Local and global Maass relations.

We characterize the irreducible, admissible, spherical representations of GSp 4 ( F ) (where F is a p -adic ﬁeld) that occur in certain CAP representations in terms of relations satisﬁed by their spherical vector in a special Bessel model. These local relations are analogous to the Maass relations satisﬁed by 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 be 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 Jacobi 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
Let F be 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 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, 1 symmetric and positive semidefinite 2 × 2-matrices S. We say that F satisfies the Maass relations if, for all S = a b/2 b/2 c , 1 Recall that a matrix is semi-integral if its off-diagonal entries are in 1 2 Z, and its diagonal entries are in Z.
The first and third authors are supported by NSF grant DMS-1100541. 1 .
(2) Such a relation was first known to be satisfied by Eisenstein series; see [20]. Maass, in [11], 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 by 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 book [3] 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 possible to construct Saito-Kurokawa liftings using automorphic representations theory. For simplicity, we only consider cuspforms. The procedure may be illustrated as follows: 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 be 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 ∼ = SO 3 , there is a theta correspondence between this group and SL 2 (A), the double 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 be any pre-image of π under this lifting. 2 Since PGSp 4 ∼ = SO 5 , there is another theta correspondence between SL 2 (A) and PGSp 4 (A). We can use it to forward τ to an automorphic representation of PGSp 4 (A). From this one can extract a Siegel modular form F of weight k. It turns out that 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 [16] and [26]. There is a marked difference between 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 with even k there exists a cuspidal Siegel eigenform F of weight k such that its spin L-function is given by L(s, F) = L(s, f )ζ (s − k + 1)ζ (s − k + 2)." In this case the Fourier coefficients of F are not readily available.
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 [4] or [13]. 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 because of the appearance of the zeta factors above.
However, it would be desirable 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 [4] or [13] are not available. For example, what happens if we replace the above condition "k even" by "k odd"? In this case it turns out that one can still do the representation theoretic construction, the difference being that the archimedean component of the automorphic representation is no longer in the holomorphic discrete series. Hence, one will obtain a certain type of non-holomorphic Siegel modular form whose adelization generates a global CAP representation. As far as we know, the full details of this construction have yet to be written out (though see [12]). 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 subgroups.
It was shown in [14] that a representation theoretic method for proving the Maass relations exists. In the present paper we take a similar, but slightly different approach. Common to both approaches is the fact that certain local Jacquet modules are one-dimensional. This may be 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 [14] makes use of certain Siegel series to derive an explicit formula for local p-adic Bessel functions, we use Sugano's formula, to be found in [28].
Our main local result, Theorem 2.1 below, asserts the equivalence of five conditions on a given irreducible, admissible, 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 1/2 (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 global CAP representations with respect to the Borel or Siegel parabolic subgroup. 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 [14] for the values of a Siegel series.
In the global 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 in the diagram (3) are of the kind covered in Theorem 2.1. Hence, the corresponding spherical Bessel functions satisfy the "local Maass relations" (this implication is all that is needed from Theorem 2.1). Since Bessel models are closely related to Fourier coefficients, one can deduce the global (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 be 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 by the second author in collaboration with Kowalski and Tsimerman [10], while the other relies on some explicit computations with Bessel functions which may be of independent interest. As explained above, what we have in mind are future applications to more general situations. Finally, in the last section, we prove a result (Theorem 8.1) that gives a new characterization of Saito-Kurokawa lifts involving a certain average of Fourier coefficients.

Notation
Let G = GSp 4 be the group of symplectic similitudes of semisimple rank 2, defined by Here, μ is called the similitude character. Let Sp 4 = {g ∈ GSp 4 : μ(g) = 1}. The Siegel parabolic subgroup P of GSp 4 consists of matrices whose lower left 2 × 2-block is zero. Its unipotent radical U consists of all elements of the block form 1 X 1 , where X is symmetric.
The standard Levi component M of P consists of all elements A u t A −1 with u ∈ GL 1 and A ∈ GL 2 .
Over the real numbers, we have the identity component G(R) + := {g ∈ GSp 4 (R) : μ(g) > 0}. Let H 2 be 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) + Given any commutative ring R, we denote by Sym 2 (R) the set of symmetric 2×2-matrices with coefficients in R. The symbol P 2 denotes the set of positive definite, half-integral symmetric 2 × 2-matrices.

Spherical Bessel functions
In this section only, F is a local non-archimedean field with ring of integers o, prime ideal p, uniformizer and order of residue field q. An irreducible, admissible 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 ) be such a representation. Then π is a constituent of a representation parabolically induced from a character γ of the standard Borel subgroup of GSp 4 (F). The numbers are called the Satake parameters of π. The conjugacy class of diag(γ (1) , γ (2) , γ (3) , γ (4) ) in GSp 4 (C) determines the isomorphism class of π. Note that γ (1) γ (3) = γ (2) γ (4) = ω π ( ), where ω π is the central character of π. Hence, in the case of trivial central character, the Satake parameters are {α ±1 , β ±1 } for some α, β ∈ C × . In this case we allow ourselves a statement like "one of the Satake parameters of π is α ±1 ".
In this work we will employ the notation of [29] and the classification of [21] for constituents of parabolically induced representations of GSp 4 (F). According to Table A.10 of [21], the spherical representations are of type I, IIb, IIIb, 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 irreducible principal series representations, and they are the only generic spherical representations.
By [26], representations of type IIb occur as local components of the automorphic representations attached to classical Saito-Kurokawa liftings. Recall that these automorphic representations are CAP (cuspidal associated to parabolic) with respect to the Siegel parabolic subgroup; this property has been defined on p. 315 of [16], 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 subgroup. By Theorem 2.2 of [16], P-CAP and B-CAP representations with trivial central character have a common characterization as being theta liftings from the metaplectic cover of SL 2 .
We will see below that spherical representations of type IIb, Vd and VId with trivial central character have a common characterization in terms of their spherical Bessel functions.
We will briefly recall the notion of Bessel model; for more details see [22] (and [17] for the archimedean case). Let ψ be a fixed character of F.
Such a matrix defines a character θ = θ S of U , the unipotent radical of the Siegel parabolic subgroup, by Every character of U is of this form for a uniquely determined S. From now on we will assume that θ is non-degenerate, by which we mean that S is invertible. 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 symbol Let T = T S be the torus defined by T = {g ∈ GL 2 : t gSg = det(g)S}. Then T (F) ∼ = L × .
We think of T (F) embedded into GSp 4 (F) via g → g det(g) t g −1 . Then T (F) is the identity component of the stabilizer of θ in the Levi component of the Siegel parabolic subgroup. We call the semidirect product R = T U the Bessel subgroup defined by S. Given a character of 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 6.1.4 of [22] every infinite-dimensional π admits a Bessel functional for some choice of θ and . The question of uniqueness is discussed in Sect. 6.3 of [22]. Bessel models with = 1 are called special.
In the case of spherical representations, one may ask about an explicit formula for the spherical vector in a ( , θ )-Bessel model. Such a formula was given by Sugano in [28] (at the same time proving that such models are unique for spherical representations). In the case that is unramified, Sugano's formula is conveniently summarized in Sect. (3.6) of [6]. We recall the result. To begin, we assume that the elements a, b, c defining the matrix S satisfy the following standard assumptions: This is not a restriction of generality; using some algebraic number theory, one can show that, after a suitable transformation S → λ t AS A with λ ∈ F × and A ∈ GL 2 (F), the standard assumptions are always satisfied. One consequence of (6) is the decomposition see Lemma 2-4 of [28]. In conjunction with the Iwasawa decomposition, this implies where

Hence, a spherical Bessel function B is determined by 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 [6]). Sugano's formula now says that 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 ; see p. 205 of [6] for details. The formula implies in particular that B(1) = 0, so that we may always normalize B(1) to be 1.
With these preparations, we may now formulate our main local theorem.

Theorem
Let π be an irreducible, admissible, 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 be the spherical vector in such a Bessel model for π, normalized such that B(1) = 1. Then the following are equivalent.
(i) One of the Satake parameters of π is q ±1/2 . (ii) π is one of the representations in the following list: (iv) The following relation is satisfied: (v) The following two conditions are satisfied: • For all m ≥ 0, where α ±1 is one of the Satake parameters of π.
Proof (i) ⇔ (ii) follows by inspecting the list of Satake parameters of all spherical representations; see Table A.7 of [21]. (10) is equivalent to the following identity between generating series, By Sugano's formula, the left hand side equals with H, P, Q as in Proposition 2-5 of [28]. For the right hand side of (13), we calculate Hence, (13) is equivalent to If one of the Satake parameters is q ±1/2 , then one can verify that (14) is satisfied. This shows that (i) ⇒ (iii). Conversely, assume that (iv) is satisfied. Let F(x, y) be the polynomial on the left hand side of (14). Then (iv) is equivalent to saying that the coefficient of y of the power series which has no constant term, vanishes. In particular, this means that the y 2 -coefficient of where α ±1 , β ±1 are the Satake parameters of π. It follows that α = q ±1/2 or β = q ±1/2 . The completes the proof of (iv) ⇒ (i).
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 be omitted, since in this exceptional situation the formula (12) holds as well. (ii) There is a certain analogy of the identity (10) 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 10 in [9] and Corollary 5.1 in [14]. In fact, formula (17) appears as Lemma 8.1 of [14].

Adelization and Fourier coefficients
We turn to classical Siegel modular forms and their adelization. Let = Sp 4 (Z) and S (2) k ( ) be the space of holomorphic cuspidal Siegel modular forms of degree 2 and weight k with respect to . Hence, if F ∈ S (2) k ( ), then for all γ ∈ we have F| k γ = F, where for g ∈ G(R) + and Z ∈ H 2 , the Siegel upper half space. Here j (g, The Fourier expansion of F is given by where the sum is taken over the set P 2 of semi-integral, symmetric and positive definite matrices S. Let A be the ring of adeles of Q. It follows from the strong approximation theorem for Sp 4 that where K 0 := p<∞ p with p = G(Z p ). Let F ∈ S (2) k ( ). Write g ∈ G(A) as g = g Q g ∞ g 0 with g Q ∈ G(Q), g ∞ ∈ G(R) + , g 0 ∈ K 0 , and define F : G(A) → C by the formula Since G(R) Here Z GL 1 is the center of GSp 4 and K ∞ U (2) is the standard maximal compact subgroup of Sp 4 (R).
Let P = MU be the Siegel parabolic subgroup of G. By the Iwasawa decomposition, Note that every character of U (Q)\U (A) is obtained in this way. For S ∈ Sym 2 (Q) we define the following adelic Fourier coefficient of F , The following result, which is standard, provides a formula for S F (g) in terms of the Fourier coefficients of F.

Proposition
k ( ) with Fourier expansion (19) and let S F be as defined in (23).
Also, for S ∈ Sym 2 (Q) and S = v t AS A, with A ∈ GL 2 (Q) and v ∈ Q × , we have The proof of this result is standard. We provide the details in the long version [18] of this article.

Special automorphic forms
We now assume that F ∈ S (2) k ( ) 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 [3]. Let F be as defined in (21) and let (π F , V F ) be the irreducible cuspidal automorphic representation of G(A) generated by right translates of F . Then π F is isomorphic to a restricted tensor product ⊗ p≤∞ π p with irreducible, admissible 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 by the weight k. Following the notation of [17], we denote this representation by E(k, k). • For a prime number p, the representation π p is a degenerate principal series representation χ1 GL(2) χ −1 with an unramified character χ of Q × p . Here, we are using the notation of [21]. In particular, π p is a representation of type IIb according to Table A.1 of [21]. Note that these are non-tempered, non-generic representations.

Lemma Let p be a prime number or p = ∞. Let S ∈ Sym 2 (Q p ) be non-degenerate. In the archimedean case, assume also that S is positive or negative definite. Let T S be the connected component of the stabilizer of the character S of U (Q p ). Explicitly,
where we embed GL 2 into GSp 4 via g → g det(g) t g −1 . Let V p be a model for π p , and consider functionals β p : V p → C with the property β p (π p (n)v) = S (n)β p (v) for all v ∈ V p and n ∈ U (Q p ). Then: (i) The space of such functionals β p is one-dimensional.
(ii) If β p satisfies (28), then it automatically satisfies Proof This follows from Lemma 5.2.2 of [22] in the non-archimedean case, and from Theorem 3.10 of [15] in the archimedean case.
In the language of Bessel models, Lemma 4.1 states that the only such model admitted by 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 [15], Sect. 2.6, for the definition in the archimedean case. Part (i) of Lemma 4.1 asserts the uniqueness of such models. We remark that property (ii) in this lemma is precisely the "U -property" of [15].
We fix a distinguished vector v 0 p ∈ V p for each of our local representations π p : If p is finite, we let v 0 p be a spherical (i.e., non-zero G(Z p )-invariant) vector, and if p = ∞ we let v 0 p be a vector spanning the one-dimensional K ∞ -type determined by 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 v 0 p for this purpose. Let S ∈ Sym 2 (Q) be positive or negative definite. We will see in a later section that β p (v 0 p ) = 0 for almost all p. For those places where this is the case, we normalize the β p such that β p (v 0 p ) = 1. The following lemma states that the automorphic forms in the space of π are special in the sense of [15], p. 310.

Lemma Let the notations be as above. Then, for any non-degenerate S ∈ Sym 2 (Q), and all ∈ V F , we have S (mg) = S (g) for all g ∈ G(A) and m ∈ T S (A). (30)
Proof We fix S. The assertion is trivial if the functional is zero. Assume that β is non-zero. For each place p, let V p be 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 [17] implies that S is positive or negative definite. By the uniqueness asserted in Lemma 4.1, it follows that there exists a non-zero constant C S such that whenever ∈ V F corresponds to the pure tensor ⊗v p via V F ∼ = ⊗V p ; note that the product on the right is finite by our normalizations. Using (ii) of Lemma 4.1, it follows that β(π(m) ) = β( ) for all m ∈ T S (A). Since is arbitrary, this implies the assertion of the lemma.

A proof of the classical Maass relations
and call c(S) the content (resp., call disc(S) the discriminant) of S. For S 1 , S 2 ∈ P 2 , we say that S 1 ∼ S 2 if there exists a matrix A ∈ SL 2 (Z) such that t AS 1 A = S 2 . For any S ∈ P 2 , let [S] denote the equivalence class of S under the above 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 1 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 Our objective in this section is to prove the following theorem.

Theorem Let F be 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 1 ) = disc(S 2 ) and c(S 1 ) = c(S 2 ). Then a(S 1 ) = a(S 2 ). . (33) Proof In the notation of the above theorem, gcd(a, b, c) 2 ; gcd(a, b, c)), a Now the corollary follows immediately from (32).
Let us now prove Theorem 5.1. As a first step, we recall a very useful characterization of the elements of H (D; L). Let d < 0 be a fundamental discriminant. 3 We define S d ∈ P 2 as follows.
For any positive integer M, we let K We define Cl d (M) as follows: It is easy to see that Cl d (M) is finite. For example, Cl d (1) 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 ) be a representative for c. By strong approximation, we can write (non-uniquely) with γ c ∈ GL(2, Q) + , and κ c ∈ p<∞ K (0) p (M). It is known (see [6, p. 209 It follows that c(φ L ,M (c)) = L, disc(φ L ,M (c)) = d L 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.

Proposition For each pair of positive integers L , M, the map c → [φ L ,M (c)] gives a well-defined bijection between Cl d (M) and H (d M 2 ; L).
Proof Let us first show that the map is well defined. Let 0 (M) (resp. 0 (M)) be the usual congruence subgroups of SL 2 (Z) consisting of matrices whose upper-right (resp. lower-left) entry is divisible by M. Let c ∈ Cl d (M). Suppose that γ (1) c , γ (2) c are two distinct elements obtained in (36) from c and that φ (1) L ,M (c), φ (2) L ,M (c) are the matrices obtained via (37). Then our definitions imply that there exists t ∈ T (Q), k ∈ 0 (M) such that γ Hence [φ Let us explicate (39) in the special case g = (H L ,M ) f . Note that for i = 1, 2, we have (20). It follows from Proposition 3.1 that Combining (39) and (40), we deduce (38). This completes the proof of the first assertion of Theorem 5.1. To prove the second assertion of Theorem 5.1, we need the following result, which is Theorem 2.10 of [10]. Cl d (1)). Let F be a cuspidal Siegel Hecke eigenform of weight k with respect to Sp 4 (Z) and let π = ⊗ p π p be the irreducible 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 be the spherical vector in the (S d , p , θ p )-Bessel model for π p , normalized so that B p (1) = 1. Then for any positive integers L = p p l p and M = p p m p the relation

Theorem (Kowalski-Saha-Tsimerman) Let d < 0 be a fundamental discriminant and = p p be a character of Cl d (1) (note that induces a character on Cl d (M) for all positive integers M via the natural surjection Cl d (M) →
holds.
Let us see what this implies in the setup of Theorem 5.1. 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 member of P 2 satisfying c(S) = L and disc(S) = DL 2 (this is well-defined by the first assertion of Theorem 5.1, which we have already proven). Then Theorem 5.4, in the special case = 1, F as above, tells us that (l p , m p )). Hence, we are reduced to showing that This equation would follow provided for each prime p | L M we could prove that But this follows from Theorem 2.1. Note here that, by our remarks in Sect. 4, the nonarchimedean local components π p associated to F are of the form χ1 GL(2) χ −1 for an unramified character χ of Q × p (type IIb). This concludes the proof of Theorem 5.1.

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 be fixed. Recall that in Sect. 4 we have fixed distinguished vectors v 0 p in each local representation V p . For any place p let β p be a non-zero functional V p → C as in Lemma 4.1. Let B S p be the Bessel function corresponding to v 0 p , i.e., B S p (g) = β p (π p (g)v 0 p ) for g ∈ G(Q p ). We are going to normalize the β p , hence the B S p , in a certain way.

Non-archimedean case
Assume that p is a prime. It follows from Sugano's formula (9) that B S p (1) = 0, provided S satisfies the standard assumptions (6). Hence, if S satisfies these conditions, then we can normalize B S p (1) = 1. For arbitrary positive definite S in P 2 we proceed as follows. Let disc(S) = N 2 1 d, where d is a fundamental discriminant. Let S d be as in (34). Then we have S d = a t AS A, where For brevity, we put S = S d . We observe that the matrix S satisfies the standard assumptions (6) (8) and then using Sugano's formula (9). It is explained in [18] how to obtain such a decomposition; here, we just state the result.

Lemma Let A, S, S , d, N 1 be as above. Let L = c(S). Then
Consequently, B S p (1) = 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 be evaluated using Sugano's formula (9).

Archimedean case
, where as before we write disc(S) = N 2 1 d, with d a fundamental discriminant. Then S = a t A S A = 1 2 , the identity matrix. We normalize so that B 1 2 ∞ (1) = e −4π ; this is possible by Theorem 3.10 of [17]. What naturally appears when we relate Bessel models to Fourier coefficients is the value Calculating this value requires decomposing the argument of B 1 2 ∞ as thk, where t ∈ T S (R), the matrix h is diagonal, and k is in the standard maximal compact subgroup of Sp 4 (R). Then one may use the explicit formula given in Theorem 3.10 of [17]. The result is as follows.

Lemma With the above notations and normalizations, we have
We refer to [18] for the details of this calculation.

Global normalization
Recall that our starting point was a Saito-Kurokawa lift F ∈ S (2) k ( ) and its associated adelic function F defined in (20). Having fixed the local vectors v 0 p at each place, we may normalize the isomorphism V ∼ = ⊗V p such that F corresponds to the pure tensor ⊗v 0 p . Given S ∈ P 2 , let C S be the constant defined by (31). Having fixed the vectors v 0 p , the functionals β p , and the isomorphism V ∼ = ⊗V p , the constants C S are well-defined. By definition, for all g = (g p ) ∈ G(A). The values of the constants C S are unknown, but the following property will be sufficient to derive the Maass relations.

Lemma
Let S, S ∈ P 2 such that S F (1), S F (1) = 0 and S = v t AS A for some A ∈ GL 2 (Q) and v ∈ Q × . Then C S = C S .
On the other hand, the left hand side equals C S p≤∞ B S p (1). The assertion follows.
In this section we will give another proof of the Maass relations satisfied by the Fourier coefficients of a Saito-Kurokawa lift using our knowledge about Bessel models for the underlying automorphic representation, without recourse to the classical construction. This proof will not use Theorem 5.4.

Theorem
Analogously, a(S (r ) ) = C S (r) det(S (r ) ) k/2 If N 1 /r is not divisible by p, then B S p (h(0, v p (N 1 /r ))) = 1 by our normalizations. Hence, the second condition under the first product sign can be omitted. Since C S = C S (r) by Lemma 6.3, we conclude from the above equations that infinitely many relations on the Fourier coefficients of F. The advantage of those conditions, on the other hand, is that they also apply to non-eigenforms. Indeed, a Siegel cusp form F of weight k with respect to Sp 4 (Z) satisfies any of those conditions if and only if it lies in the Maass Spezialschar, i.e., it is a linear combination of eigenforms that are Saito-Kurokawa lifts.
In [5], some new characterizations for a Siegel eigenform being a Saito-Kurokawa lift were proved. These involved a single condition on the Hecke eigenvalues at a single prime. Unlike the conditions referred to in the previous paragraph, the new characterizations of [5] are only applicable to eigenforms.
Below, we prove yet another condition that if satisfied implies that a Siegel eigenform is a Saito-Kurokawa lift. Like in [5], this new condition only applies to Hecke eigenforms (though we need this only at a single prime) and involves checking a single condition. However, unlike in [5], this new condition is phrased purely in terms of Fourier coefficients; thus, it is closer in spirit to the original relations of Maass, Andrianov, and Zagier. Indeed, this new condition, as phrased in (50) below, is nothing but a single "Maass relation on average". Let be a Siegel cusp form of weight k with respect to Sp 4 (Z). Let D < 0 be a discriminant and L > 0 an integer. Recall that H (D; L) denotes the set of equivalence classes of matrices in P 2 whose content is equal to L and whose discriminant is equal to DL 2 . We define In other words, a(D; L) is the average of the Fourier coefficients of F over matrices of discriminant DL 2 and content L. We now state our result.

Theorem Let F be a Siegel cusp form of weight k with respect to Sp 4 (Z).
Suppose that there is a prime p such that F is an eigenform for the local Hecke algebra at p (equivalently, F is an eigenform for the Hecke operators T ( p) and T ( p 2 )). For each discriminant D < 0 and each positive integer L, let a(D; L) be defined as in (49). Then the following are equivalent.
(i) F lies in the Maass Spezialschar.
Proof Suppose that F lies in the Maass Spezialschar. Then a(D; L) = a(D; L) where a(D; L) is defined as in Theorem 5.1. It follows immediately from Theorem 5.1 that (50) is satisfied for all d and p. A special case of the main result of [24] tells us that there exists a fundamental discriminant d < 0 such that a(d; 1) = 0. Next, suppose that there exists a fundamental discriminant d < 0 such that a(d; 1) = 0. Since F is an eigenform of the local Hecke algebra at p, there exists a well-defined representation π p of GSp 4 (Q p ) attached to F. Indeed, for each irreducible subrepresentation π of the representation of G(A) generated by F , the local component of π at p is isomorphic to π p . By writing F as a linear combination of Hecke eigenforms, applying Theorem 5.4 on each of these eigenforms and then putting it all together, we conclude that, for all non-negative integers l, m, a(dp 2m ; p l ) = p (l+m)k a(d; 1)B p (h(l, m)), where B p is the spherical vector in the (S d , 1, θ p )-Bessel model for π p . The relation (50) together with the non-vanishing of a(d; 1) now implies that B p (h(1, 0)) = B p (h(0, 1)) + p −1 . (51) Using Theorem 2.1, it follows that one of the Satake parameters of π p is p ±1/2 . Thus, each irreducible subrepresentation of the representation of G(A) generated by F is non-tempered at p. By a result of Weissauer [30], it follows that each irreducible subrepresentation of the representation of G(A) generated by F is of CAP-type. This is equivalent to saying that F lies in the Maass Spezialschar.
Theorem 8.1 tells us that if an eigenform F lies in the orthogonal complement of the Maass Spezialschar and satisfies a(d; 1) = 0 for some fundamental discriminant d < 0, then a(d; p) = a(dp 2 ; 1) + p k−1 a(d; 1) for every prime p. We end this section with a slightly weaker version of this result that applies to non-eigenforms.

Theorem
Let F be a Siegel cusp form of weight k with respect to Sp 4 (Z). Suppose that F lies in the orthogonal complement of the Maass Spezialschar. For each discriminant D < 0 and each positive integer L, let a(D; L) be defined as in (49). Let d < 0 be a fundamental discriminant such that a(d; 1) = 0. Then, for all sufficiently large primes p, we have a(d; p) = a(dp 2 ; 1) + p k−1 a(d; 1). (52) Proof Write F = s i=1 F i such that each F i is a Hecke eigenform. Let a i (D; L) be the quantity corresponding to F i . Denote c i = a i (d; 1). So we have s i=1 c i = a(d; 1) = 0. Now, as in the proof of Theorem 8.1, let π i, p be the representation of GSp 4 (Q p ) attached to F i at each prime p. Note that by our assumption, and by the result of Weissauer [30], the representation π i, p is tempered. As before, we have a i (dp 2m ; p l ) = p (l+m)k a i (d; 1)B i, p (h(l, m)), where B i, p is the spherical vector in the (S d , 1, θ p )-Bessel model for π i, p . This implies that a(d; p) − a(dp 2 ; 1) − p k−1 a(d; p (h(1, 0)) − B p (h(0, 1)) − p −1 ). (53) Using Sugano's formula, and using the fact that the local parameters of π i, p lie on the unit circle, it is easy to check that It follows that a(d; p) − a(dp 2 ; 1) − p k−1 a(d; 1) = 0 for all sufficiently large primes p.