ( N r ))). (5)