Algebraicity of L-values attached to quaternionic modular forms

Abstract In this paper, we prove the algebraicity of some L-values attached to quaternionic modular forms. We follow the rather well-established path of the doubling method. Our main contribution is that we include the case where the corresponding symmetric space is of non-tube type. We make various aspects very explicit, such as the doubling embedding, coset decomposition, and the definition of algebraicity of modular forms via CM-points.


Introduction
Special values of L-functions attached to automorphic forms have a long history in modern number theory.Their importance is difficult to overestimate, and for this reason, they have been the subject of intense study in recent decades.There is no doubt that it is important to study L-values of automorphic forms whose underlying symmetric space does not have hermitian structure (for example, automorphic forms for GL n ); however, in this paper, we will be dealing with a kind of automorphic form where the corresponding symmetric space has a hermitian structure.To go a bit further, we now introduce some notation.
Let D be a division algebra over Q, and let V be a free left D-module of finite rank n.Denote End(V , D) be the ring of D-linear endomorphism of V and GL(V , D) = End(V , D) × .For a nondegenerate hermitian (or skew-hermitian) form ⟨ , ⟩ ∶ V × V → D, we define a generalized unitary group G ∶= G n ∶= {g ∈ GL(V , D) ∶ ⟨gx, g y⟩ = ⟨x, y⟩}.
One can define automorphic forms associated with such group as in [2].These can be seen as functions on a symmetric space G(R)/K, where K is a maximal compact subgroup of G(R).In addition, when the associated symmetric space can be given a hermitian structure, one can define holomorphic automorphic forms which is what we refer as modular forms in this paper.The symmetric spaces G(R)/K have been classified [12, Chapter X] (see also [17]).For positive integers n and m, we denote by C n m the set of n × m matrices with entries in C.There are four infinite families of irreducible hermitian symmetric spaces of non-compact type as follows: 2 T. Bouganis and Y. Jin The spaces above are the so-called bounded realizations of the symmetric spaces, and one can with the use of the Cayley transform show that they are biholomorphic to unbounded domains.For example, when D = Q and ⟨ , ⟩ is skew-hermitian, then G is the symplectic group and we have the notion of Siegel modular forms defined over symmetric spaces of type C. The unbounded realization is the classical Siegel upper space.When D is an imaginary quadratic field, G is the unitary group and we have the notion of Hermitian modular forms defined over symmetric spaces of type A. For these two types of domains (and their groups), there has been an intensive study on the algebraic properties of their attached special L-values.We will not cite here the vast literature that has grown in the past few decades, so we will only mention here the book [32], the more recent article [21] and the references therein for a more complete account of the Siegel case, and the work of Harris [11] in the Hermitian modular forms case.
The focus of this paper is on the domains of type D above.This domain arises when we select D to be a definite quaternion algebra and the form ⟨ , ⟩ skew-hermitian (see the next section for details).There are already some works for these modular forms (for example, [3,16,35,37]), but it is fair to say that these modular forms are not as intensively studied as the Siegel or Hermitian ones.Even more importantly, most, if not all, of the works are restricted to the case when the dimension of V is even.The importance of this restriction is related to the unbounded realization of the corresponding symmetric domain.In particular, when n is even, the unbounded domain is biholomorphic to a tube domain, or what is usually called a domain of Siegel Type I.When n is odd, the domain is not any more of tube type (a similar aspect is seen also for Hermitian modular forms in the non-split case U(n, m) with n ≠ m).The significance of this distinction will become clear later in this paper, since the nontube case is considerably more technical.For example, as we will see, the notion of an algebraic modular form cannot be the usual one (algebraic Fourier coefficients) or the doubling embedding which is needed in the doubling method is considerably more complicated to write explicitly in the unbounded realization.
As we have indicated, we will be studying the special values of L-functions by using the doubling method of Garrett, Shimura, Piatetski-Shapiro, and Rallis.Without going here into details but referring later to the paper, the key idea is to obtain an integral representation relating the L-function to the pullback of a Siegel-type Eisenstein series.Then the analytic and algebraic properties of the L-function can be studied from those of the Eisenstein series.The latter is well understood thanks to the rather explicitly known Fourier expansion.
Our starting point is a cuspidal Hecke eigenform f ∈ S k (K 1 (n)).We then consider two copies of our group G n with an embedding G n × G n → G N with N = 2n, and hence G N splits (see Section 4 for notation).For P N the Siegel parabolic subgroup of G N , we describe in Proposition 4.3 the double coset P N /G N /G n × G n .Then a Siegeltype Eisenstein series over G N can be decomposed into several orbits, and except for https://doi.org/10.4153/S0008414X23000184Published online by Cambridge University Press Algebraicity of L-values attached to Quaternionic modular forms 3 one "main orbit, " all orbits vanish when considering an inner product (in one variable) with the cusp form f. This allows us to prove the following formula (see Section 4 and, in particular, Theorem 4.7 for details and further notation): where χ is a Dirichlet character, c k (s) is an explicit function on s, and D(s, f, χ) is a Dirichlet series which is related (see equation (3.1)) to the twisted standard L-function L(s, f, χ).
In Section 5, we review the definition of algebraic modular forms and differential operators.It is well known how to define algebraic modular forms on hermitian symmetric space.There are several different definitions, and we will mainly follow the one via CM-point as in [32].Using the Maass-Shimura differential operators, we discuss the notion of a nearly holomorphic modular form in our setting.These differential operators for all four types of symmetric spaces mentioned above have been studied in [27,29].We will summarize the result there and apply it to the Siegel-type Eisenstein series mentioned above.Based on this and thanks to the wellunderstood Fourier expansion of Siegel-type Eisenstein series, we will prove our main algebraic result for L-functions by the same method as in [32].Our main result is Theorem 6.3 which gives the following.Theorem 1. 1 Let n be an ideal in Z, and assume that all finite places v with v ∤ n are split in B. Let f ∈ S k (K 1 (n), Q) be an algebraic cuspidal Hecke eigenform, and let χ be a Dirichlet character whose conductor divides the ideal n.Assume that k > 2n − 1, and let μ ∈ Z such that 2n − 1 < μ ≤ k.Then L(μ, f, χ) π n(k+μ)− 3 2 n(n−1) ⟨f, f⟩ ∈ Q.
Remark 1. 2 We note here that the condition on the conductor of the Dirichlet character is not restrictive.Indeed, for Most of our arguments to prove the above are straightforward generalization of [32] from the unitary and symplectic setting to our setting.Our main contribution is making some of the not always obvious generalizations as explicit as possible, such as the diagonal embedding, especially in the non-tube case (see Section 2.3), and the coset decomposition Section 4).Another contribution of the present paper is in the definition of algebraic modular forms (see Section 5), especially in the non-tube case, where we follow a rather more explicit approach by using the theory of CM-points developed by Shimura [32] rather than simply referring to the more advanced and general theory of Harris [9,10], Deligne [5], and Milne [20] on automorphic vector bundles of Shimura varieties.Finally, we should add that our computations are done mainly using the adelic language (in comparison to the more classical in [32]), which is also inspired by [21].

Groups and symmetric spaces
In this and the next section, we introduce the notion of a quaternionic modular form and discuss some main properties.Such modular forms have been already studied (see, for example, [3,16]), but we extend the discussion to include also the case of non-split groups.For most of our notation here, we follow the one introduced in the books [31,32], where the case of Siegel and Hermitian modular forms is considered.

Quaternionic unitary groups
We start by fixing some notation.For more details on quaternion algebras, the reader is referred to [36].In this work, a quaternion algebra will mean a central simple algebra of dimension 4 over Q.After selecting a basis, we can write it in the form where with α, β nonzero square-free integers.We assume in this paper that B is definite, i.e., α, β < 0. The main involution of B is given by We warn the reader that we may, by abusing the notation, denote ⋅ various involution of algebras (for example, complex conjugation on quadratic imaginary fields), but it will be always clear from the context what is meant.The trace and the norm are defined by tr(x) = x + x, N(x) = xx for x ∈ B. As usual, we write M n (B) for the set of n × n matrices with entries in B. We also use the notation B m n for the set of m × n matrices with entries in B. For X ∈ M n (B), we write X * = t X, X = (X * ) −1 for the conjugate transpose and its inverse (if makes sense).
Identify ζ, ξ with √ α, √ β ∈ Q, and let K = Q(ξ).We define the embedding One easily checks that for x ∈ B, and i induces an isomorphism We extend this map to an embedding i ∶ M n (B) → M 2n (K) by sending x = (x i j ) to (i(x i j )).Denote For a matrix with entries in quaternion algebra, the determinant det and the trace tr will mean the reduced norm and the reduced trace.That is taking the determinant and the trace for its image under i.It is well known that the definition of reduced norm and trace is indeed independent of the choice of such embedding and the field Let A be the adele ring of Q.By a place v, we mean either a finite place corresponding to a prime or the archimedean place ∞.The set of finite places is denoted by h.We write The previous definition of trace, norm, and determinant naturally extends locally or adelically.Fix a maximal order O of B and set and definite otherwise.In particular, B is definite if α, β < 0, and indefinite otherwise.That is, for the infinite place, by our assumption, B ∞ is the Hamilton quaternion and the map i above induces an isomorphism In this paper, we consider the following algebraic groups: Here, n = 2m + r and we assume that m > 1 is the global Witt index of the group G.Such a group is usually called a quaternionic unitary group, and has with some anisotropic matrix θ = t θ ∈ SL rv (Q v ) (that is, the corresponding quadratic form does not represent zero), and 2n In these two cases, we have m v = n, r v = 0, and

T. Bouganis and Y. Jin
We remark here that the condition on m being the Witt index of our group G implies that r ≤ 3. Indeed, using a result in [13], we know that for r ≥ 4, ζ ⋅ 1 r is isotropic if and only if it is locally isotropic for all finite places v and infinity.But the latter (i.e., locally isotropic for all finite places and infinity) is always the case for r ≥ 4. Indeed, for v split, this follows from [33,Theorem 7.6] and for v nonsplit (including ∞) from [34].
We will discuss the local archimedean group G(R) and the associated symmetric spaces in the next subsection.
We fix an integral two-sided ideal n = (N) of O generated by It is well known (see, for example, [22, p. 251]) that we have a finite decomposition Moreover, thanks to the weak approximation which is valid for our group (see [22,Proposition 7.11]), we can take t j such that (t j ) v = 1 for v|n (compare with [31,Lemma 8.12]).For finite places v not in the support of n, the Iwasawa decomposition is valid, and hence we can take t j to be upper triangular.Let Γ We can take t 0 = 1 so that

Abstract symmetric spaces
To motivate our definition of symmetric spaces, we start with a rather general and abstract setting before giving explicit realizations of our symmetric spaces.Let i be any embedding M n (H) → M 2n (C).Then, by the Skolem-Noether theorem, there exists We call it a realization of G(R).Suppose that we are given two such data Again, by Skolem-Noether, there exists Following [23], we will define the associated symmetric space via its Borel embedding into its compact dual symmetric space.In our case, we have that the semisimple compact dual of our group is the group SO(2n) (see [12, p. 330]), and the corre-sponding dual symmetric space is SO(2n)/U(n).This space may be identified (see, for example, [28, p. 6]) with the space V = L/GL n (C) where n ∶ −iU * HU > 0, t UKU = 0} ⊂ L, with the action of GL n (C) by right multiplication and G by left multiplication.The symmetric space H is defined as for some fixed suitable u 0 , which we make explicit later.The following lemma is a direct consequence of our definition for H.

Lemma 2.1 There is a bijection
Note that G acts on Ω by left multiplication.By the above lemma, it follows that for any element α ∈ G, we can find a z ′ ∈ H and an λ(α, z) We then define the action of G(R) on H by α.z ∶= αz ∶= z ′ and λ(α, z) satisfies the cocycle relation We set j(α, z) ∶= det(λ(α, z)) ∈ C × .We call λ(α, z) or j(α, z) automorphy factors.

More explicitly
That is, αz = (az + bu 0 )(cz + du 0 ) −1 u 0 ,and λ(α, z) = u −1 0 (cz + du 0 ).For z 1 , z 2 ∈ H, we set We now note that In particular, we obtain that T. Bouganis and Y. Jin and after taking the determinant, we have In particular, We now discuss the relation between different realizations of the symmetric space H.Given H 1 , K 1 and H 2 , K 2 as above, we have seen at the beginning of this subsection that we can find an R such that and the isomorphism can be given by ρ(z 1 ) = z 2 .

The symmetric space Z
We now apply the above considerations to some explicit realizations of G(R).We first define a symmetric space Z which can be directly obtained from G(R).This realization is useful in the computations of the doubling map and Lemma 4.6.
Note that the map i defined above induces the following isomorphism on Q-groups: This induces the following isomorphism on R-groups: and define the symmetric space by Explicitly, The action of G ∞ on Z is given by We note that we are using the same notation K ∞ for maximal subgroup of G ∞ and its preimage in G(R).

The symmetric spaces H and B
We now give another two useful realizations.They are much simpler than the symmetric space Z and are useful in studying CM-points in Section 5.1.However, as the isomorphism between G ∞ above and G ′ ∞ below is rather complicated, the action of the Q-group G on the symmetric space is difficult to compute.

T. Bouganis and Y. Jin
Note that G is isomorphic to .
By changing rows and columns, the map i induces isomorphism and Take u 0 = 1, and the symmetric space associated with this group is This is an unbounded realization of type-D domain in [17].The action of G ′ ∞ on H and the automorphy factor is given by For ], and sending g ↦ T ′−1 gT ′ , we have isomorphism Take u 0 = 1, and the symmetric space associated with this group is defined as [14].The action of G ∞ " on B and the automorphy factor is given by For . We take z 0 = 0 to be the origin of B and

The relation between H and B can be given explicitly by the Cayley transform
as above, and dz = (dz hk ) be a matrix of the same shape as z ∈ C n n whose entries are 1-forms dz hk .Comparing and using the fact (which can be obtained from the property of we have Therefore, Since the jacobian of the map z ↦ αz is j(α, z) −n+1 , the differential form is an invariant measure.If we have another realization H (e.g., Z, H) with identification ρ ∶ H → B, we then define dz ∶= d(ρ(z)) with z ∈ H to be the differential form on H. Clearly, this is also an invariant measure.

Doubling embedding
We keep the notation as before and consider two groups

and consider the map
Composing the above map with g ↦ R −1 gR, we obtain an embedding We can thus view G n1 × G n2 as a subgroup of G N .To ease notation, we write Now we consider a special case of this embedding, namely the case where To ease the notation, we always omit the subscript "n" and keep the subscript N = 2n.The embedding ρ then induces , where the entries with ±1, ±1/2 should be understood as ±I r , ± 1 2 I r .Let Z, Z N be symmetric spaces associated with G ∞ , G N∞ .We are now going to define an associated embedding of symmetric space ι ∶ Z × Z → Z N .We first define the embedding We then define the embedding of symmetric space as where .
Here, we note that we have "normalized" our embedding by S so that ι maps the origin of Z × Z to the "origin" of Z N .That is, ι(z 0 × z 0 ) = i ⋅ 1 N =∶ Z 0 , where z 0 and Z 0 are the origins of Z and Z N , respectively.
For example, in the case where r = 0, then the embedding is quite simple, namely , where in the case of r = 1, it is given by (note that in this case .
We now show that the embedding of the symmetric spaces is compatible with the embedding of the groups.

T. Bouganis and Y. Jin
Proof By our definition of embedding and the action, We must have and the desired result follows.Taking the determinant, we also obtain (2). Suppose

Quaternionic modular forms, Hecke operators, and L-functions
In this section, we introduce the notion of a modular form of scalar weight and define the Hecke operators in our setting.We then define the associated standard L-function.
We keep writing G for G n with n = 2m + r as above.

Modular forms and Fourier-Jacobi expansion
Fix an integral ideal n as in the previous section.
We note here that since we are assuming m ≥ 2, we do not need any condition at the cusps due to Koecher's principle (see [16,Lemma 1.5]).
Denote the space of such functions by M k (Γ).Here, we are using the realization (G ∞ , Z) for our symmetric space.In fact, the definition is independent of the choice of realizations in the following sense.If we choose another realization H (e.g., H, B) with identification ρ ∶ H → Z, then with notation as in equation (2.2), to a function We write S ∶= S(Q) ∶= {X ∈ M m (B) ∶ X * = X} for the (additive) algebraic group of hermitian matrices.We use S + (resp.S + ) denote the subgroup of S consisting of positive-definite (resp.positive) elements.For a fractional ideal a ⊂ B, we set S(a) = S ∩ M m (a).Denote e ∞ (z) ∶= exp(2πiz) for z ∈ C and λ = 1 2 tr.For f ∈ M k (Γ) and γ ∈ G, there is a Fourier-Jacobi expansion of the form In particular, for γ = 1, we simply write We call f a cusp form if c(τ, γ, f ; v, w) = 0 for every γ ∈ G and every τ such that det(h) = 0.The space of cusp form is denoted by S k (Γ).
Given a function f ∶ G(A) → C, we can, by abusing the notation, also view it as a We will denote the space of such functions by for all unipotent radicals U of all proper parabolic subgroups of G.The space of cusp forms will be denoted by It is well known that the above two definitions are related by Write f ↔ ( f 0 , f 1 , . .., f h ) for the correspondence under above maps.Here, When n = 2m, r = 0, then the Fourier-Jacobi expansion becomes the usual Fourier expansion.For We call c(τ, q; f) the Fourier coefficients of f.For two modular forms f , h ∈ M k (Γ), we define the Petersson inner product by whenever the integral converges.For example, this is well defined when one of f , g is a cusp form.Adelically, for f, h Here, dg, an invariant differential of G(A), is given as follows: Again, these integrals are well defined if one of f, h is a cusp form.

Hecke operators and L-functions
In the rest of the paper, we make the assumption that all finite places v with v ∤ n are split in B. We define the groups Define the Hecke algebra T = T(K 1 (n), X) be the Q-algebra generated by double coset acts on f by where Y is a finite subset of G h such that We say that f ∈ S k (K 1 (n)) is an eigenform if there exist some numbers λ f (ξ) ⊂ C called eigenvalues such that We use the notation l(ξ For a Hecke character χ, we define the series Here, χ * is the associated Dirichlet character of χ.We further define the L-function by Here, the subscript n means the Euler factors at v|n are removed. Define Obviously, T v is trivial if v|n.For v ∤ n, by our assumptions, we can identify the local group G v with local orthogonal group, as in equation (2.1).Such local Hecke algebra is discussed in [33] where a Satake map is constructed

1).
Given an eigenform f, the map ξ ↦ λ f (ξ) induces homomorphism T v → C which are parameterized by Satake parameters α ±1  1,v , . .., α ±1 mv ,v .The L-function then has an Euler product expression In particular, if Here, we write p for the prime corresponding to some place v in the notation above.Finally, it is known that L(s, f, χ) is absolutely convergent for Re(s) > 2n − 1 (see [ In this paper, our starting point is an eigenform f for the Hecke algebra related to K 1 (n) (i.e., the analog of Γ 1 (N) in the classical GL 2 setting).As it is noted above, this Hecke algebra is locally trivial for primes dividing n and, hence, the corresponding Euler factor is also trivial.On the other hand, if we denote by K any congruence subgroup with the property that there is an n such that K 1 (n) ⊂ K (as, for example, of the form K 0 (n), the analo of the classical Γ 0 (N)) and we further assume that our f is an eigenform for the Hecke algebra corresponding to such a K, then since However, in such a situation, our L-function will be only the partial L-function for the Hecke algebra with respect to K since we will be simply setting L p (s, f, χ) = 1 for p|n.
It is a delicate matter to define the missing Euler factor in such a situation.Indeed, it is a conjecture of Langlands [18] that one can associate with all places a local Lfactor and local root number such that the global complete L-function satisfies a functional equation.In such a situation and using the doubling method, Yamana [38] gives a definition of local L-factors and proves the functional equation for cuspidal representations over classical groups.However, his local L-factors are not given explicitly, but rather an existential result is proved [38,Theorem 5.2].
We will return to this matter (complete vs. incomplete L-function) again after we prove our main theorem (see Remark 6.4).

Siegel Eisenstein series and its Fourier expansion
We fix an integer 0 ≤ t ≤ m, and for x ∈ G, we write In particular, if r = 0, then n = 2m.In this case, the parabolic subgroup for t = 0, is called Siegel parabolic subgroup.We now fix weight l ∈ N, and let χ be a Hecke character whose conductor divides n.The Siegel-type Eisenstein series is defined as Then we have the following proposition (see, for example, [3]).

Proposition 4.1 E *
l (x, s) has a Fourier expansion of form where q ∈ GL m (B A ) and σ ∈ S(A).The Fourier coefficient c(h, q, s) ≠ 0 only if ) for all v ∈ h.In this case, we have Here: (1) A(n) ∈ Q × is a constant depending on n.
(2) If h has rank r, then where P h,q, p (X) ∈ O[X] and P h,q, p = 1 if det(h) ∈ 2 m+1 Z × p .Here, p is the prime corresponding to v.
(3) Let p be the number of positive eigenvalues of h, q the number of negative eigenvalues of h, and t = m − p − q, then for y where and ω is holomorphic with respect to s + l , s − l.In particular, when p = m, and ]. Set q = q j q ∞ , σ = σ j σ ∞ , then by modularity property, E * l (g, s) can be written as with c(h, q, s) in the above propositions.We are interested in the special values E * l (g, s) for s = l.From the Fourier expansion, we have the following proposition by counting poles and degree of π in those Gamma functions.Proposition 4.2 Assume that l > n − 1.Then the Fourier coefficients c(h, q, l) ≠ 0 unless h > 0 and in this case det(q ∞ ) −l c(h, q, l) = C ⋅ e ∞ (iλ(q * hq)).
Here, up to a constant in

is holomorphic in the sense that when viewed as a function on G(A
We note, in particular, that the proposition implies that also E l (g, l) is holomorphic in the above sense since

Coset decompositions
Let be the doubling embedding defined before.To ease the notation, we may omit the subscript n.Denote P N for the Siegel parabolic subgroup of G N and P t = P t n the tparabolic subgroup of G. Proposition 4.3 For 0 ≤ t ≤ m, let τ t be the element of G N given by

Then τ t form a complete set of representatives of P
Proof This can be proved similarly to the proofs of Lemmata 4.1 and 4.2 in [30].Let Multiplying by some element in GL N , we can assume that Let V = B n with standard basis {ε i } and denote x i be the ith row of x.Let Ũ be the subspace of V spanned by basis ε i with i ≤ t or m + r ≤ i ≤ m + r + t and Ũ⊥ the subspace spanned by other basis, so V = Ũ ⊕ Ũ⊥ .Let θ be the restriction of ϕ 1 on Ũ and η the restriction on Ũ⊥ , then we can write (V , ϕ 1 ) = ( Ũ , θ) ⊕ ( Ũ⊥ , η).Assume xϕx * is given by the first matrix as above, let U be the subspace of V spanned by vector x i with i ≤ m or m + r ≤ i ≤ m + r + t, and let U ′ be the subspace spanned by other x i .We also denote U ⊥ = {v ∈ V ∶ uϕv * = 0 for any u ∈ U}, then V = U ⊕ U ⊥ .Then there exists an automorphism γ of (V , ϕ) such that Uγ = Ũ , U ⊥ γ = Ũ⊥ .Now U ′ γ ⊂ Ũ⊥ is a totally η-isotropic subspace; thus; there exists an automorphism γ ′ of ( Ũ⊥ , η) such that U ′ γγ ′ ⊂ ∑ m+t+r+1≤i≤n Bε i .Viewing γ ′ as an automorphism of (V , ϕ) and putting g 1 = γγ ′ , we have (similarly for y) We further modify x * equals the second matrix as above, then by the same argument, we can obtain a similar result but without the term 1 in the middle of x g 1 and yg 2 , which contradicts the assumption T. Bouganis and Y. Jin , and multiplying on the left, we then get the desired form in the proposition.∎ Then, by straightforward computation, To simplify the computation, we may also use the modified representatives τt One easily shows that (compare with Lemma 4.3 in [30]) where ξ runs over G 2t+r and β, γ run over P t n /G n .In particular, Proposition 4. 4 Assume n is coprime to (2) and (ζ).Then, for v|n, τm ∈ G n where blocks has size (m, r, m) × (m, r, m).We calculate that  The second line shows that p 22 ≡ e ≡ 1 mod n and then p 12 ≡ p 32 ≡ 0 mod n by other formulas.Therefore, From the above identities, we already have The claim ξ ∈ K v then follows from the above identities together with

Integral representation
Let χ be a Hecke character whose conductor divides the fixed ideal n.Recall that we write N = 2n.We define σ ∈ G N (A h ) as σ v ∶= 1, the identity matrix, if v ∤ n, and σ v = τm if v|n.
We then consider the weight k Siegel-type Eisenstein series (twisted by τm ) on G N defined as We are going to study the integral Proof Let U t be the unipotent radical of P t n .Then the integral equals Since ξ normalizes U t and τt (n × 1) ∈ P n , this equals which vanishes by the cuspidality of f. ∎ Therefore, The infinite part is calculated in following lemma.

Lemma 4.6
For k + Re(s) > 2n + 1, we have , where n = 2q + t with q ∈ N and t ∈ {0, 1} and α(s This kind of integral is calculated in [31, Appendix A2] and [14].In particular, it is shown there that for k + Re(s) where ck (s) is a function on s which does not depend on f.Indeed, as it is explained in [31], the quantity ck (s) is independent of f and it is equal to where dz is the invariant measure on the bounded domain and is given as dz = det(I + z z) −n+1 dz, and α(s) is a holomorphic function on s such that α(λ) ∈ Q for all λ ∈ Q (actually it can be made precise, but we do not need it here).But this last integral has been computed in [14, p. 46] from which we obtain that , Note that for v|n, ϕ v is nonzero unless h v ∈ K v .Hence, it remains to consider the integral over ∏ v∤n K v /G(Q v ).These unramified integrals are well known by [19,33,37].Indeed, by the Cartan decomposition, we can write where m v is the local Witt index of G(Q v ) and p v the prime corresponds to v. Note that by definition of ϕ, In conclusion, we obtain We summarize the discussion in the following theorem.
Theorem 4.7 Let f ∈ S n k (K 1 (n)) be an eigenform, and assume that n is coprime to

Algebraic modular forms and differential operators
In order to move from the analytic considerations discussed so far to algebraic questions, we need to discuss the notion of an algebraic modular form in our setting.The notion of algebraic modular forms on Hermitian symmetric space is well understood.There are mainly four characterizations of algebraic modular forms: via Fourier-Jacobi expansion, CM-points, pullback to elliptic modular forms, and canonical model of automorphic vector bundle.For example, in [20,Section III.7], automorphic forms are interpreted as sections of certain automorphic vector bundles.The canonical model of automorphic vector bundles then defines a subspace of algebraic automorphic forms (see also [9,10]).It is also proved there that this definition is equivalent to the definition in terms of values at CM-points.In [6], Garrett gives three characterizations of algebraicity for scalar-valued modular forms via CMpoints, Fourier-Jacobi expansion, and pullback to elliptic modular forms.They are also proved to be equivalent.
However, in this work, instead of simply referring to the results of Harris as in [9,10], we have decided to offer a definition of algebraic modular forms via CMpoints using the rather more explicit language of Shimura as in [32], without need to refer to the more advanced and general theory as developed by Deligne, Milne, and others.Indeed, our approach of the definition of CM-points and the underlying periods follows an idea in the first works of Shimura on the subject [25], where one "tensors" a given embedding h ∶ K 1 × ⋯ × K n ↪ G, of CM field K i , with another CM field K, disjoint to the K i 's to obtain a point whose associated abelian variety is of CM type (see also [5,Proof of Theorem 6.4]).In this way, we will be able to define and study the CM-points in our case by considering an embedding of our group into a unitary group, after a choice of an imaginary quadratic field.However, we will show that our definition of CM-points and the attached periods is independent of the choice of the auxiliary imaginary quadratic field.This should be seen as our main contribution in this section, which we believe it is worth appearing in the literature and could be helpful to other researchers, thanks to its rather explicit nature and basic background, Finally, we will show that in certain case, when the underlying symmetric space is a tube domain, i.e., a Siegel Domain of Type I, our definition is equivalent to standard definition using the Fourier expansion.

CM-points
We introduce the following setting, with some small repetition of what we have discussed so far.
We let B be a definite quaternion algebra over Q, T * = −T ∈ M n (B) a skewhermitian matrix, and define the algebraic group Let K i , i = 1, . . ., n, be imaginary quadratic fields and consider the CM algebra Y = K 1 × ⋯ × K n and Y 1 = {y ∈ Y ∶ yy ρ = 1} with ρ induced by the nontrivial involutions (i.e., complex conjugation) on each K i .We are interested in embeddings Let us show that there always exists such an embedding.
Without loss of generality, we may write T = diag[a 1 , . .., a n ] in diagonal form.We then select as imaginary quadratic fields K i ∶= Q(a i ), for i = 1, . . ., n, and define the embedding Back to our general considerations, we select an imaginary quadratic field K which is different from K i 's above, and splits B. It is easy to see that that there exists always such a field K.We now fix an embedding M n (B) → M 2n (K).Denote the image of T in M 2n (K) by T and the unitary group

T. Bouganis and Y. Jin
We note that the group of R-points of U(T) is isomorphic to Its action on the bounded domain (see, for example, [32]), with the obvious block matrices.
The two factors of automorphy are given by λ(g, z) = c t z + d, and μ(g, z We will view G(T) (resp.B) as a subgroup (resp.subspace) of U(T) (resp.B) under this embedding.Proof The first part can be shown exactly as [32,Lemma 4.12], and for the second part, we adapt an idea of the proof of that lemma.Without loss of generality, we can assume that the origin 0 of B is a fixed point for h(Y 1 ) and our task is to show that it is the unique fixed point.We note that the maximal compact subgroup in G(T)(R) fixing the origin is isomorphic to U(n), and hence with respect to the embedding In particular, we have an embedding h(Y 1 ) ↪ U(n) ↪ U(n, n).Assume now that there is another point z ∈ B which is a fixed point of h(Y 1 ).Then we must have that z = aza −1 for every element diag[a, ā] ∈ (U(n) × U(n)) ∩ h(Y 1 ).But for such a point we have that a * a = 1 and hence a −1 = t a.That is, z = az t a.Since a ∈ U(n) ↪ U(n) × U(n), we may diagonalize it, say with eigenvalues λ i , i = 1, . . ., 2n, and hence we must have z i j = 0 for every λ i ≠ λ j .Taking a to be the element obtained from β above, we have that z has to be the origin.∎ We call a point fixed by some h(Y 1 ) as above a CM-point, and we note that this definition does not depend on the choice of the field K.For example, take . This is the group G ′ in ection 2, and we have described its embedding into unitary group and the action on B explicitly there.Let h and Y be as above, and one easily checks that 0 is the fixed point of h(Y 1 ) and thus a CM-point.
We now want to attach some CM periods to our CM-points.We will do this by relating our definition with the notion of CM-points of unitary groups.Indeed, our selection of the field K allows us to view our group as a subgroup of a unitary group, and hence an embedding B ↪ B. Our next aim is to relate the just-defined CM-points in B with the well-studied, as in [32], CM-points of H.It is here that we employ the idea of Shimura, which was used in [25] (see also [26,Section 7]) to study CM-points in general type-C domains.
Let w ∈ B ⊂ B be a CM-point fixed by h(Y 1 ) ⊂ G ⊂ U(n, n).Then, for such a point, we have that where Λ(α, w) ∈ GL 2n (C) and p(x, z) ∶ C 2n × B → C 2n are the maps defined in [32, §4.7].In this way, we can obtain an embedding Y → End C (C 2n ) by sending α ↦ Λ(α, w) where we have used the fact that Y is spanned by Y 1 over Q.We now extend this to an injection That is, the point w can be seen as a fixed point of S 1 ⊗ Q R, where S 1 = {s ∈ S | ss ρ = 1} with ρ the involution on S induced by the complex conjugation on KK i .Hence, w is a CM-point in B defined in [32, §4.11] for unitary groups.In particular, w has entries in Q by [32,Lemma 4.13].
In the following lemma, we use the notation I Y , J Y , J S j as defined in [32, p. 77].
Lemma 5.3 With notation as above, for all 1 ≤ i ≤ n, we have that where Proof Let us write Φ = ∑ n j=1 Φ j with Φ j ∈ I S j and Φ ′ = ∑ n j=1 Φ ′ j , with Φ ′ j ∈ I K j .Then we have that Φ j = Inf S j /K j (Φ ′ j ).Indeed, first, we observe that Ψ = ∑ n j=1 Res S j /K Φ j ∈ I K (see [32, p. 85]), where Ψ as in Remark 5.2.Moreover, we know that Φ = ϕ + ψ with ϕ, ψ ∈ I S as above, and we have seen that ψ = ϕ when restricted to K via K ↪ Y ⊗ Q K = S. But, on the other hand, we have seen that ψ = ϕ when restricted to Y, from which we obtain that Φ j = Φ ′ j ⊗ τ + Φ ′ j ⊗ τ, where τ is a fixed embedding of K ↪ C (i.e., a CM type for K).Since S j = K j ⊗ Q K, the claim that Φ j = Inf S j /K j (Φ ′ j ) now follows.The statement of the lemma is now obtained from the inflation-restriction properties of the periods (see [32, p. 84]): where ψ i j ∈ J S j induced by ψ i ∈ J S = ⋃ n j=1 J S j .Similarly follows also the other equality. ∎ The above lemma shows that we have p ∞ (w) = p ∞ρ (w) for w ∈ B and they are independent of the choice of the imaginary quadratic field K we chose above (and hence of the embedding to the unitary group).We then simply define p(w) = p ∞ (w) = p ∞ρ (w) for the period attached to CM-point w ∈ B. By [32,Proposition 11.5] and the definition of periods, we immediately have: (1) The coset p(w)GL n (Q) is determined by the point w ∈ B independently of the embedding (Y , h) chosen above.4 Even though the definition of a CM-point in B given above is enough for our applications, we mention here that there is a more general definition as follows.We may take Y above as Y = M n1 (K 1 ) × ⋯ × M ns (K s ) with K i CM fields and the condition that n = ∑ s i=1 n i [K i ∶ Q] and assume that there exists an embedding h ∶ Y 1 → G(T) where Y 1 ∶= {y ∈ Y | yy ρ = 1} with the involution on Y induced by complex conjugation and transpose.Then one can show as above that h(Y 1 ⊗ Q R) has a unique fixed point w ∈ B. Picking as before an imaginary quadratic field K disjoint from all K i , we can see that the point w ∈ B ↪ B corresponds to an abelian variety A w with endomorphism ring equal to Y ⊗ Q K.In particular, we have that A w is isogenous to A n1 1 × ⋯ × A ns s , where the abelian variety A i has CM by the field S i ∶= KK i .

Algebraic modular forms
We keep the notation from before.In particular, we write G for G(T) and we have an embedding i ∶ G → U(T) as above.For the following considerations, we need to augment our definition of modular forms from scalar-valued to vector-valued.
We start with a Q-rational representation ω ∶ GL n (C) → GL(V ).Given a function f ∶ B → V and g ∈ G, define ( f | ω g)(z) = ω(λ(g, z)) −1 f (gz).For a congruence subgroup Γ, the space of modular forms M ω (Γ) consists of holomorphic function with the property f | ω γ = f for all γ ∈ Γ.Put M ω = ⋃ M ω (Γ) where the union is over all congruence subgroups, and where e runs over Z and τ e denotes the representation defined by τ e (x) = det(x) e ω(x).Definition 5.1 Let W be a set of CM-points which is dense in B. Put P ω (w) = ω(p(w)) for w ∈ W. ( We can compare the definition for our group with the unitary group.Let ω ∶ GL n (C) × GL n (C) → GL(V ) be Q-rational representation.Denote A ω , A ω (Γ) for modular function spaces for unitary group as in [32, §5.3].The composition of ω with the diagonal embedding GL n → GL n × GL n gives a representation ω ∶ GL n (C) → GL(V ).Clearly, if f ∈ A ω , then its pullback f ○ ι ∈ A ω is a quaternionic modular form.Moreover, f ∈ A ω (Q) and f ○ ι is finite, then the pullback f ○ ι ∈ A ω (Q).
Even though we have provided a definition of algebraicity for modular forms on the bounded domain B, we can transfer it also to the other realization of the symmetric spaces discussed in Section 2. Indeed, with the notation of Section 2.2, suppose we are given two of these realizations (i 1 , Φ 1 , H 1 , K 1 , G 1 ) and (i 2 , Φ 2 , H 2 , K 2 , G 2 ), with i 1 = i 2 both induced from an algebraic embedding M n (B) → M 2n (K), where K is an imaginary quadratic field which splits B. In particular, the matrix R in equation (2.2) has algebraic entries, and hence we obtain that the bijective map ρ ∶ H 1 → H 2 as defined there is algebraic, in the sense that maps algebraic points of H 1 to algebraic points of H 2 .We also conclude from this that μ(z), as defined in the same equation, is algebraic if z is.In particular, given any realization H, there is a bijection ρ ∶ H → B. We define the CM-points on H to be the inverse image with respect to ρ of the CMpoints of B.
As we have discussed, every vector-valued modular form g ∶ H → V corresponds uniquely to a modular form f ∶ B → V by the rule g(z) = ω(μ(z)) −1 f (ρ(z)).So it is enough to now observe that if w is a CM-point of H, which by definition means ρ(w) is a CM-point of B, and we have established above μ(w) ∈ GL n (Q).Hence, we can use the same periods P ω (w) for both f and g.

T. Bouganis and Y. Jin
In particular, the algebraicity as defined for bounded domains can be transferred to the unbounded domain Z.Let now f ∶ Z → C be a weight k modular form defined in Section 3, we can take its Fourier-Jacobi expansion.We denote the Fourier-Jacobi coefficients by c(τ, f ; v, w).When r = 0, n = 2m, we simply denote it by c(τ, f ).Proposition 5.5 (1) (2) For every f ∈ M k and σ ∈ Aut(C/Q), we have c(τ, f σ ; 0, w) = c(τ, f ; 0, w) σ , for all τ and w.
We briefly explain the proof for (3).Let r = 0, n = 2m, and f ∈ M k (Γ) for a congruence subgroup Γ.Let V be the model of Γ/Z defined over Q, then A 0 (Γ) can be identified with the function field of V.By the same method in [32,Sections 6,7], one can show that g ∈ A 0 (Γ) if and only if g has algebraic Fourier coefficients.We will reduce our problem for f ∈ M k to A 0 similarly to what is done in the proof of [32,Proposition 11.11].
Let W be a dense subset of CM-points in Z.We first assume that det(p(w)) −k f (w) ∈ Q for all w ∈ W where f is finite.Note that there exists a function U ∈ A k (Q) on Z holomorphic in w with det(U)(w) ≠ 0. Indeed, denote H for the unbounded realization of B via the Cayley transform, and we can simply put U(z) = R(z) for z ∈ Z ↪ H with R the function in [32,Proposition 9.11].We set g ∶= det(U) −k f , and note that g(w) = det (U(w) −1 p(w)) k det(p(w)) −k f (w).
But now we have that U(w) −1 p(w) is Q -rational since this holds for the function R in unitary case.That is, g(w) is Q-rational for every CM-point w where g is finite; thus, g ∈ A 0 (Q).Since f = det(U) −k g, we obtain that f also has algebraic Fourier expansion.
For the other direction, we keep the same notation.If f ∈ M k has algebraic Fourier expansion, then g ∈ A 0 (Q).For every CM-point w, we may choose the function U above such that U is finite at w and U(w) is invertible.If f is finite at w, then so is g and g(w) is Q rational.The equality of g(w) as above then shows that det(p(w) We end this subsection by giving a definition for adelic modular forms.Let f ∈ M k be an (adelic) modular form of scalar weight k, i.e., we are taking ω = det k .We say that f is algebraic, denoted by f ∈ M k (Q) if for a dense subset W of CM-points in Z, P k (w) −1 f(g h g) ∈ Q for all w ∈ W. Since j(g, w) ∈ Q for CM-point w, this is the same as all component f j under correspondence f ↔ ( f 0 , . .., f h ) are algebraic.
Let r = 0, n = 2m, and keep the notation for Eisenstein series in previous sections.Let E l (g, s) be a Siegel Eisenstein series for group G n .By explicit computation of Fourier expansion, we have E * l (g, l) ∈ M l (Q).Clearly, we also have E l (g, l) ∈ M l (Q).

Differential operators and nearly holomorphic functions
In this subsection, we summarize some of the result of [29] (see also [32,Chapter 3]) on differential operators on type-D domains and then apply these operators to Siegel-type Eisenstein series.We will be working with the bounded realization of our symmetric space, but thanks to the remark above, we can transfer the definitions from one realization to the other.We set Here, T is the tangent space of B at the origin 0. Given a positive integer d and two finite-dimensional complex vector spaces W and V, we denote by Ml d (W, V ) the vector space of all C-multilinear maps of W × ⋯ × W (d copies) into V and S d (W, V ) the vector space of all homogeneous polynomial maps of W into V of degree d.We omit the symbol In particular, taking d = 1 and ω the trivial representation, we define the representation {τ, S 1 (T)} of GL n (C) by [τ(a)h](u) = h( t aua) for h ∈ S 1 (T), u ∈ T.
Take an R-rational basis {ε ν } of T over C and for u We further define We now recall the important fact, due to Hua, Schmid, Johnson, and Shimura (see, for example, [27]), that the representation {τ d , S d (T)} is the direct sum of irreducible representations and each irreducible constituent has multiplicity one.In particular, for each GL n (C)-stable subspace Z ⊂ S d (T), we can define the projection map T. Bouganis and Y. Jin Lemma 5.7 With notation as above, we have: ( is Q-rational for any CM-point w. Proof The proof is the same as the one in [32, Theorems 14.5 and 14.7] (see also [27,Sections 5 and 6]).Indeed, as we have a natural inclusion B → B, we can reduce our problem to unitary case.For example, for (1), denote D f for the differential operators in unitary case.The lemma is proved for this case in [32,Theorem 14.5].Let p be the complex dimension of B, and we can take p elements g 1 , . .., g p ∈ A 0 (Q) such that g 1 ○ ε, . .., g p ○ ε are algebraically independent.Put f j = g j ○ ε.As shown in the last section, This proves our assertion.∎ We now set r(z) ∶= −η(z) −1 z.Let d be a nonnegative integer and {ω, V } the representation as before.A function f ∈ C ∞ (B, V ) is called nearly holomorphic of degree d if it can be written as a polynomial in r, of degree less than d, with Vvalued holomorphic functions on B as coefficients.We denote the space of such functions by N d (B, V ).Let N d ω be the space consisting of functions satisfying the modular properties as in M ω but now replacing the holomorphic condition with nearly holomorphic.For a congruence subgroup Γ, we can similarly define the space N d ω (Γ).An exact same argument in the proof of [32,Lemma 14.3] shows that this space is finite-dimensional over C.
Suppose V is Q-rational.A function f ∈ N d ω is called algebraic, denoted by The proof of the following lemma is the same as the one in [32,Theorem 14.9].

Lemma 5.8 Let Z be an irreducible subspace of S p (T). Then π
Here, τ Z is the restriction of τ p to Z.
We now extend the above definitions to adelic modular forms.Let f ∈ M k and viewing it as a function on G(A h ) × B by setting f(g h , z) = j(g z , z 0 ) k f(g h g z ) with z = g z ⋅ 0 ∈ B. Then D k f, D Z k f is defined as applying differential operators on z ∈ B. A function f ∶ G(A h ) × B → C is called nearly holomorphic if it is nearly holomorphic in z ∈ B. We can then define the space N d k and of nearly holomorphic modular forms as before.Similarly, we can define subspace N d k (Q).These definitions are equivalent to all components in the correspondence f ↔ ( f 1 , . .., f h ) being nearly holomorphic or algebraic nearly holomorphic.
We now apply the differential operators to Siegel-type Eisenstein series and show that it is nearly holomorphic for certain values of s.We will keep the notation of Section 4, and so in particular E l (g, s) is the Siegel-type Eisenstein series associated with group G n , n = 2m of weight l and character χ.Proposition 5.9 Assume l > n − 1, and let μ ∈ Z such that n − 1 < μ ≤ l.Then: (1) E l (g, μ) ∈ π α N m(l −μ) l (Q) with α = m(l − μ).
Proof For this, we use [27,Theorem 2D], which classifies the irreducible representations of (τ m p , S m p (T)).In particular, for p ∈ Z and a weight q, we can define the operator Δ p q by Δ p q f = (D Z ω f)(ψ) with ω = det q , Z = Cψ ⊂ S m p (T), and ψ = det p/2 .Here, the square root of the determinant denotes the Pfaffian of the skew-symmetric matrix.Then Δ p q N t q (Q) ⊂ π m p N t+m p q+p (Q).
We have shown that E l (g, l) ∈ M l (Q), so Δ This concludes the proof of the proposition.∎

Main results
We now recall that we have established the integral representation of the L-function: Here, E k (g, s) = Λ n (s, χ)E(g, s); E(g, s) = E k (gσ −1 , s) for g ∈ G N (A) and σ = 1 if v ∤ n, σ = τm if v|n, where E k (g, s) is the Siegel-type Eisenstein series defined on G N of weight k and N = 2n.We first prove a lemma which is the analog of [32,Lemma 26.12] in our setting.Proof Write with z = g ∞ z 0 , w = h ∞ z 0 ∈ Z m,r .By definition, f (z, w) ∶= f(g h × h h , z × w) is nearly holomorphic in z × w.Similarly to the proof of [32,Lemma 26.12], one can show that it is also nearly holomorphic in z and f (z, w) is nearly holomorphic in w.Therefore, as a function in g or h.Let {g j } e j=1 be a Q-rational basis of N d k (Q).For each fixed h, we have f(g × h) = ∑ e j=1 g j (g)h j (h) with h j (h) ∈ C. Since g j are linearly independent, we can find e points g 1 , . .., g e such that det(g j (z k )) e j,k=1 ≠ 0. Solving the linear equations f(g, h) = ∑ e j=1 g j (z k )h j (h), we find functions h j ∈ N d k .
https://doi.org/10.4153/S0008414X23000184Published online by Cambridge University Press It suffices to prove that {h j } are algebraic.Since W = {g ⋅ z 0 ∶ g ∈ G(Q)} is a dense subset of Z m,r , we can take g j such that g j z 0 ∈ W. We easily calculate the period P k (z 0 × z 0 ) = P k (z 0 )P k (z 0 ); hence, P k (z 0 × z 0 ) −1 f(g h × h h ) = e ∑ j=1 P k (z 0 ) −1 g j (g h )P k (z 0 ) −1 h j (h).
By algebraicity of f, g j , we have P k (z 0 ) −1 h j (h) ∈ Q, and thus for all w = h ∞ z 0 ∈ W, we have P k (w) −1 h j (h) ∈ Q.Hence, h j ∈ N d k (Q), which completes the proof.∎ Before stating the main theorem, we need to establish one more result.Proof This is the analog of Lemma 29.3 proved in [32] in the unitary case.Actually, it is even simpler in our case since we do not need to involve some more complicated differential operators needed in the unitary case.Here, we simply indicate some changes to the proof in [32] to cover our case.We follow the notation of Appendix A8 in [32] and write g for the real Lie algebra of G ∶= G(R).We then have the familiar decomposition of the complexification g C = t C ⊕ p + ⊕ p − where t is the Lie algebra of the fixed maximal compact subgroup K ≅ U(n).Finally, we write U for the universal enveloping Lie algebra of g C , and K c = GL n (C) for the complexification of K. Given a representation (ρ, V ) of K c , we write C ∞ (ρ) for the functions f ∈ C ∞ (G, V ) such that f (xk −1 ) = ρ(k) f (x) for all k ∈ K ⊂ K c , and x ∈ G.As in [32], there is a bijection between C ∞ (Z, V ) and C ∞ (ρ) which we denote by f ↦ f ρ .We also write H(ρ) for the functions in C ∞ (ρ) such that Y f = 0 for all Y ∈ p − .These functions correspond to holomorphic functions in C ∞ (Z, V ).
Recall (see [32]) that a U-module Y is called unitarizable if there exists a positivedefinite hermitian form { , } ∶ Y × Y → C such that {X g, h} = −{g, Xh} for every g, h ∈ Y and X ∈ g.

Lemma 6 . 1
Let f ∈ N d k (Q) be an algebraic nearly holomorphic form associated with group G N .Then there exist g j ,h j ∈ N d k (Q) associated with group G n such that f(g × h) = e ∑ j=1g j (g)h j (h).
[37,n (C) ↪ GL 2n (C) (with a suitable normalization on s).Here, L G o denotes the connected component of the L-group of G, which can be identified with SO 2n (C) (see, for example,[37, Appendix B]).
[33,rk 3.1We first note that for p not dividing n, the local L-factors defined above are given in[31, Theorem 16.16]or[33, Proposition 17.14].These agree with the Euler factors of the Langlands L-function defined by the (standard) embeddingL G o = 11 p 12 p 13 0 0 0 p 21 p 22 p 23 0 0 0 p 31 p 31 p 33