Zeta functions and Eisenstein series on classical groups

Section 1.

Given a reductive algebraic group G over an algebraic number field, we denote by G _A, G _a, and G _h its adelization, the archimedean factor of G _A, and the nonarchimedean factor of G _A. We take an open subgroup D of G _A of the form D = D ₀ G _a with a compact subgroup D ₀ such that D ₀ ∩ G _a is maximal compact in G _a. Choosing a specific type of representation of D ₀ ∩ G _a, we can define automorphic forms on G _A as usual. For simplicity we consider here the forms invariant under D ₀ ∩ G _h. Each Hecke operator is given by DτD, with τ in a subset 𝔛 of G _A, which is a semigroup containing D and the localizations of G for almost all nonarchimedean primes. Taking an automorphic form f such that f|DτD = λ(τ)f with a complex number λ(τ) for every τ ∈ 𝔛 and a Hecke ideal character χ of F, we put where ν₀(τ) is the denominator ideal of τ and N(ν₀(τ)) is its norm. Now our first main result is that if G is symplectic, orthogonal, or unitary, then where Λ(s, χ) is an explicitly determined product of L-functions depending on χ, W _p is a polynomial determined for each v ∈ h whose constant term is 1, and p runs over all the prime ideals of the basic number field. This is a purely algebraic result concerning only nonarchimedean primes.

Let Z(s, f, χ) denote the right-hand side of Eq. 1.2. As our second main result, we obtain a product 𝔊(s) of gamma factors such that 𝔊Z can be continued to the whole s-plane as a meromorphic function with finitely many poles, when G is a unitary group of an arbitrary signature distribution over a CM field, and f corresponds to holomorphic forms.

Now these problems are closely connected with the theory of Eisenstein series E on a group G′ in which G is embedded. To describe the series, let ℨ′ denote the symmetric space on which G′ acts. Then the series as a function of (z, s) ∈ ℨ′ × C can be given (in the classical style) in the form where Γ is a congruence subgroup of G′, and P is a parabolic subgroup of G′ which is a semidirect product of a unipotent group and G × GL _m with some m. The adelized version of δ will be explicitly described in Section 5. Now our third main result is that there exists an explicit product 𝔊′ of gamma factors and an explicit product Λ′ of L-functions such that 𝔊′(s)Λ′(s)Z(s, f, χ)E(z, s; f, χ) can be continued to the whole s-plane as a meromorphic function with finitely many poles.

Though the above results concern holomorphic forms, our method is applicable to the unitary group of a totally definite hermitian form over a CM field. In this case, we can give an explicit class number formula for such a hermitian form, which is the fourth main result of this paper.

Section 2.

For an associative ring R with identity element, we denote by R ^× the group of all its invertible elements and by R _n ^m the R-module of all m × n matrices with entries in R. To indicate that a union X = ∪_i∈I Y _i is disjoint, we write X = ⊔_i∈I Y _i.

Let K be an associative ring with identity element and an involution ρ. For a matrix x with entries in K, we put x* = ^t x ^ρ, and x̂ = (x*)⁻¹ if x is square and invertible. Given a finitely generated left K-module V, we denote by GL(V) the group of all K-linear automorphisms of V. We let GL(V) act on V on the right; namely we denote by wα the image of w ∈ V under α ∈ GL(V). Given ɛ = ±1, by an ɛ-hermitian form on V, we understand a biadditive map ϕ:V × V → K such that ϕ(x, y)^ρ = ɛϕ(y, x) and ϕ(ax, by) = aϕ(x, y)b ^ρ for every a, b ∈ K. Assuming that ϕ is nondegenerate, we put Given (V, ϕ) and (W, ψ), we can define an ɛ-hermitian form ϕ ⊕ ψ on V ⊕ W by We then write (V ⊕ W, ϕ ⊕ ψ) = (V, ϕ) ⊕ (W, ψ). If both ϕ and ψ are nondegenerate, we can view G ^ϕ × G ^ψ as a subgroup of G ^ϕ⊕ψ. The element (α, β) of G ^ϕ × G ^ψ viewed as an element of G ^ϕ⊕ψ will be denoted by α × β or by (α, β). Given a positive integer r, we put H _r = I′_r ⊕ I _r, I _r = I′_r = K _r ¹ and We shall always use H _r, I′_r, I _r, and η_r in this sense. We understand that H ₀ = {0} and η₀ = 0.

Hereafter we fix V and a nondegenerate ϕ on V, assuming that K is a division ring whose characteristic is different from 2. Let J be a K-submodule of V which is totally ϕ-isotropic, by which we mean that ϕ(J, J) = 0. Then we can find a decomposition (V, ϕ) = (Z, ζ) ⊕ (H, η) and an isomorphism f of (H, η) onto (H _r, η_r) such that f(J) = I _r. In this setting, we define the parabolic subgroup of G ^ϕ relative to J by and define homomorphisms → G ^ζ and → GL(J) such that zα − zπ_ζ ^ϕ (α) ∈ J and wα = wλ_J ^ϕ(α) if z ∈ Z, w ∈ J, and α ∈

Taking a fixed nonnegative integer m, we put We can naturally view G ^ψ × G ^ϕ as a subgroup of G ^ω. Since W = V ⊕ H _m, we can put X = V ⊕ H _m ⊕ V with the first summand V in W, and write every element of X in the form (u, h, v) with (u, h) ∈ V ⊕ H _m = W and v ∈ V. Put Observing that U is totally ω-isotropic, we can define

Proposition 1. Let λ(ϕ) be the maximum dimension of totally ϕ-isotropic K-submodules of V. Then has exactly λ(ϕ) orbits. Moreover, with ξ running over G ^ϕ and β over G ^ψ, where H = H _m and I = I_m.

In fact, we can give an explicit set of representatives {τ_e}_e=1 ^λ(ϕ) for Eq. 2.6 and also an explicit set of representatives for P _U ^ω/P _U ^ωτ_e[G ^ψ × G ^ϕ] in the same manner as in Eq. 2.7. This proposition plays an essential role in the analysis of our Eisenstein series E(z, s; f, χ).

Section 3.

In this section, K is a locally compact field of characteristic 0 with respect to a discrete valuation. Our aim is to establish the Euler factor W _p of Eq. 1.2. We denote by r and q the valuation ring and its maximal ideal; we put q = [r:q] and |x| = q ^−ν if x ∈ K and x ∈ π^ν r^× with ν ∈ Z. We assume that K has an automorphism ρ such that ρ² = 1, and put F = {x ∈ K | x ^ρ = x}, g = F ∩ r, and d⁻¹ = {x ∈ K | Tr_K/F (xr) ⊂ g} if K ≠ F. We consider (V, ϕ) as in Section 2 with V = K _n ¹ and ϕ defined by ϕ(x, y) = xϕy* for x, y ∈ V with a matrix ϕ of the form where t = n − 2r. We assume that θ is anisotropic and also that Thus our group G ^ϕ is orthogonal, symplectic, or unitary. The element δ of Eq. 3.2b can be obtained by putting δ = u − u ^ρ with u such that r = g[u]. We include the case rt = 0 in our discussion. If t = 0, we simply ignore θ; this is always so if K = F and ɛ = −1. We have ϕ = θ if r = 0.

Denoting by {e _i} the standard basis of K _n ¹, we put Then G ^ϕ = P _J ^ϕ C. We choose {e _r+i}_i=1 ^t so that N = ∑_i=1 ^tre _r+i. Then we can find an element λ of r_t ^t such that Put We can write every element of P _J ^ϕ in the form If t = 0, we simply ignore b, e, and f, so that ξ = [  ]; we have ξ = e if r = 0.

We consider the Hecke algebra ℜ(E, GL _r(K)) consisting of all formal finite sums ∑c_xExE with c _x ∈ Q and x ∈ GL _r(K), with the law of multiplication defined as in ref. 1. Taking r indeterminates t ₁, … , t _r, we define a Q-linear map as follows; given ExE with x ∈ GL _r(K), we can put ExE = ⊔_y Ey with upper triangular y whose diagonal entries are π^e₁, … , π^e_r with e _i ∈ Z. Then we put Next we consider the Hecke algebra ℜ(C, G ^ϕ) consisting of all formal finite sums ∑c_τCτC with c _τ ∈ Q and τ ∈ G ^ϕ. We then define a Q-linear map as follows; given CτC with τ ∈ G ^ϕ, we can put CτC = ⊔_ξ Cξ with ξ ∈ P of form Eq. 3.5. We then put where ω₀ is given by Eq. 3.6 and d _ξ is the d-block in Eq. 3.5. We can prove that this is well defined and gives a ring-injection.

Given x ∈ K _n ^m, we denote by ν₀(x) the ideal of r which is the inverse of the product of all the elementary divisor ideals of x not contained in r; we put then ν(x) = [r:ν₀(x)]. We call x primitive if rank(x) = Min(m, n) and all the elementary divisor ideals of x are r.

Proposition 2. Given ξ as in Eq. 3.5, suppose that both e and (δθ)⁻¹ (e − 1) have coefficients in r if t > 0. Let a = g ⁻¹ h with primitive [g h] ∈ r _2r ^r and gb = j ⁻¹ k with primitive [j k] ∈ r _r+t ^r. Then where we take j = 1_r if t = 0.

We now define a formal Dirichlet series 𝔗 by This is a formal version of the Euler factor of Eq. 1.2 at a fixed nonarchimedean prime.

Theorem 1. Suppose that δϕ ∈ GL _n(r); put p = [g:g ∩ q]. (Thus p = q if K = F.) Then Here θⁱ = 1 if i is even; when i is odd, θⁱ is −1 or 0 according as d = r or d ≠ r.

This can be proved in the same manner as in ref. 2 by means of Proposition 2.

Since we are going to take localizations of a global unitary group, we have to consider G ^ϕ = G(V, ϕ) of Eq. 2.1 with V = K _n ¹, K = F × F, and ρ defined by (x, y)^ρ = (y, x), where F is a locally compact field of characteristic 0 with respect to a discrete valuation. Let g and p be the valution ring of F and its maximal ideal; put r = g × g and p = [g:p]. We consider ℜ(C, G ^ϕ) with C = G ^ϕ ∩ GL _n(r). Then the projection map pr of GL _n(K) onto GL _n(F) gives an isomorphism η:ℜ(C, G ^ϕ) → ℜ(E ₁, GL _n(F)), where E ₁ = GL _n(g). To be explicit, we have η(C(x, ^tx ⁻¹)C) = E ₁ xE ₁. Let ω₁ denote the map of Eq. 3.6 defined with n, E ₁, and F in place of r, E, and K. Putting ω = ω₁ ○ η, we obtain a ring-injection For z = (x, y) ∈ K _n ⁿ with x, y ∈ F _n ⁿ put ν₁(z) = ν(x) and ν₂(z) = ν(y), where ν is defined with respect to g instead of r. We then put Then we obtain

Section 4.

We now take a totally imaginary quadratic extension K of a totally real algebraic number field F of finite degree. We denote by a (resp. h) the set of archimedean (resp. nonarchimedean) primes of F; further we denote by g (resp. r) the maximal order of F (resp. K). Let V be a vector space over K of dimension n. We take a K-valued nondegenerate ɛ-hermitian form ϕ on V with ɛ = 1 with respect to the Galois involution of K over F, and define G ^ϕ as in Section 2. For every v ∈ a ∪ h and an object X, we denote by X _v its localization at v. For v ∈ h not splitting in K and for v ∈ a, we take a decomposition with anisotropic θ′_v and a nonnegative integer r _v. Put t _v = dim(T _v). Then n = 2r _v + t _v. If n is odd, then t _v = 1 for every v ∈ h. If n is even, then t _v = 0 for almost all v ∈ h and t _v = 2 for the remaining v ∈ h. If n is odd, by replacing ϕ by cϕ with a suitable c ∈ F, we may assume that ϕ is represented by a matrix whose determinant times (−1)^(n−1)/2 belongs to N _K/F(K).

We take and fix an element κ of K such that κ^ρ = −κ and iκ_vϕ_v has signature (r _v + t _v, r _v) for every v ∈ a. Then G(iκ_vϕ_v) modulo a maximal compact subgroup is a hermitian symmetric space which we denote by ℨ_v ^ϕ. We take a suitable point i _v of ℨ_v ^ϕ which plays the role of “origin” of the space. If r _v = 0, we understand that ℨ_v ^ϕ consists of a single point i _v. We put ℨ^ϕ = ∏_v∈a ℨ_v ^ϕ. To simplify our notation, for x ∈ K _A ^× or x ∈ (C ^×)^a, a ∈ Z ^a, and c ∈ (C ^×)^a, we put For ξ ∈ G _v ^ϕ and w ∈ ℨ_v ^ϕ, we define ξw ∈ ℨ_v ^ϕ in a natural way and define also a scalar factor of automorphy j _ξ(w) so that det(ξ)^r_v j _ξ(w)⁻ⁿ is the jacobian of ξ. Given k, ν ∈ Z ^a, z ∈ ℨ^ϕ, and α ∈ G _A ^ϕ, we put Then, for a function f:ℨ^ϕ → C, we define f∥_k,να:ℨ^ϕ → C by Now, given a congruence subgroup Γ of G ^ϕ, we denote by 𝔐_k,ν ^ϕ(Γ) the vector space of all holomorphic functions f on ℨ^ϕ which satisfy f∥_k,νγ = f for every γ ∈ Γ and also the cusp condition if G ^ϕ is of the elliptic modular type. We then denote by 𝔖_k,ν ^ϕ(Γ) the set of all cusp forms belonging to 𝔐_k,ν ^ϕ(Γ). Further, we denote by 𝔐_k,ν ^ϕ resp. 𝔖_k,ν ^ϕ the union of 𝔐_k,ν ^ϕ(Γ) resp. 𝔖_k,ν ^ϕ(Γ) for all congruence subgroups Γ of G. If ϕ is anisotropic, we understand that 𝔖_0,ν ^ϕ = C.

Let D be an open subgroup of G _A ^ϕ such that D ∩ G _h ^ϕ is compact. We then denote by 𝔖_k,ν ^ϕ(D) the set of all functions f: G _A ^ϕ → C satisfying the following conditions: for every  p ∈ G _h ^ϕ there exists an element f _p ∈ 𝔖_k,ν ^ϕ such that We now take D in a special form. We take a maximal r-lattice M in V whose norm is g in the sense of ref. 3 (p. 375) and put where d is the different of K relative to F and c is a fixed integral g-ideal. Clearly M̃ is an r-lattice in V containing M, and we easily see that D ^ϕ is an open subgroup of G _A ^ϕ. We assume that Define a subgroup 𝔛 of G _A ^ϕ by We then consider the algebra ℜ(D, 𝔛) consisting of all the finite linear combinations of DτD with τ ∈ 𝔛 and define its action on 𝔖_k,ν ^ϕ (D) as follows. Given τ ∈ 𝔛 and f ∈ 𝔖_k,ν ^ϕ(D), take a finite subset Y of G _h ^ϕ so that DτD = ⊔_η∈Y Dη and define f|DτD:G _A ^ϕ → C by These operators form a commutative ring of normal operators on 𝔖_k,ν ^ϕ(D).

For x ∈ G _A ^ϕ, we define an ideal ν₀(x) of r by where ν₀(x _v) is defined as in Section 3 with respect to an r_v-basis of M _v. Clearly ν₀(x) depends only on CxC.

Let f be an element of 𝔖_k,ν ^ϕ(D) that is a common eigenfunction of all the DτD with τ ∈ 𝔛, and let f|DτD = λ(τ)f with λ(τ) ∈ C. Given a Hecke ideal character χ of K such that |χ| = 1, define a Dirichlet series 𝔗(s, f, χ) by where χ* is the ideal character associated with χ and N(a) is the norm of an ideal a. Denote by χ₁ the restriction of χ to F _A ^×, and by θ the Hecke character of F corresponding to the quadratic extension K/F. For any Hecke character ξ of F, put From Theorem 1 and Eq. 3.13, we see that with a polynomial W_q of degree n whose constant term is 1, where q runs over all the prime ideals of K prime to c. Let Z(s, f, χ) denote the function of Eq. 4.16. Put Theorem 2. Suppose that χ_a(b) = b ^μ|b|^iκ−μ with μ ∈ Z^a and κ ∈ R^a such that ∑_v∈a κ_v = 0. Put m = k + 2ν − μ and with γ_v defined by Then ℜ(s, f, χ) can be continued to the whole s-plane as a meromorphic function with finitely many poles, which are all simple. It is entire if χ₁ ≠ θ^ν for ν = 0, 1.

We can give an explicitly defined finite set of points in which the possible poles of ℜ belong. Notice that p _v and q _v are polynomials; in particular, p _v = 1 if 0 ≤ m _v ≤ k _v and q _v = 1 if |μ_v − 2ν_v| ≥ n − 1.

The results of the above type and also of the type of Theorem 3 below were obtained in refs. 2, 4, and 5 for the forms on the symplectic and metaplectic groups over a totally real number field. The Euler product of type Z, its analytic continuation, and its relationship with the Fourier coefficients of f have been obtained by Oh (6) for the group G ^ϕ as above when ϕ = η_r.

Section 5.

We now put (W, ψ) = (V, ϕ) ⊕ (H _m, η_m) as in Eq. 2.5 with (V, ϕ) of Section 4 and m ≥ 0. Writing simply I = I _m, we can consider the parabolic subgroup P _I ^ψ of G ^ψ. We put P ^ψ = P _I ^ψ for simplicity, λ₀(α) = det(λ_I ^ψ(p)) for p ∈ P ^ψ, and with M of Section 4 and the standard basis {ɛ_i, ɛ_m+n+i}_i=1 ^m of H _m. We can define the space ℨ^ψ and its origin i ^ψ in the same manner as for G ^ϕ. We then put Here e _v is the element of End(V _v) defined for x _v by wx _v − we _v ∈ (H _m)_v for w ∈ V _v. We define an R-valued function h on G _A ^ψ by Taking f ∈ 𝔖_k,ν ^ϕ(D ^ϕ) and χ as in Section 4, we define μ:G _A ^ψ → C as follows: μ(x) = 0 if x ∉ P _A ^ψ D ^ψ; if x = pw with p ∈ P _A ^ψ and w ∈ D ^ψ ∩ C ₀ ^ψ, then we put where χ_c = ∏_v|c χv. Then we define E(x, s) for x ∈ G _A ^ψ and s ∈ C by This is meaningful if χ_a(b) = b ^k+2ν|b|^{iκ−k−2ν} with κ ∈ R ^a, ∑_v∈a κ_v = 0, and the conductor of χ divides c. We take such a χ in the following theorem. The series of Eq. 5.6 is the adelized version of a collection of several series of the type in Eq. 1.3.

Theorem 3. Define γ_v as in Theorem 2 with m = 0. Put Then the product can be continued to the whole s-plane as a meromorphic function with finitely many poles, which are all simple.

We can give an explicitly defined finite set of points in which the possible poles of the above product belong.

Section 6.

Let G be an arbitrary reductive algebraic group over Q. Given an open subgroup U of G _A containing G _a and such that U ∩ G _h is compact, we put U ^a = aUa ⁻¹ and Γ^a = G ∩ U ^a for every a ∈ G _A. We assume that G _a acts on a symmetric space 𝔚, and we let G act on 𝔚 via its projection to G _a. We also assume that Γ^a/𝔚 has finite measure, written vol(Γ^a/𝔚), with respect to a fixed G _a-invariant measure on 𝔚. Taking a complete set of representatives 𝔅 for G/G _A/U, we put where T is the set of elements of G which act trivially on 𝔚, and we assume that [Γ^a ∩ T:1] is finite. Clearly σ(U) does not depend on the choice of 𝔅. We call σ(G, U) the mass of G with respect to U. If G _a is compact, we take 𝔚 to be a single point of measure 1 on which G _a acts trivially. Then we have We can show that σ(U′) = [U:U′]σ(U′) if U′ is a subgroup of U. If strong approximation holds for the semisimple factor of G, then it often happens that both [Γ^a ∩ T:1] and vol(Γ^a/𝔚) depend only on U, so that If G _a is compact and U is sufficiently small, then Γ^a = {1} for every a, in which case we have σ(U) = #(G/G _A/U). If U is the stabilizer of a lattice L in a vector space on which G acts, then #(G/G _A/U) is the number of classes in the genus of L. Therefore, σ(U) may be viewed as a refined version of the class number in this sense.

Coming back to the unitary group G ^ϕ of Section 4, we can prove the following theorem.

Theorem 4. Suppose that G _a ^ϕ is compact. Let M be a g-maximal lattice in V of norm g and let d be the different of K relative to F. Define an open subgroup D of G _A ^ϕ by Eq. 4.9 with an integral ideal c. If n is odd, assume that ϕ is represented by a matrix whose determinant times (−1)^(n−1)/2 belongs to N _K/F(K); if n is even, assume that c is divisible by the product e of all prime ideals for which t_v = 2. Then where d = [F:Q], D_F is the discriminant of F, and A = 1 or A = N(e)ⁿ N(d)^−n/2 according as n is odd or even.

If n is odd, we can also consider σ(D′) for with an arbitrary integral ideal c. Then σ(D′) = 2^−τσ(D), where τ is the number of primes in F ramified in K.

Section 7.

Let us now sketch the proof of the above theorems. The full details will be given in ref. 7. We first take 𝔅 ⊂ so that = ⊔_b∈𝔅 G ^ϕ bD ^ϕ. Given E(x, s) as in Eq. 5.6, for each q ∈ G _h ^ψ we can define a function E _q(z, s) of (z, s) ∈ ℨ^ψ × C by The principle is the same as in Eq. 4.6, and so it is sufficient to prove the assertion of Theorem 3 with E _q(z, s) in place of E(x, s). In particular, we can take q to be q = b × 1_2m with b ∈ 𝔅. Define (X, ω) as in Eq. 2.5. Then there is an isomorphism of (X, ω) to (H _m+n, η_m+n) which maps P _U ^ω of Proposition 1 to the standard parabolic subgroup P of G(η_m+n). Therefore, we can identify ℨ^ω with the space h^a with We can also define an Eisenstein series E′(x, s; χ) for x ∈ G _A ^ω and s ∈ C, which is defined by Eq. 5.6 with (G(η_m+n)_A, P, 1) in place of (G _A ^ψ, P ^ψ, f). Taking E′ and (q, a) ∈ G _h ^ω (with a ∈ 𝔅) in place of E(x, s) and q, we can define a function E′_q,a(z, s) of (z, s) ∈ h^a × C in the same manner as in Eq. 7.1. There is also an injection ι of ℨ^ψ × ℨ^ϕ into h^a compatible with the embedding G ^ψ × G ^ϕ → G(η_m+n). We put then for every function g on h^a, where δ(w, z) is a natural factor of automorphy associated with the embedding ι. Take a Hecke eigenform f as in Section 4 and define f _a by the principle of Eq. 4.6. Then, employing Proposition 1, we can prove where q = b × 1_2m, A is a certain gamma factor, and Φ_a = Γ^a/ℨ^ϕ. The computation is similar to, but more involved than, that of ref. 4 (Section 4). Since the analytic nature of E′_q,a can be seen from the results of ref. 8, we can derive Theorem 3 from Eq. 7.4.

Take m = 0. Then ψ = ϕ and E _q(z, s) = f _b(z). Then the analytic nature of 𝔗 (s, f, χ), and consequently that of Z(s, f, χ), can be derived from Eq. 7.4. However, here we have to assume that χ_a(b) = b ^k+2ν|b|^{iκ−k−2ν} with κ ∈ R ^a, ∑_v∈a κ_v = 0, and the conductor of χ divides c. The latter condition on c is a minor matter, but the condition on χ_a is essential. To obtain Z(s, f, χ) with an arbitrary χ, we have to replace E′_q,a by 𝔇E"_q,a, where E" is a series of type E′ with 2ν − μ in place of k and 𝔇 is a certain differential operator on h^a.

As for Theorem 4, we take again ψ = ϕ and observe that a constant function can be taken as f if G _a ^ϕ is compact. The space ℨ^ϕ consists of a single point. The integral on the right-hand side of Eq. 7.4 is merely the value (E′_q,a)°(z, w; s). We can compute its residue at s = n explicitly. Comparing it with the residue on the left-hand side, we obtain Theorem 4 when c satisfies Eq. 4.10. If n is odd, we can remove this condition by computing a group index of type [U:U′].