On the standard $L$-function for $GSp_{2n} \times GL_1$ and algebraicity of symmetric fourth $L$-values for $GL_2$

We prove an explicit integral representation -- involving the pullback of a suitable Siegel Eisenstein series -- for the twisted standard $L$-function associated to a holomorphic vector-valued Siegel cusp form of degree $n$ and arbitrary level. In contrast to all previously proved pullback formulas in this situation, our formula involves only scalar-valued functions despite being applicable to $L$-functions of vector-valued Siegel cusp forms. The key new ingredient in our method is a novel choice of local vectors at the archimedean place which allows us to exactly compute the archimedean local integral. By specializing our integral representation to the case $n=2$ we are able to prove a reciprocity law -- predicted by Deligne's conjecture -- for the critical special values of the twisted standard $L$-function for vector-valued Siegel cusp forms of degree 2 and arbitrary level. This arithmetic application generalizes previously proved critical-value results for the full level case. By specializing further to the case of Siegel cusp forms obtained via the Ramakrishnan--Shahidi lift, we obtain a reciprocity law for the critical special values of the symmetric fourth $L$-function of a classical newform.


Critical L-values
The critical values of L-functions attached to cohomological cuspidal automorphic representations of algebraic groups are objects of deep arithmetic significance. In particular, it is expected that these values are algebraic numbers up to multiplication by suitable automorphic periods. This is closely related to a famous conjecture of Deligne [12] on the algebraicity of critical values of motivic L-functions up to suitable periods (however, it is often a non-trivial problem to relate the automorphic periods to Deligne's motivic periods).
The simplest case of algebraicity of critical L-values is the (classical) fact that ζ(2n) π 2n is a rational number for all positive integers n (the Riemann zeta function ζ(s) being the L-function associated to the trivial automorphic representation of GL 1 ). In the case of GL 2 , Shimura [51,52,53,54] and Manin [34] were the first to study the arithmetic of critical L-values. For higher rank groups, initial steps were taken by Harris [21] and Sturm [60] in 1981, who considered automorphic representations of GSp 2n × GL 1 whose finite part is unramified and whose infinity type is a holomorphic discrete series representation with scalar minimal K-type. Since then, there has been considerable work in this area and algebraicity results for L-values of automorphic representations on various algebraic groups have been proved; the general problem, however, remains very far from being resolved.
In this paper we revisit the case of the standard L-function on GSp 2n × GL 1 . As mentioned earlier, this is the first case outside GL 2 that was successfully tackled, with the (independent) results of Harris and Sturm back in 1981. However, the automorphic representations considered therein were very special and corresponded, from the classical point of view, to scalar-valued Siegel cusp forms of full level. Subsequent works on the critical L-values of Siegel cusp forms by Böcherer [4], Mizumoto [35], Shimura [58], Böcherer-Schmidt [7], Kozima [31], Bouganis [9] and others strengthened and extended these results in several directions. Nonetheless, a proof of algebraicity of critical L-values for holomorphic forms on GSp 2n × GL 1 in full generality has not yet been achieved, even for n = 2.
We prove the following result for n = 2, which applies to representations whose finite part is arbitrary and whose archimedean part can be (almost) any holomorphic discrete series. 1 1.1 Theorem. Let k 1 ≥ k 2 ≥ 3, k 1 ≡ k 2 (mod 2) be integers. For each cuspidal automorphic representation π on GSp 4 (A Q ) with π ∞ isomorphic to the holomorphic discrete series representation with highest weight (k 1 , k 2 ), there exists a real number C(π) with the following properties.
Above, G(χ) denotes the Gauss sum, L S (s, π ⊠χ, ̺ 5 ) denotes the degree 5 L-function (after omitting the local factors in S) associated to the representation π ⊠ χ of GSp 4 × GL 1 , and Q(π, χ) denotes the (CM) field generated by χ and the field of rationality for π.
Before going further, we make a few remarks pertaining to the statement above.
1.2 Remark. Note that we can take S = {∞}, which gives an algebraicity result for the full finite part of the global L-function. This is an important point because some previous works on this topic [7,55,58] either omit the bad local L-factors, impose some extra conditions on them, or define these bad factors in an ad-hoc manner.
1.3 Remark. Our proofs show that C(π) = π 2k 1 F, F where F equals a certain nearly holomorphic modular form of scalar weight k 1 . Alternatively one can take C(π) = π k 1 +k 2 F 0 , F 0 where F 0 equals a certain holomorphic vector-valued modular form of weight det k 2 sym k 1 −k 2 .
Classically, Theorem 1.1 applies to vector-valued holomorphic Siegel cusp forms of weight det k 2 sym k 1 −k 2 with respect to an arbitrary congruence subgroup of Sp 4 (Q). The only previously known result for critical L-values of holomorphic Siegel cusp forms in the vector-valued case (k 1 > k 2 ) is due to Kozima [31]. Kozima's result only applies to full-level Siegel cusp forms,

INTRODUCTION
4 omits some low-weight cases, and also only deals with the case χ = 1. In contrast, our theorem, which relies on an adelic machinery to separate out the difficulties place by place, is more general, especially in that it applies to arbitrary congruence subgroups.
We present an application of Theorem 1.1 to critical values of the symmetric fourth Lfunction of elliptic newforms twisted by odd Dirichlet characters.
1.5 Theorem. Let k ≥ 2 be even. For each cuspidal, non-dihedral, automorphic representation η on PGL 2 (A Q ) with η ∞ isomorphic to the holomorphic discrete series representation of lowest weight k, there exists a real number C(η) with the following properties.
ii) Let χ be an odd Dirichlet character and r be an odd integer such that 1 ≤ r ≤ k − 1.
Furthermore, if χ 2 = 1, we assume that r = 1. Then, for any finite subset S of places of Q that includes the archimedean place, and any σ ∈ Aut(C), we have Our proof of Theorem 1.5 relies on a result of Ramakrishnan and Shahidi [43] which states that given an elliptic cuspidal newform of even weight and trivial nebentypus, there exists a holomorphic vector-valued Siegel cusp form of genus 2 such that the degree 5 standard Lfunction of the Siegel modular form is equal to the symmetric fourth L-function of the elliptic newform. This allows us to derive Theorem 1.5 from Theorem 1.1. One of the reasons to prove Theorem 1.1 with no restrictions on the non-archimedean components is the incomplete information regarding the congruence subgroup associated to the Siegel modular form in [43]. Theorem 1.1 follows from an explicit integral representation (Theorem 1.6 below) for the standard L-function L(s, π ⊠ χ, ̺ 2n+1 ) on GSp 2n × GL 1 , which may be viewed as the main technical achievement of this paper. While Theorem 1.6 is formally similar to the well-known pullback formula (or doubling method) mechanism, what distinguishes it from previous works is the generality of the setup and the fact that all constants are completely explicit. We remark here that Theorem 1.6 applies to any n; we restrict to n = 2 only in Section 7 of this paper, where we prove Theorem 1.1.
In the rest of this introduction we will explain our approach to some of the points mentioned above.

Integral representations for GSp 2n × GL 1 and the pullback formula
The first integral representation for the standard (degree 2n + 1) L-function for automorphic representations of GSp 2n × GL 1 was discovered by Andrianov and Kalinin [1] in 1978. The integral representation of Andrianov-Kalinin applied to holomorphic scalar-valued Siegel cusp forms of even degree n with respect to Γ 0 (N ) type congruence subgroups, and involved a certain theta series. The results of Harris [21] and Sturm [60] mentioned earlier relied on this integral representation.
A remarkable new integral representation, commonly referred to as the pullback formula, was discovered in the early 1980s by Garrett [17] and involved pullbacks of Eisenstein series.
Roughly speaking, the pullback formula in the simplest form says that where F is a Siegel cusp form of degree n and full level that is an eigenform for all the Hecke operators, E k (Z, s) is an Eisenstein series of degree 2n and full level (which becomes a holomorphic Siegel modular form of weight k when s = 0), L(s, π, ̺ 2n+1 ) denotes the degree 2n + 1 L-function for π, and the symbol ≈ indicates that we are ignoring some unimportant factors. The pullback formula was applied by Böcherer [4] to prove various results about the functional equation and algebraicity of critical L-values that went well beyond what had been possible by the Andrianov-Kalinin formula. Subsequently, Shimura generalized the pullback formula to a wide variety of contexts (including other groups). We refer to Shimura's books [57,58] for further details.
In the last two decades, the pullback formula has been used to prove a host of results related to the arithmetic and analytic properties of L-functions associated to Siegel cusp forms. However, most of these results involve various kinds of restrictions. To get the most general results possible, it is necessary to extend (3) to a) incorporate characters, b) include Siegel cusp forms with respect to arbitrary congruence subgroups, and c) cover the case of vectorvalued Siegel cusp forms. While the first two of these objectives have to a large extent been achieved for scalar-valued Siegel cusp forms, the situation with vector-valued forms is quite different. Following the important work of Böcherer-Satoh-Yamazaki [6], there have been a few results about vector-valued forms by Takei [63], Takayanagi [61,62], Kozima [31,32], and others. However, all these works are valid only for full level Siegel cusp forms and involve strong restrictions on the archimedean type. 2 On the other hand, Piatetski-Shapiro and Rallis [38,19] discovered a very general identity (the doubling method) on classical groups. When the group is Sp 2n , this is essentially a generalized, adelic version of the pullback formula described above. 3 However, Piatetski-Shapiro and Rallis computed the relevant local integrals only at the unramified places (where they chose the vectors to be the unramified vector). Thus, to get a more explicit integral representation in the general case, it is necessary to make specific choices of vectors at the ramified and archimedean primes such that one can exactly evaluate all the associated local integrals. So far, this has been carried out in very few situations.
In this paper, we begin by reproving the basic identity of Piatetski-Shapiro and Rallis in a somewhat different manner that is more convenient for our purposes. Let π be a cuspidal automorphic representation of GSp 2n (A) and χ a Hecke character. Our central object of investigation is the global integral Here, E χ (−, s, f ) is an Eisenstein series on GSp 4n (A) associated to a choice of section f ∈ I(χ, s), the pair (h, g) represents an element of GSp 4n (A) corresponding to the diagonal embedding of

INTRODUCTION
6 elements of GSp 2n (A) × GSp 2n (A) with the same multiplier, and φ is an automorphic form in the space of π. The integral (4) represents a meromorphic function of s on all of C due to the analytic properties of the Eisenstein series. As we will prove, away from any poles, the function g → Z(s; f, φ)(g) is again an automorphic form in the space of π. Next, assume that φ = ⊗ v φ v and f = ⊗ v f v are factorizable. We define, for each place v, local zeta integrals where Q n is a certain explicit matrix in GSp 4n (A). It turns out (see Proposition 3.2) that Z v (s; f v , φ v ) converges to an element in the space of π v for real part of s large enough.
Our "Basic Identity" (Theorem 3.6) asserts that the automorphic form Z(s; f, φ) corresponds to the pure tensor ⊗ v Z v (s; f v , φ v ). Now assume that, for all v, the vectors φ v and the sections f v can be chosen in such a way that In this way the Euler product v c v (s), convergent for Re(s) large enough, inherits analytic properties from the left hand side of (6). If this Euler product, up to finitely many factors and up to "known" functions, represents an interesting L-function, one can thus derive various properties of said L-function.
Our main local task is therefore to choose the vectors φ v and the sections f v such that Z v (s, f v , φ v ) = c v (s)φ v for an explicitly computable function c v (s). For a "good" non-archimedean place v we will make the obvious unramified choices. The unramified calculation, Proposition 4.1, then shows that If v is non-archimedean and some of the data is ramified, it is possible to choose φ v and f v such that c v (s) is a non-zero constant function; see Proposition 4.3. The idea is to choose φ v to be invariant under a small enough principal congruence subgroup, and make the support of f v small enough. This idea was inspired by [36].

The choice of archimedean vectors and our main formula
We now explain our choice of φ v and f v at a real place v, which represents one of the main new ideas of this paper. We only treat the case of π v being a holomorphic discrete series representation, since this is sufficient for our application to Siegel modular forms. Assume first that π v has scalar minimal K-type of weight k, i.e., π v occurs as the archimedean component attached to a classical Siegel cusp form of weight k. Then it is natural to take for φ v a vector of weight k (spanning the minimal K-type of π v ), and for f v a vector of weight k in I(χ v , s). Both φ v and f v are unique up to scalars. The local integral in this case is calculated in Proposition 5.5 and turns out to be an exponential times a rational function. The calculation is made possible by the fact that we have a simple formula for the matrix coefficient of φ v ; see (68). Now assume that π v is a holomorphic discrete series representation with more general minimal K-type ρ (k 1 ,...,kn) , where k 1 ≥ . . . ≥ k n > n; see Sect. 5 for notation. In this case we will not work with the minimal K-type, but with the scalar K-type ρ (k,...,k) , where k := k 1 . We will show in Lemma 5.3 that ρ (k,...,k) occurs in π v with multiplicity one. Let φ v be the essentially unique vector spanning this K-type, and let f v again be the vector of weight k in I(χ v , s). The function c v (s) in this case is again an exponential times a rational function; see Proposition 5.8. We note that our calculation for general minimal K-type uses the result of the scalar case, so the scalar case cannot be subsumed in the more general case. One difficulty in the general case is that we do not know a formula for the matrix coefficient of φ v , or in fact any matrix coefficient. Instead, we use a completely different method, realizing φ v as a vector in an induced representation.
Our archimedean calculations require two different integral formulas, which are both expressions for the Haar measure on Sp 2n (R). The first formula, used in the scalar minimal K-type case, is with respect to the KAK decomposition; see (27). The second formula, used in the general case, is with respect to the Iwasawa decomposition; see (88). We will always normalize the Haar measure on Sp 2n (R) in the "classical" way, characterized by the property (12). It is necessary for us to determine the precise constants relating the KAK measure and the Iwasawa measure to the classical normalization. This will be carried out in Appendix A.
Finally, combining the archimedean and the non-archimedean calculations, we obtain an explicit formula for the right hand side of (6), which is our aforementioned pullback formula for L-functions on GSp 2n × GL 1 . This is Theorem 6.1 in the main body of the paper; below we state a rough version of this theorem.
The above formula can be reformulated in a classical language, which takes a similar form to (3) and involves functions F (Z) and E χ k,N (Z, s) that correspond to φ and E χ (−, s, f ) respectively. In fact F and E χ k,N (−, s) are (scalar-valued) smooth modular forms of weight k (and degrees n and 2n respectively) with respect to suitable congruence subgroups. We refer the reader to Theorem 6.4 for the full classical statement. In all previously proved classical pullback formulas [6,61,62] for L(s, π ⊠ χ) with π ∞ a general discrete series representation, the analogues of F and E χ k,N (−, s) were vector-valued objects; in contrast, our formula involves only scalar-valued functions. This is a key point of the present work.
We hope that our pullback formula will be useful for arithmetic applications beyond what we pursue in this paper. A particularly fruitful direction might be towards congruence primes and the Bloch-Kato conjecture, extending work of Agarwal, Berger, Brown, Klosin, and others. Initial steps towards this application have already been made by us in [40] where we build upon the results of this paper, and prove p-integrality and cuspidality of pullbacks of the Eisenstein series E χ k,N (Z, s). It also seems worth mentioning here the recent work of Zheng Liu [33] who uses the doubling method for vector-valued Siegel modular forms and constructs a p-adic L-function.

Nearly holomorphic modular forms and arithmeticity
To obtain results about the algebraicity of critical L-values, we delve deeper into the arithmetic nature of the two smooth modular forms given above. The general arithmetic principle here is that whenever a smooth modular/automorphic form is holomorphic, or close to being holomorphic, it is likely to have useful arithmetic properties. In this case, if 0 ≤ m 0 ≤ k−n−1 2 is an integer, then Shimura has proved that E χ k,N (Z, −m 0 ) is a nearly holomorphic Siegel modular form (of degree 2n) with nice arithmetic properties.
The next step is to prove that the inner product of F (Z) and E χ k,N (Z, −m 0 ) is Aut(C) equivariant. It is here that we are forced to assume n = 2. In this case, our recent investigation of lowest weight modules [39] of Sp 4 (R) and in particular the "structure theorem" proved therein allows us to define an Aut(C) equivariant isotypic projection map from the space of all nearly holomorphic modular forms to the subspace of cusp forms corresponding to a particular infinitytype. Once this is known, Theorem 1.1 follows by a standard linear algebra argument going back at least to Garrett [18].
It is worth contrasting our approach here with previously proved results on the critical Lvalues of holomorphic vector-valued Siegel cusp forms such as the result of Kozima [31] mentioned earlier. In Kozima's work, the modular forms involved in the integral representation are vectorvalued and the cusp form holomorphic; ours involves two scalar-valued modular forms that are not holomorphic. Our approach allows us to incorporate everything smoothly into an adelic setup and exactly evaluate the archimedean integral. But the price we pay is that the arithmeticity of the automorphic forms is not automatic (as we cannot immediately appeal to the arithmetic geometry inherent in holomorphic modular forms). In particular, this is the reason we are forced to restrict ourselves to n = 2 in the final section of this paper, where we prove Theorem 1.1. We expect that an analogue of the structure theorem for nearly holomorphic forms proved in [39] for n = 2 should continue to hold for general n. This is the topic of ongoing work of the authors and will lead to an extension of Theorem 1.1 for any n.

Acknowledgements
We would like to thank A. Raghuram for helpful comments regarding the subject of this paper. A.S. acknowledges the support of the EPSRC grant EP/L025515/1.

Basic notations and definitions
Let F be a totally real algebraic number field and A the ring of adeles of F . For a positive integer n let G 2n be the algebraic F -group GSp 2n , whose F -points are given by The symplectic group Sp 2n consists of those elements g ∈ G 2n for which the multiplier µ n (g) is 1. Let P 2n be the Siegel parabolic subgroup of G 2n , consisting of matrices whose lower left n × n-block is zero. Let δ P 2n be the modulus character of P 2n (A). It is given by and | · | denotes the global absolute value, normalized in the standard way. Fix the following embedding of H 2a,2b : We will also let H 2a,2b denote its image in G 2a+2b . Let G be any reductive algebraic group defined over F . For a place v of F let (π v , V v ) be an admissible representation of G(F v ). If v is non-archimedean, then this means that every vector in V v is smooth, and that for every open-compact subgroup Γ of G(F v ) the space of fixed vectors V Γ v is finite-dimensional. If v is archimedean, then it means that V v is an admissible (g, K)-module, where g is the Lie algebra of G(F v ) and K is a maximal compact subgroup of G(F v ). We say that π v is unitary if there exists a G(F v )-invariant (non-archimedean case) resp. g-invariant (archimedean case) hermitian inner product on V v . In this case, and assuming that π v is irreducible, we can complete V v to a unitary Hilbert space representationV v , which is unique up to unitary isomorphism. We can recover V v as the subspace ofV v consisting of smooth (non-archimedean case) resp. K-finite (archimedean case) vectors.
We define automorphic representations as in [8]. In particular, when we say "automorphic representation of G(A)", we understand that at the archimedean places we do not have actions of G(F v ), but of the corresponding (g, K)-modules. All automorphic representations are assumed to be irreducible. Cuspidal automorphic representations are assumed to be unitary. Any such representation π is isomorphic to a restricted tensor product ⊗π v , where π v is an irreducible, admissible, unitary representation of G(F v ).
For a place v of F , let σ, χ 1 , · · · , χ n be characters of F × v . We denote by χ 1 × · · · × χ n ⋊ σ the representation of G 2n (F v ) parabolically induced from the character  of the standard Borel subgroup of G 2n (F v ). Restricting all functions in the standard model of this representation to Sp 2n (F v ), we obtain a Borel-induced representation of Sp 2n (F v ) which is denoted by χ 1 × . . . × χ n ⋊ 1.
We also define parabolic induction from P 2n (F v ). Let χ and σ be characters of F × v . Then χ ⋊ σ is the representation of G 2n (F v ) parabolically induced from the character of P 2n (F v ). The center of G 2n (F v ) acts on χ⋊σ via the character χ n σ 2 . Restricting the functions in the standard model of χ ⋊ σ to Sp 2n (F v ), we obtain the Siegel-induced representation of Sp 2n (F v ) denoted by χ ⋊ 1.
We fix a Haar measure on Sp 2n (R), as follows. Let H n be the Siegel upper half space of degree n, consisting of all complex, symmetric n × n matrices X + iY with X, Y real and Y positive definite. The group Sp 2n (R) acts transitively on H n in the usual way. The stabilizer of the point I := i1 n ∈ H n is the maximal compact subgroup K = Sp 2n (R) ∩ O(2n) ∼ = U (n). We transfer the classical Sp 2n (R)-invariant measure on H n to a left-invariant measure on Sp 2n (R)/K. We also fix a Haar measure on K so that K has volume 1. The measures on Sp 2n (R)/K and K define a unique Haar measure on Sp 2n (R). Let F be a measurable function on Sp 2n (R) that is right K-invariant. Let f be the corresponding function on H n , i.e., F (g) = f (gI). Then, by these definitions, We shall always use the Haar measure on Sp 2n (R) characterized by the property (12). Haar measures on Sp 2n (F ), where F is a non-archimedean field with ring of integers o, will be fixed by requiring that the open-compact subgroup Sp 2n (o) has volume 1. The Haar measure on an adelic group Sp 2n (A) will always be taken to be the product measure of all the local measures.

Some coset decompositions
For 0 ≤ r ≤ n, let α r ∈ Sp 4n (Q) be the matrix where the n × n matrix I r is given by I r = 0 n−r 0 0 Ir . For our purposes, it is nicer to work with the coset representatives Q r := α r · (I 4n−2r , J r ) where J r = Ir −Ir . It is not hard to see that (I 4n−2r , J r ) ∈ H 4n−2r,2r is actually an element of H 2n,2n , so that P 4n (F )α r H 2n,2n (F ) = P 4n (F )Q r H 2n,2n (F ).
One can write down the matrix Q r explicitly as follows, where I ′ n−r = I n −Ĩ r = I n−r 0 0 0 . For 0 ≤ r ≤ n, let P 2n,r be the maximal parabolic subgroup (proper if r = n) of G 2n consisting of matrices whose lower-left (n + r) × (n − r) block is zero. Its Levi component is isomorphic to GL n−r × G r . Note that P 2n,0 = P 2n and P 2n,n = G 2n . Let N 2n,r denote the unipotent radical of P 2n,r .
The next lemma expresses the reason why {Q r } is more convenient than {α r } for the double coset representatives. Let d : P 4n → GL 1 be the homomorphism defined by
iii) Let g ∈ G 2n . Then the matrix X = Q n (g, g)Q −1 n lies in P 4n and satisfies d(X) = 1.
Proof. This follows by direct verification.
Next we provide a set of coset representatives for P 4n (F )\P 4n (F )Q r H 2n,2n (F ).

Proposition.
For each 0 ≤ r ≤ n, we have the coset decomposition

Degenerate principal series representations
Let χ be a character of F × \A × . We define a character on P 4n (A), also denoted by χ, by . For a complex number s, let for all p ∈ P 4n (A) and g ∈ G 4n (A). Note that these functions are invariant under the center of G 4n (A). Let I v (χ v , s) be the analogously defined local representation. Using the notation introduced in Sect. 2.1, we have We have We will mostly use this observation in the following form. Let f v ∈ I v (χ v , s) and K a maximal compact subgroup of Sp 2n (F v ). Then and all h ∈ Sp 2n (F v ).
In preparation for the next result, and for the unramified calculation in Sect. 4, we recall some facts concerning the unramified Hecke algebra at a non-archimedean place v of F . We fix where K e 1 ,··· ,en = K diag(̟ e 1 , · · · , ̟ en , ̟ −e 1 , · · · , ̟ −en )K.
Consider the spherical Hecke algebra H n consisting of left and right K-invariant compactly supported functions on Sp 2n (F ). The structure of this Hecke algebra is described by the Satake isomorphism S : where the superscript W indicates polynomials that are invariant under the action of the Weyl group of Sp 2n . Let T e 1 ,...,en be the characteristic function of the set K e 1 ,...,en defined in (21).
Then T e 1 ,...,en is an element of the Hecke algebra H n . The values S(T e 1 ,...,en ) are encoded in the rationality theorem This identity of formal power series is the main result of [5]. Let χ 1 , . . . , χ n be unramified characters of F × v . Let π be the unramified constituent of . . , n, are called the Satake parameters of π. Let v 0 be a spherical vector in π. It is unique up to scalars. Hence, if we act on v 0 by an element T of H n , we obtain a multiple of v 0 . This multiple is given by evaluating ST at the Satake parameters, i.e., Now assume that v is a real place. Let K = Sp 2n (R) ∩ O(2n) ∼ = U (n) be the standard maximal compact subgroup of Sp 2n (R). Let g be the Lie algebra of Sp 2n (R), and let a be the subalgebra consisting of diagonal matrices. Let Σ be the set of restricted roots with respect to a. If e i is the linear map sending diag(a 1 , . . . , a n , −a 1 , . . . , −a n ) to a i , then Σ consists of all ±(e i − e j ) for 1 ≤ i < j ≤ n and ±(e i + e j ) for 1 ≤ i ≤ j ≤ n. As a positive system we choose (what is more often called a negative system). Then the positive Weyl chamber is a + = {diag(a 1 , . . . , a n , −a 1 , . . . , −a n ) : a 1 < . . . < a n < 0}.
By Proposition 5.28 of [29], or Theorem 5.8 in Sect. I.5 of [24], we have the integration formula which we will use for continuous, non-negative functions φ on Sp 2n (R). The measure dH in (27) is the Lebesgue measure on a + ⊂ R n . The positive constant α n relates the Haar measure given by the integration on the right hand side to the Haar measure dh on Sp 2n (R) we fixed once and for all by (12). We will calculate α n explicitly in Appendix A.1.

Local zeta integrals
ii) Let f ∈ I(χ, s). Then, for Re(s) large enough, the function on Proof. Since ii) follows from i) by definition of the adelic measure, we only have to prove the local statement. To ease notation, we will omit all subindices v. Define a function f ′ (g, s) by From (16), we see that for all p ∈ P 4n (F ) and g ∈ G 4n (F ). Equation (19) implies that Now assume that v is a non-archimedean place. It follows from (20) that From (29), we find with A = diag(̟ e 1 , . . . , ̟ en ). By smoothness, the term f ′ (. . .) in the second line of (32) takes only finitely many values, and can therefore be estimated by a constant C independent of e 1 , . . . , e n . Thus where Since it is a subrepresentation of | · | −n × . . . × | · | −1 , its Satake parameters are α i = q i for i = 1, . . . , n. Let v 0 be a spanning vector of 1 Sp 2n (F ) . Then T e 1 ,...,en v 0 = vol(K e 1 ,··· ,en )v 0 . By (24) it follows that vol(K e 1 ,··· ,en ) = S(T e 1 ,...,en )(α 1 , . . . , α n ), where α i = q i . Substituting α i = q i into (23), we get an identity of formal power series in Y . We see that (33) is convergent if c(s) is sufficiently small, i.e., if Re(s) is sufficiently large.
Next assume that v is a real place. By (27), (19) and (30), It follows from (29) that, with |χ| = | · | d and H = diag(a 1 , . . . , a n , −a 1 , . . . , −a n ), with A = diag(e a 1 , . . . , e an ). Since the a i 's are negative, the term f ′ (. . .) on the right hand side can be estimated by a constant C. Hence where c(s) = e d+(2n+1)(Re(s)+1/2) . Writing out the expressions for sinh(λ(H)), it is easy to see that the integral on the right converges for real part of s large enough.
converges absolutely to an element ofV v , for any w v in the Hilbert space ii) Let π ∼ = ⊗π v be a cuspidal, automorphic representation of G 2n (A). Let V be the space of automorphic forms on which π acts. If Re(s) is sufficiently large, then the function , and if φ corresponds to the pure tensor ⊗w v , then the function (39) corresponds to the pure tensor ⊗Z v (s; f v , w v ).
Proof. i) The absolute convergence follows from Lemma 3.1 i). The second assertion can be verified in a straightforward way using (19). ii) Lemma 3.1 ii) implies that the integral where R denotes right translation, converges absolutely to an element in the Hilbert space completionV of V . With the same argument as in the local case we see that this element has the required smoothness properties that make it an automorphic form, thus putting it into V . Evaluating at g, we obtain the first assertion. The second assertion follows by applying a unitary isomorphism π ∼ = ⊗π v to (40).

The basic identity
Let I(χ, s) be as in (15), and let f (·, s) be a section whose restriction to the standard maximal compact subgroup of G 4n (A) is independent of s. Consider the Eisenstein series on G 4n (A) which, for Re(s) > 1 2 , is given by the absolutely convergent series and defined by analytic continuation outside this region. Let π be a cuspidal automorphic representation of G 2n (A). Let V π be the space of cuspidal automorphic forms realizing π. For any automorphic form φ in V π and any s ∈ C define a function 3.3 Remark. Note that g · Sp 2n (A) = {h ∈ G 2n (A) : µ n (h) = µ n (g)}.
3.4 Remark. E(g, s, f ) is slowly increasing away from its poles and φ is rapidly decreasing. So Z(s; f, φ)(g) converges absolutely for s ∈ C away from the poles of the Eisenstein series and defines an automorphic form on G 2n . We will see soon that Z(s; f, φ) in fact belongs to V π .
Since the element x normalizes N 2n,r (A) and N 2n,r (F ), we can commute x and n. Then n can be omitted by i) of Lemma 2.2. Hence x,z f (Q r · (xh, zg), s) φ(nh) dn dh, and the cuspidality of φ implies that this is zero.
For the following theorem, which is the main result of this section, recall the local integrals defined in (38).
3.6 Theorem. (Basic identity) Let φ ∈ V π be a cusp form which corresponds to a pure tensor (42). Then Z(s; f, φ) also belongs to V π and corresponds to the pure tensor where the last step follows from (18). The theorem now follows from Proposition 3.2.
Our goal will be to choose, at all places, the vectors φ v and the sections f v in such a way that

The local integral at finite places
In this section we define suitable local sections and calculate the local integrals (38) for all finite places v. We will drop the subscript v throughout. Hence, let F be a non-archimedean local field of characteristic zero. Let o be its ring of integers, ̟ a uniformizer, and p = ̟o the maximal ideal.

Unramified places
We begin with the unramified case. Let χ be an unramified character of F × , and let π be a spherical representation of G 2n (F ). Let f ∈ I(χ, s) be the normalized unramified vector, i.e., for A ∈ GL 2n (F ), u ∈ F × and k ∈ G 4n (o). Let v 0 be a spherical vector in π. We wish to calculate the local integral Let σ, χ 1 , · · · , χ n be unramified characters of F × such that π is the unique spherical con- . Consequently we get a tensor product map from GSpin 2n+1 (C) × GL 1 (C) into GL 2n+1 (C) which we denote also by ̺ 2n+1 . The L-function L(s, π ⊠ χ, ̺ 2n+1 ) is then given as follows, We also define L(s, 4.1 Proposition. Using the above notations and hypotheses, the local integral (44) is given by for real part of s large enough.

Ramified places
Now we deal with the ramified places. For a non-negative integer m, From the last two rows and last three columns we get where M n (p m ) is the set of n × n matrices with entries in p m . Hence g ∈ I 2n + M 2n (p m ) = Γ 2n (p m ). Multiplying (46) from the left by p −1 and looking at the lower left block, we see that Let m be a positive integer such that χ| (1+p m )∩o × = 1. Let f (g, s) be the unique function on It is easy to see that f is well-defined. Evidently, f ∈ I(χ, s). Furthermore, for each h ∈ G 4n (o), define f (h) ∈ I(χ, s) by the equation

Proposition.
Let π be any irreducible admissible representation of G 2n (F ). Let m be a positive integer such that χ| (1+p m )∩o × = 1 and such that there exists a vector φ in π fixed by ) is a non-zero rational number depending only on m.
The main difference between the above description and our original definition is that it uses the Siegel type congruence subgroup rather than the principal congruence subgroup. The fact that makes this alternate description possible is that P 4n (F )Γ 0,4n (p m ) = P 4n (F )Γ 4n (p m ). In particular, this shows that our local section f (g, s) is essentially identical to that used by Shimura [55,58] in his work on Eisenstein series, which will be a key point for us later on. 5 The local integral at a real place

Holomorphic discrete series representations
We provide some preliminaries on holomorphic discrete series representations of Sp 2n (R). We fix the standard maximal compact subgroup and Z in the Siegel upper half space H n . Then, for any integer k, the map is a character of K.
Let h be the compact Cartan subalgebra and e 1 , . . . , e n the linear forms on the complexification h C defined in [2]. A system of positive roots is given by e i ± e j for 1 ≤ i < j ≤ n and 2e j for 1 ≤ j ≤ n. The positive compact roots are the e i − e j for 1 ≤ i < j ≤ n. The K-types are parametrized by the analytically integral elements k 1 e 1 + . . . + k n e n , where the k i are integers with k 1 ≥ . . . ≥ k n . We write ρ k , k = (k 1 , . . . , k n ), for the K-type with highest weight k 1 e 1 + . . . + k n e n . If k 1 = . . . = k n = k, then ρ k is the K ∞ -type given in (51); we simply write ρ k in this case.
The holomorphic discrete series representations of Sp 2n (R) are parametrized by elements λ = ℓ 1 e 1 + . . . + ℓ n e n with integers ℓ 1 > . . . > ℓ n > 0. The representation corresponding to the Harish-Chandra parameter λ contains the K-type ρ k , where k = λ + n j=1 je j , with multiplicity one; see Theorem 9.20 of [29]. We denote this representation byπ λ or by π k ; sometimes one or the other notation is more convenient. If k = (k, . . . , k) with a positive integer k > n, then we also write π k for π k .
Let G 2n (R) + be the index two subgroup of G 2n (R) consisting of elements with positive multiplier. We may extend a holomorphic discrete series representationπ λ of Sp 2n (R) in a trivial way to G 2n (R) + ∼ = Sp 2n (R) × R >0 . This extension induces irreducibly to G 2n (R). We call the result a holomorphic discrete series representation of G 2n (R) and denote it by the same symbolπ λ (or π k ). These are the archimedean components of the automorphic representations corresponding to vector-valued holomorphic Siegel modular forms of degree n. i) The holomorphic discrete series representationπ λ of Sp 2n (R) embeds into and in no other principal series representation of Sp 2n (R).
ii) The holomorphic discrete series representationπ λ of G 2n (R) embeds into where ε can be either 0 or 1, and in no other principal series representation of G 2n (R).
Proof. i) follows from the main result of [64]. Part ii) can be deduced from i), observing that the holomorphic discrete series representations of G 2n (R) are invariant under twisting by the sign character.

Lemma.
Let k be a positive integer. Consider the degenerate principal series representation of G 2n (R) given by where ε ∈ {0, 1}. Then J(s) contains the holomorphic discrete series representation π k of G 2n (R) as a subrepresentation if and only if s = k.
Proof. By infinitesimal character considerations, we only need to prove the "if" part. Since π k is invariant under twisting by sgn, we may assume that ε = 0. Consider the Borel-induced representation By Lemma 5.1 ii), π k is a subrepresentation of J ′ (k). Since | · | s−n × | · | s−n+1 × . . . × | · | s−1 contains the trivial representation of GL n (R) twisted by | · | s− n+1 2 , it follows that J(s) ⊂ J ′ (s). Let f s be the function on G 2n (R) given by for A ∈ GL n (R), u ∈ R × and g ∈ K. Then f s is a well-defined element of J(s). Since f s is the unique up to multiples vector of weight k in J ′ (s), π k lies in the subspace J(k) of J ′ (k).
Our method to calculate the local archimedean integrals (38) will work for holomorphic discrete series representationsπ λ , where λ = ℓ 1 e 1 + . . . + ℓ n e n with ℓ 1 > . . . > ℓ n > 0 satisfies Equivalently, we work with the holomorphic discrete series representations π k , where k = k 1 e 1 + . . . + k n e n with k 1 ≥ . . . ≥ k n > n and all k i of the same parity; this last condition can be seen to be equivalent to (57). An example for λ satisfying (57) is (k − 1)e 1 + . . . + (k − n)e n , the Harish-Chandra parameter of π k . The next lemma implies that whenever (57) is satisfied, theṅ π λ contains a convenient scalar K-type to work with.
Proof. By Theorem 8.1 of [29] we need only show that ρ m occurs inπ λ . We will use induction on n. The result is obvious for n = 1. Assume that n > 1, and that the assertion has already been proven for n − 1.
Using standard notations as in [2], we have g C = p − C ⊕ k C ⊕ p + C . The universal enveloping algebra of p + C is isomorphic to the symmetric algebra S(p + C ). We haveπ λ ∼ = S(p + C ) ⊗ ρ λ as Kmodules. Let I be the subalgebra of S(p + C ) spanned by the highest weight vectors of its K-types. By Theorem A of [27], there exists in I an element D + of weight 2(e 1 + . . . + e n ). By the main result of [25], the space of K-highest weight vectors ofπ λ is acted upon freely by I. It follows that we need to prove our result only for m = 0.
We will use the Blattner formula proven in [22]. It says that the multiplicity with which ρ m occurs inπ λ is given by Here, W K is the compact Weyl group, which in our case is isomorphic to the symmetric group S n , and ε is the sign character on W K ; the symbols ρ c and ρ n denote the half sums of the positive compact and non-compact roots, respectively; and Q(µ) is the number of ways to write µ as a sum of positive non-compact roots. In our case Hence Now assume that m = 0. Then If σ(1) = 1, then the coefficient of e 1 is negative, implying that Q(. . .) = 0. Hence If we set e ′ j = e j+1 and m ′ = ℓ 1 − ℓ 2 − 1, then this can be written as We see that this is the formula (61), with n − 1 instead of n and m ′ instead of m; note that m ′ is even and non-negative by our hypotheses. There are two different Q-functions involved, for n and for n − 1, but since the argument of Q in (63) has no e 1 , we may think of it as the Q-function for n−1. Therefore (64) represents the multiplicity of the K (n−1) -type (ℓ 2 +1+m ′ )(e ′ 1 +. . .+e ′ n−1 ) in the holomorphic discrete series representationπ λ ′ of Sp 2(n−1) (R), where λ ′ = ℓ 2 e ′ 1 + . . . + ℓ n e ′ n−1 . By induction hypothesis, this multiplicity is 1, completing our proof.

Calculating the integral
In the remainder of this section we fix a real place v and calculate the local archimedean integral (38) for a certain choice of vectors f and w. To ease notation, we omit the subscript v. We assume that the underlying representation π of G 2n (R) is a holomorphic discrete series representatioṅ π λ , where λ = ℓ 1 e 1 + . . . + ℓ n e n with ℓ 1 > . . . > ℓ n > 0 satisfies (57). Set k = ℓ 1 + 1. By Lemma 5.3, the K-type ρ k appears in π with multiplicity 1. Let w λ be a vector spanning this one-dimensional K-type. We choose w = w λ as our vector in the zeta integral Z(s, f, w).
To explain our choice of f , let J(s) be the degenerate principal series representation of G 4n (R) defined in Lemma 5.2 (hence, we replace n by 2n in (54)). We see from (17) that I(sgn k , s) equals J((2n + 1)(s + 1 2 )) for appropriate ε ∈ {0, 1}. Let f k (·, s) be the vector spanning the K-type K (2n) ∋ g −→ j(g, I) −k and normalized by f k (1, s) = 1. Explicitly, for A ∈ GL 2n (R), u ∈ R × and g ∈ K (2n) . Then f = f k is the section which we will put in our local archimedean integral Z(s, f, w). Thus consider Z(s, f k , w λ ). By Proposition 3.2 i), this integral is a vector in π. The observation (19), together with the transformation properties of f k , imply that for g ∈ K π(g)Z(s, f k , w λ ) = j(g, I) −k Z(s, f k , w λ ). (66) Since the K-type ρ k occurs only once in π, it follows that for a constant B λ (s) depending on s and the Harish-Chandra parameter λ. The rest of this section is devoted to calculating B λ (s) explicitly.

The scalar minimal K-type case
We first consider the case λ = (k − 1)e 1 + . . . + (k − n)e n with k > n. Thenπ λ = π k , the holomorphic discrete series representation of G 2n (R) with minimal K-type ρ k . Let w k be a vector spanning this minimal K-type. Let , be an appropriately normalized invariant hermitian inner product on the space of π k such that w k , w k = 1. As proved in the appendix of [30], we have the following simple formula for the corresponding matrix coefficient, Here, h = A B C D ∈ G 2n (R). We will need the following result.

Lemma. For a complex number z, let
where dt is the Lebesgue measure and Then, for real part of z large enough, Proof. After some straightforward variable transformations, our integral reduces to the Selberg integral; see [50] or [14].
Proof. Taking the inner product with w k on both sides of (67), we obtain Since the integrand is left and right K-invariant, we may apply the integration formula (27). Thus B λ (s) = α n a + λ∈Σ + sinh(λ(H)) f k (Q n · (exp(H), 1), s) π k (exp(H))w k , w k dH.
The function f k in (74) can be evaluated as follows, It follows that Substituting this and (68) into (74), we get after some simplification Now introduce the new variables t j = cosh(a j ). The domain a + turns into the domain T defined in (70). We get Thus our assertion follows from Lemma 5.4 and the value of α n given in (142).
In this induced model, the weight k vector w λ inπ λ has the formula for a 1 , . . . , a n ∈ R × and g ∈ K (n) . Evaluating (67) at 1, we get Recall the beta function One possible integral representation for the beta function is For s ∈ C and m ∈ Z, let β(m, s) = B n + 1 2 where C k (s) = depends only on k = ℓ 1 + 1. Here,Ñ 1 is the space of upper triangular nilpotent matrices of size n × n, andÑ 2 is the space of symmetric n × n matrices.

sin(θn)
. Then and thus f k (Q n · (an, 1)) = i nk n j=1 1 a j + a −1 Write U = I n + V , so that V is upper triangular nilpotent. A calculation confirms that where Z = −(1 + SV S) −1 SV C and p = B * t B −1 with det(B) = 1. Hence In In ).
(98) LetÑ 1 be the Euclidean space of upper triangular nilpotent real matrices of size n × n. Then it is an exercise to verify that where dV is the Lebesgue measure, defines a Haar measure on the group of upper triangular unipotent real matrices. (Use the fact that V → U V defines an automorphism ofÑ of determinant 1, for every upper triangular unipotent U .) Therefore, as we integrate (98) over N 1 , we may treat V as a Euclidean variable. We then have to consider the Jacobian of the change of variables V → Z. It is not difficult to show that this Jacobian is n j=1 sin(θ j ) j−n cos(θ j ) 1−j . Substituting from (93), we find Using the above and some more matrix identities, we get This last integral is the C k (s) defined in (87). Going back to (89) and using (84), we have This concludes the proof.
5.7 Lemma. The function C k (s) defined in (87) is given by where γ n is the rational function from Lemma 5.4, and β(m, s) is defined in (85).
Inductively one confirms the identity for any integer m ≥ 0. Use the abbreviation t = (n + 1 2 )s + 1 4 . Applying the above formula with m = k−k j 2 , which by (57) is a non-negative integer, and replacing i by m − 1 − i, we get The result follows by using formula (71) for γ n .
5.9 Remark. Using (106), one can check that A k (t) is a non-zero rational number for any integer t satisfying 0 ≤ t ≤ k n − n.
6 The global integral representation

The main result
Consider the global field F = Q and its ring of adeles A = A Q . All the results are easily generalizable to a totally real number field. Let π ∼ = ⊗π p be a cuspidal automorphic representation of G 2n (A). We assume that π ∞ is a holomorphic discrete series representation π k with k = k 1 e 1 + . . . + k n e n , where k 1 ≥ . . . ≥ k n > n and all k i have the same parity. (From now on it is more convenient to work with the minimal K-type k rather than the Harish-Chandra parameter λ.) We set k = k 1 . Let χ = ⊗χ p be a character of Q × \A × such that χ ∞ = sgn k . Let N = p|N p mp be an integer such that • For each finite prime p ∤ N both π p and χ p are unramified.
• For a prime p|N , we have χ p | (1+p mp Zp)∩Z × p = 1 and π p has a vector φ p that is right invariant under the principal congruence subgroup Γ 2n (p mp ) of Sp 2n (Z p ).
Let φ be a cusp form in the space of π corresponding to a pure tensor ⊗φ p , where the local vectors are chosen as follows. For p ∤ N choose φ p to be a spherical vector; for a p|N choose φ p to be a vector right invariant under Γ 2n (p mp ); and for p = ∞ choose φ ∞ to be a vector in π ∞ spanning the K ∞ -type ρ k ; see Lemma 5.3. Let f = ⊗f p ∈ I(χ, s) be composed of the following local sections. For a finite prime p ∤ N let f p be the spherical vector normalized by f p (1) = 1; for p|N choose f p as in Sect. 4.2 (with the positive integer m of that section equal to the m p above); and for p = ∞, choose f ∞ by (65). Define L N (s, π ⊠ χ, ̺ 2n+1 ) = where the local factors on the right are given by (45).
Next, for any h ∈ p<∞ Sp 4n (Z p ), define f (h) (g, s) = f (gh −1 , s). Let Q denote the element Q n embedded diagonally in p<∞ Sp 4n (Z p ), and for any τ We can now state our global integral representation.
Proof. By Theorem 3.6, Proposition 4.1, Proposition 4.3 and Proposition 5.8, the equation (110) is true for all Re(s) sufficiently large. Since the left side defines a meromorphic function of s for each g, it follows that the right side can be analytically continued to a meromorphic function of s such that (110) always holds.

A classical reformulation
We now rewrite the above theorem in classical language. For any congruence subgroup Γ of Sp 2n (R) with the symmetry property −In whenever the integral converges. Above, dZ is any Sp 2n (R)-invariant measure on H n (it is equal to c det(Y ) −(n+1) dX dY for some constant c). Note that our definition of the Petersson inner product does not depend on the normalization of measure (the choice of c), and is also not affected by different choices of Γ. Note also that Now, let Φ be an automorphic form on G 2n (A) such that Φ(gh) = j(h, I) −k Φ(g) for all h ∈ K (n) ∞ ∼ = U (n). Then we can define a function F (Z) on the Siegel upper half space H n by where g is any element of Sp 2n (R) with g(I) = Z. If Γ p is an open-compact subgroup of G 2n (Q p ) such that Φ is right invariant under Γ p for all p, then it is easy to check that We apply this principle to our Eisenstein series E(g, s, f ), where f is the global section constructed in Sect. 6.1. Consider E χ k,N (Z, s) := j(g, I) k E g, where g is any element of Sp 4n (R) with g(I) = Z. Since the series defining E(g, s, f ) converges absolutely for Re(s) > 1 2 , it follows that the series defining E χ k,N (Z, s) converges absolutely whenever 2Re(s) + k > 2n + 1. More generally, for any h ∈ p<∞ Sp 4n (Z p ) and g as above, define E χ k,N (Z, s; h) = j(g, I) k E g, By the invariance properties of our local sections and by analytic continuation, it follows that for all s and all h as above, we have E χ k,N (Z, s; h) ∈ C ∞ k (Γ 4n (N )). As usual, Γ 4n (N ) = {g ∈ Sp 4n (Z) : g ≡ I 4n (mod N )} is the principal congruence subgroup of level N .
It is instructive to write down the functions E χ k,N (Z, s; h) classically. First of all, a standard calculation shows that where (using strong approximation) we let h 0 be an element of Sp 4n (Z) such that h −1 0 h ∈ p|N Γ 4n (p mp ) p∤N Sp 4n (Z p ). This enables us to reduce to the case h = 1 for many properties. We now write down the classical definition in this case, i.e., for E χ k,N (Z, s). Let P ′ 4n (Z) = P 4n (Q)∩Sp 4n (Z). Using the relevant definitions and the explication of the local sections f p (γ p , s) at the end of Section 4.2, it follows that for 2Re(s) + k > 2n + 1, 6.3 Remark. Shimura defined certain Eisenstein series on symplectic and unitary groups over number fields [55,56,57,58] and proved various properties about them. Following the notation of [58, Sect. 17], we denote Shimura's Eisenstein series by E(z, s; k, χ, N ). A comparison of [58, (16.40)] and (120) shows that E χ k,N (Z, s) = E(z, s + k/2; k, χ, N ).
The above identity can also be proved adelically, by comparing the alternate description of our section at ramified places (see end of Section 4.2) with Shimura's section (see Sect. 16.5 of [58]).
Combining (120) with (119), we can now write down a similar expansion for E χ k,N (Z, s; h) for each h ∈ p<∞ Sp 4n (Z p ). Note that if Q is the element defined immediately above Theorem 6.1, then E χ k,N (Z, s; Q) = j(Q n , Z) −k E χ k,N (Q n Z, s). Let π and φ be as in Sect. 6.1. Let F (Z) be the function on H n corresponding to the automorphic form φ via (116). Then F,F ∈ C ∞ k (Γ 2n (N )) and both these functions are rapidly decreasing at infinity (as φ is a cusp form). For Z 1 , Z 2 ∈ H n , and h ∈ p<∞ Sp 4n (Z p ), write Using adelic-to-classical arguments similar to Theorem 6.5.1 of [45], we can now write down the classical analogue of Theorem 6.1.
6.4 Theorem. Let the element Q τ for each τ ∈Ẑ × be as defined just before Theorem 6.1, and let E χ k,N (Z, s; Q τ ) be as defined in (118). Let F (Z),F (Z) be as defined above. Then we have the relation with the rational function A k (z) defined as in Proposition 5.8.

Near holomorphy and rationality of Eisenstein series
In this section, we will prove two important properties of the Eisenstein series E χ k,N (Z, −m 0 ; h) for certain non-negative integers m 0 . These are stated as Propositions 6.6 and 6.8.
For each positive integer r, let N (H r ) be the space of nearly holomorphic functions on H r . By definition, these are the functions which are polynomials in the entries of Im(Z) −1 with holomorphic functions on H r as coefficients. For each discrete subgroup Γ of Sp 2r (R), let N k (Γ) be the space of all functions F in N (H r ) that satisfy F (γZ) = j(γ, Z) k F (Z) for all Z ∈ H r and γ ∈ Γ (if r = 1, we also need an additional "no poles at cusps" condition, as explained in [39]). The spaces N k (Γ) are known as nearly holomorphic modular forms of weight k for Γ. We let M k (Γ) ⊂ N k (Γ) denote the usual space of holomorphic modular forms of weight k for Γ. The following important result is due to Shimura.
Proof. If h = 1, this is a special case of Theorem 17.9 of [58] (see Remark 6.3). The proof for general h is now an immediate consequence of (119).

Remark.
In the absolutely convergent range k ≥ 2n + 2, 0 ≤ m 0 ≤ k 2 − n − 1, the above proposition can also be proved directly, using the expansion (120). To go beyond the realm of absolute convergence, one needs delicate results involving analytic behavior of Fourier coefficients of Eisenstein series, which have been done by Shimura.
Next, for any nearly holomorphic modular form F ∈ N k (Γ) and σ ∈ Aut(C) we let σ F denote the nearly holomorphic modular form obtained by letting σ act on the Fourier coefficients of F . Note that if σ is complex conjugation then σ F equalsF . Denote We will prove the following result.
6.8 Proposition. Let the setup be as in Proposition 6.6. Let σ ∈ Aut(C), and let τ ∈Ẑ × be the element corresponding to σ via the natural map Aut(C) → Gal(Q ab /Q) ≃Ẑ × . (Concretely, this means that for each positive integer m, σ(e 2πi/m ) = e 2πit/m where t ≡ τ mod m.) Then Proof. We will prove the result in several steps.
The next step is to extend the above lemma to the case of general h. This follows from a very general lemma of Shimura. For any Siegel modular form F ∈ M k (Γ 2r (N )) and any 6.10 Lemma. (Shimura) Let σ, τ be as in Proposition 6.8. Then, for all h ∈ p<∞ Sp 2r (Z p ), Proof. This is immediate from Lemma 10.5 of [58] and its proof.
Combining the above two lemmas, since k − 2m 0 ≥ n + 1, we see now that for all k, m 0 , χ as in Proposition 6.6. Next, we need the Maass-Shimura differential operator ∆ p k which is defined in [56, 4.10a] or [58, p. 146]. Note that ∆ p k takes N k (Γ) to N k+2p (Γ). By [58, (17.21)] we obtain where d is a non-zero rational number. (Note here that the differential operator ∆ p k commutes with the action of h). Finally, we have the identity (see equation (1) of [9]): Combining (122), (123), and (124), we conclude the proof of Proposition 6.8.

Preliminaries
For the algebraicity results of critical values of L-functions, we will use [39]. Since the results of [39] are available only for n = 2, we will assume n = 2 throughout this section. Let ℓ, m be nonnegative integers with m even and ℓ ≥ 3. We put k = (ℓ + m)e 1 + ℓe 2 and k = ℓ + m. For each integer N = p p mp , we let Π N (k) denote the set of all cuspidal automorphic representations π ∼ = ⊗π p of G 4 (A) such that π ∞ equals the holomorphic discrete series representation π k and such that for each finite prime p, π p has a vector right invariant under the principal congruence subgroup Γ 4 (p mp ) of Sp 4 (Z p ). We put Π(k) = N Π N (k). We say that a character χ = ⊗χ p of Q × \A × is a Dirichlet character if χ ∞ is trivial on R >0 . Any such χ gives rise to a homomorphismχ : (Z/N χ Z) × → C × , where N χ denotes the conductor of χ. Concretely 4 the mapχ is given byχ(a) = p|Nχ χ −1 p (a). Given a Dirichlet character χ, we define the corresponding Gauss sum by G(χ) = n∈(Z/NχZ) ×χ(n)e 2πin/Nχ . 7.1 Lemma. Let χ, χ ′ be Dirichlet characters. Given σ ∈ Aut(C), let τ ∈Ẑ × be as in Proposition 6.8. The following hold: Proof. This is a special case of Lemma 8 of [53].
The reader should not be confused by the two different π (one the constant, the other an automorphic representation) appearing in the above definition.
For the rest of this paper, we also make the following assumption (which is forced upon us as we need to use results of Shimura where this assumption appears): The reader might wonder if C N (π, χ, r) can be infinite. It turns out that (125) eliminates that possibility. First of all, any π ∈ Π N (k) is either of generic type (meaning, it lifts to a cusp form on GL 4 ), or of endoscopic (Yoshida) type, or of P-CAP (Saito-Kurokawa) type. (CAP representations of Soudry type or Howe-Piatetski-Shapiro type do not occur if ℓ ≥ 3.) In each case, we have precise information about the possible poles of L(r, π ⊠ χ, ̺ 5 ); see [49]. It follows that if π is generic, then L N (r, π ⊠ χ, ̺ 5 ) is finite for all r ≥ 1. On the other hand, for π either endoscopic or P-CAP, L N (r, π ⊠ χ, ̺ 5 ) is finite for r > 1 and L N (1, π ⊠ χ, ̺ 5 ) = ∞ ⇒ χ = 1. In particular, assumption (125) implies that C N (π, χ, r) is finite in all cases considered by us.
Recall that N k (Γ 4 (N )) denotes the (finite-dimensional) space of nearly holomorphic modular forms of weight k for the subgroup Γ 4 (N ) of Sp 4 (Z). Let V N be the subset of N k (Γ 4 (N )) consisting of those forms F which are cuspidal and for which the corresponding function Φ F on Sp 4 (R) generates an irreducible representation isomorphic to π k . By Theorem 4.8 and Proposition 4.28 of [39], V N is a subspace of N k (Γ 4 (N )) and isomorphic to the space S ℓ,m (Γ 4 (N )) of holomorphic vector-valued cusp forms of weight det ℓ sym m for Γ 4 (N ); indeed V N = U m/2 (S ℓ,m (Γ 4 (N ))), where U is the differential operator defined in Section 3.4 of [39]. We put V = N V N and N k = N N k (Γ 4 (N )).
As in [39], we let p • ℓ,m denote the orthogonal projection map from N k to V ; note that it takes N k (Γ 4 (N )) to V N for each N . Let 1 ≤ r ≤ ℓ − 2, r ≡ ℓ (mod 2) be an integer and χ ∈ X N ; also suppose that (125) holds.
is cuspidal in each variable using methods very similar to [18], but we will not need this. We define If F ∈ V is such that (the adelization of) F generates a multiple of an irreducible (cuspidal, automorphic) representation of G 4 (A), then we let π F denote the representation associated to F . Note that the set of automorphic representations π F obtained this way as F varies in V N is precisely equal to the set Π N (k) defined above. For each π ∈ Π(k), we let V N (π) denote the π-isotypic part of V N . Precisely, this is the subspace consisting of all those F in V N such that all irreducible constituents of the representation generated by (the adelization of) F are isomorphic to π. Note that V N (π) = {0} unless π ∈ Π N (k). We have an orthogonal direct sum decomposition We define V (π) = N V N (π). Therefore we have V = π∈Π(k) V (π). Now, let B be any orthogonal basis of V N formed by taking a union of orthogonal bases from the right side of (127). Thus each F ∈ B belongs to V N (π) for some π ∈ Π N (k). From Corollary 6.5, Proposition 6.6, and (115), we deduce the following key identity:

Lemma.
For all σ ∈ Aut(C), Proof. Recall that G χ k,N (Z 1 , Z 2 , r; Q τ ) is defined by (126). So the corollary follows from Proposition 6.8 (taking h = Q) and the fact that the map p • ℓ,m commutes with the action of Aut(C) (see Proposition 5.17 of [39]). Note that the power of π is introduced in (126) precisely to cancel with the power of π in Proposition 6.8.
Let σ ∈ Aut(C). For π ∈ Π N (k), we let σ π ∈ Π N (k) be the representation obtained by the action of σ, and we let Q(π) denote the field of rationality of π; see the beginning of Section 3.4 of [46]. If σ is the complex conjugation, then we denote σ π =π. It is known that Q(π) is a CM field and Q(π) = Q(π). We use Q(π, χ) to denote the compositum of Q(π) and Q(χ). Note that This follows from Theorem 4.2.3 of [3] (see the proof of Proposition 3.13 of [46]) together with the fact that the U operator commutes with σ. In particular, the space V N (π) is preserved under the action of the group Aut(C/Q(π)). Using Lemma 3.17 of [46], it follows that the space V N (π) has a basis consisting of forms whose Fourier coefficients are in Q(π). In particular there exists some F satisfying the conditions of the next proposition.
7.3 Proposition. Let π ∈ Π N (k), χ ∈ X N , and F ∈ V N (π). Suppose that the Fourier coefficients of F lie in a CM field. Then for any σ ∈ Aut(C) we have Proof. Let us complete F to an orthogonal basis B = {F = F 1 , F 2 , . . . , F r } of V N (π). Let B ′ = {G 1 , . . . , G r } be any orthogonal basis for V N ( σ π). Given σ, let τ be as in Proposition 6.8. Using (128), (129), and Lemma 7.2, and comparing the V N ( σ π) components, we see that Taking inner products of each side with σ F 1 (in the variable Z 1 ) we deduce that Note that σ F 1 = σF 1 by our hypothesis on the Fourier coefficients of F being in a CM field. Comparing the coefficients of σF 1 (Z 2 ) on each side and using Lemma 7.1, we conclude the desired equality.

The main result on critical L-values
For each p|N , we define the local L-factor L(s, π p ⊠χ p , ̺ 5 ) via the local Langlands correspondence [15]. In particular, L(s, π p ⊠ χ p , ̺ 5 ) is just a local L-factor for GL 5 × GL 1 . This definition also works at the good places, and indeed coincides with what we previously defined. For any finite set of places S of Q, including the archimedean place, we define the global L-function L S (s, π ⊠ χ, ̺ 5 ) = p / ∈S L(s, π p ⊠ χ p , ̺ 5 ).
Using the Langlands parameter given in Sect. 3.2 of [48] and the explicit form of the map ̺ 5 given in Appendix A.7 of [44], one finds that the archimedean factor is given by with Γ R , Γ C as in (111), and The completed L-function satisfies a functional equation with respect to s → 1 − s according to Theorem 60 of [10]. Hence, the critical points of L(s, π ⊠ χ, ̺ 5 ) are precisely those integers r for which neither L(s, π ∞ ⊠ χ ∞ , ̺ 5 ) nor L(1 − s, π ∞ ⊠ χ ∞ , ̺ 5 ) have a pole at s = r. Using the well known information on poles of gamma functions, we conclude that the set of critical points for L(s, π ⊠ χ, ̺ 5 ) are given by integers r such that 7.4 Remark. The critical points as written above in (131) crucially use the fact that m is even, ℓ ≥ 3, χ ∞ (−1) = (−1) ℓ . Without these assumptions, the critical points can change.
In the following theorem, we will obtain an algebraicity result for the special value of the Lfunction at the critical points in the right half plane. The analogous result for the critical points in the left half plane can be obtained from the functional equation in Theorem 60 of [10].
Hence we can replace L N by L S in (136), obtaining the desired identity.
7.6 Remark. In view of (131), Theorem 7.5 obtains an algebraicity result for the special value of the L-function at all the critical points in the right half plane, except in the special case where ℓ is odd and χ is quadratic, in which case our theorem cannot handle the critical point s = 1.
The reason for this omission is subtle, and is related to the fact that the normalization of the Eisenstein series corresponding to this point involves the factor L(1, χ 2 ) which has a pole when χ 2 = 1. Consequently the required arithmetic results for the Eisenstein series are unavailable in this case. Further, as mentioned earlier, the analogous result for the critical points in the left half plane can be obtained from the functional equation in Theorem 60 of [10].
In summary, Theorem 7.5 (together with the functional equation) covers all the critical Lvalues, except in the special case when ℓ is odd and χ is quadratic, in which case the critical L-values at s = 0 and s = 1 are not covered. 7.7 Remark. Let F be as in the Theorem 7.5. By the results of [39], we know that F = U m/2 F 0 where F 0 is a holomorphic vector-valued Siegel cusp form. Using Lemma 4.16 of [39], we have moreover the equality F 0 , F 0 = c ℓ,m F, F for some constant c ℓ,m that depends only on ℓ and m. By restricting to the special case of a full-level vector valued Siegel cusp form of weight det ℓ sym m , and comparing Theorem 7.5 with the result of [31], we see that c ℓ,m is a rational multiple of π m . Hence in the theorem above, the term F, F can be replaced by π −m F 0 , F 0 . 7.8 Definition. For two representations π 1 , π 2 in Π(k), we write π 1 ≈ π 2 if there is a Hecke character ψ of Q × \A × such that π 1 is nearly equivalent to π 2 ⊗ ψ.
Note that if such a ψ as above exists, then ψ ∞ must be trivial on R >0 and therefore ψ must be a Dirichlet character. The relation ≈ clearly gives an equivalence relation on Π(k). For any π ∈ Π(k), let [π] denote the class of π, i.e., the set of all representations π 0 in Π(k) satisfying π 0 ≈ π. For any integer N , we define the subspace V N ([π]) of V N to be the (direct) sum of all the subspaces V N (π 0 ) where π 0 ranges over all the inequivalent representations in [π] ∩ Π N (k). 7.9 Corollary. Let π 1 , π 2 ∈ Π(k) be such that π 1 ≈ π 2 . Let F 1 ∈ V (π 1 ) and F 2 ∈ V (π 2 ) have coefficients in a CM field. Then for all σ ∈ Aut(C), we have Proof. By assumption, there is a character ψ and a set S of places containing the infinite place, such that π 1,p ≃ π 2,p ⊗ ψ p for all p / ∈ S. We fix any character χ ∈ X . Note that L(s, π 1,p ⊠ χ p , ̺ 5 ) = L(s, π 1,p ⊠ χ p , ̺ 5 ) for all p / ∈ S, as the representation ̺ 5 factors through PGSp 4 and therefore is blind to twisting by ψ. Applying Theorem 7.5 twice at the point r = ℓ−2, first with (π 1 , F 1 ), and then with (π 2 , F 2 ), and dividing the two equalities, we get the desired result.
We do not require that the adelization of G [π] should generate an irreducible representation.
Finally, for any π ∈ Π(k) we define By construction, C(π) depends only on the class [π] of π. So we only need to prove (1). The following lemma is key.
Note that the spaces V (π i ) are mutually orthogonal and hence So the desired result would follow immediately from Corollary 7.9 provided we can show that each F i has coefficients in a CM field. Indeed, let K be the compositum of all the fields Q(π i ). Thus K is a CM field containing Q([π]). For any σ ∈ Aut(C/K), we have σ G [π] = G [π] and σ F i ∈ V (π i ). As the spaces V (π i ) are all linearly independent, it follows that σ F i = F i and therefore each F i has coefficients in a CM field.
The proof of (1) follows by combining Theorem 7.5 and Lemma 7.10.

Symmetric fourth L-function of GL 2
Let k be an even positive integer and M any positive integer. Let f be an elliptic cuspidal newform of weight k, level M and trivial nebentypus that is not of dihedral type. According to Theorem A' and Theorem C of [43], there exists a cuspidal automorphic representation π of GSp 4 (A), the so-called sym 3 lift, such that i) π ∞ is the holomorphic discrete series representation with highest weight (2k − 1, k + 1), ii) for p ∤ M , the local representation π p is unramified, iii) the L-functions have the following relation.
The condition that f has trivial nebentypus and even weight k is an essential hypothesis in the results of [43]. Note that π corresponds to a holomorphic vector-valued Siegel cusp form F 0 with weight det k+1 sym k−2 . Hence, ℓ = k + 1 and m = k − 2 in this case. Let χ be a Dirichlet character in X . Since k is even, we get χ(−1) = (−1) k+1 = −1, i.e., χ ∞ = sgn. We have L(s, π ⊠ χ, ̺ 5 ) = L(s, χ ⊗ sym 4 f ). (138) Here, on the right hand side, we have the L-function of GL 5 given by the symmetric fourth power of f (see [28]), twisted by the character χ. By Lemma 1.2.1 of [47], the archimedean L-factor of L(s, χ ⊗ sym 4 f ) coincides with (130) with ℓ = k + 1, m = k − 2, as expected. By (131), the critical points for L(s, χ ⊗ sym 4 f ) are given by 7.11 Remark. As pointed out in Remark 7.4, the above calculation of the critical set uses the fact that χ is an odd Dirichlet character. If instead χ were an even Dirichlet character (for example if we were to take χ to be trivial), then the critical set would become {−k + 3, −k + 5, .., −1; 2, 4, , . . . , k − 2}, which involves a shift from (139) in each half-plane.
In the following theorem, we will obtain an algebraicity result for the special value of the Lfunction at all the critical points in the right half plane, except possibly for the point 1. The analogous result for the critical points in the left half plane can be obtained from the standard functional equation [20] of GL 5 L-functions.

7.12
Theorem. Let f be an elliptic cuspidal newform of even weight k and trivial nebentypus; assume that f is not of dihedral type. Let π be the Ramakrishnan-Shahidi lift of f to GSp 4 , and let F ∈ V (π) be such that its Fourier coefficients lie in a CM field. Let S be any finite set of places of Q containing the infinite place, χ be an odd Dirichlet character, and r be an odd integer satisfying 1 ≤ r ≤ k − 1. If r = 1, assume χ 2 = 1. Then for any σ ∈ Aut(C), we have Proof. This theorem follows from Theorem 7.5 and (138).
Proof of Theorem 1.5. This follows similarly, only using Theorem 1.1 rather than Theorem 7.5.
7.13 Remark. As noted earlier, the hypothesis that k is even and f has trivial nebentypus is necessary since we are using the results of [43]. The hypothesis that χ is odd is a consequence of our definition of X and ultimately goes back to our construction of the Eisenstein series (specifically, the definition of the vector f k from Section 5.2, which is otherwise not well-defined).
7.14 Remark. Deligne's famous conjecture on critical values of motivic L-functions predicts an algebraicity result for the critical values of L(s, χ⊗sym m f ) for each positive integer m. For m = 1 this was proved by Shimura [54], for m = 2 by Sturm [59], and for m = 3 by Garrett-Harris [16]. In the case m = 4, and f of full level, Ibukiyama and Katsurada [26] proved a formula for L(s, sym 4 f ) which implies algebraicity. Assuming functoriality, the expected algebraicity result for the critical values of L(s, χ ⊗ sym m f ) was proved for all odd m by Raghuram [41]. To the best of our knowledge, the results of this paper represent the first advances in the case m = 4 for general newforms f . However, Deligne's conjecture is in the motivic world and it is a non-trivial problem to relate Deligne's motivic period to our period in (140) which involves the Petersson norm F, F . One way to ask for compatibility of our result with Deligne's conjecture is via twisted L-values. 5 Let f be as in Theorem 7.12, χ 1 , χ 2 be two odd Dirichlet characters, and let r be an integer as in Theorem 7.12. Then Deligne's conjecture, together with expected properties on the behavior of periods of motives twisted by Artin motives implies that (see [42,Conjecture 7.1]): On the other hand, (141) is also an immediate consequence of (140). This shows the compatibility of Theorem 7.12 with Deligne's conjecture. Finally, we note that Theorem 7.12 does not cover the case of dihedral forms; however, Deligne's conjecture is known for all symmetric power L-functions of a dihedral form as explained in Section 4 of [42].

A Haar measures on Sp 2n (R)
This appendix will furnish proofs for the constants appearing in the integration formulas (27) and (88). The symbol K denotes the maximal compact subgroup Sp 2n (R) ∩ O(2n) of Sp 2n (R).

A.1 The KAK measure
Recall that we have fixed the "classical" Haar measure on Sp 2n (R) characterized by the property (12). There is also the "KAK measure" given by the integral on the right hand side of (27).
Note that this function is the square of the absolute value of the matrix coefficient appearing in (68). Hence the integrals will be convergent as long as k > n. The function f (Z) on H n corresponding to F is given by Hence, by (12), Hn det(Y ) k−n−1 | det (1 n + Y − iX)| 2k dX dY.
We now employ the following integral formula. For a matrix X, denote by [X] p the upper left block of size p × p of X. For j = 1, . . . , n let λ j , σ j , τ j be complex numbers, and set λ n+1 = σ n+1 = τ n+1 = 0. Then, by (0.11) of [37], provided the integral is convergent. We will only need the special case where all λ j are equal to some λ, all σ j are equal to some σ, and all τ j are equal to some τ . In this case the formula says that Next we evaluate the right hand side RHS of (27). With the same calculation as in the proof of Proposition 5.5, we obtain RHS = α n 2 n T 1≤i<j≤n Our assertion now follows by comparing (148) and (150).

A.2 The Iwasawa measure
In this section we will prove the integration formula (88) in the proof of Lemma 5.6. It is well known that the formula holds up to a constant, but we would like to know this constant precisely. Let T be the group of real upper triangular n × n matrices with positive diagonal entries. Then T = AN 1 , where A is the group of n × n diagonal matrices with positive diagonal entries, and N 1 is the group of n × n upper triangular matrices with 1's on the diagonal. We will put the following left-invariant Haar measure on T , where dn is the Lebesgue measure, and da = da 1 a 1 . . . dan an for a = diag(a 1 , . . . , a n ). Let P + be the set of positive definite n × n matrices. We endow P + with the Lebesgue measure dY , which also occurs in (12).
A.2 Lemma. The map α : is an isomorphism of smooth manifolds. For a measurable function ϕ on P + , we have Proof. By Proposition 5.3 on page 272/273 of [23], the map (152) is a diffeomorphism. The proof of formula (153) is an exercise using the tranformation formula from multivariable calculus.
A.3 Proposition. Let dh be the Haar measure on Sp 2n (R) characterized by the property (12).
Proof. It is well known that the right hand side defines a Haar measure d ′ h on Sp 2n (R). To prove that d ′ h = dh, it is enough to consider K-invariant functions F . For such an F , let f be the corresponding function on H n , i.e., f (gI) = F (g). Let N 1 and N 2 be as in (90), so that N = N 1 N 2 . By identifying elements of A and N 1 with their upper left block, our notations are consistent with those used in (151). We calculate G F (g) d ′ g = 2 n A N 1 N 2 K F (an 1 n 2 k) dk dn 2 dn 1 da = 2 n A N 1 N 2 F (n 2 an 1 ) det(a) −(n+1) dn 2 dn 1 da.
In the last step we think of a as its upper left n × n block when we write det(a). Continuing, we get f (X + i(an 1 ) t (an 1 )) det(a) −(n+1) dX dn 1 da.
In the last step, again, we identify a and n 1 with their upper left blocks. By (151), we obtain f (X + iY ) det(Y ) −(n+1) dX dY.
where in the last step we applied Lemma A.2. Using (12), we see G F (g) d ′ g = G F (g) dg, as asserted.