MAASS THETA-SERIES AND HECKE OPERATORS

Anatoli Andrianov

To the Memory of Hans Maass (1911–1992)

Abstract. The survey part contains definitions and basic properties of harmonic theta-series, Siegel modular forms, and Hecke operators, reminder of the transformation formalism of general harmonic theta-series under modular transformations and Hecke operators with specialization to the case of Maass theta-seies (*). The main new results: infinite sequences of eigenfunctions for all regular Hecke operators on spaces of Maass theta-series are constructed; in the case of Maass theta-series of genus 2 all the eigenfunctions are determined and the relevant zeta functions are expressed as products of two L-functions of corresponding imaginary quadratic rings.

Introduction

The evident importance of the celebrated Shimura–Taniyama conjecture on explicit relation of Hasse zeta functions of elliptic curves and Hecke zeta functions of elliptic modular forms makes it quite natural to look for generalizations.

In 1980, H.Yoshida ([11], p. 243) had stated a two-dimentional analog of the Shimura–Taniyama conjecture. It asserts that the zeta-function of a two-dimentional Abelian variety over the field of rational numbers is (up to a finite number of Euler factors) the (Andrianov) zeta function of a cusp form of weight 2 for a congruence subgroup of the Siegel modular group of genus 2. Yoshida had also given an example supporting the conjecture. In 1993, R. Salvati-Manni and J. Top [10] had considered others interesting examples confirming the Yosida conjecture.

In view of possible relations of zeta functions of algebraic varieties and zeta functions of modular forms, it would be interesting to comput more zeta functions of concrete modular forms. The zeta functions of modular forms are defined as certain Euler products associated to common eigenfunctions of sufficiently complete families of Hecke operators. The most common sources for the search of eigenfunctions give spaces of theta-series of integral positive definite quadratic forms weighted by harmonic polynomials. General formulae for modular transformation of harmonic theta series and for their transformation under Hecke operators were obtained in our works [5] and [6], respectively, and in §1–§3 we propose a survey of the transformation formulae.

2000 Mathematics Subject Classification. Primary 11F27, 11F46, 11F60, 11F66.

Key words and phrases. Harmonic theta-series, Hecke operators, Maass theta-series, Siegel modular forms, zeta functions of Siegel modular forms.

The author was supported in part by Russian Fund for Fundamental Research, grant 02-01-00087

In 1977, H. Maass [9] had introduced important class of harmonic theta-series of the form

(*)  Θ_{ν}(Z, Q, χ) = ∑_{N∈ℤ_{m}^{m}} χ(det N)(det N)^{ν} exp(πitrace(^{t}NQNZ)),

where Z belongs to the Siegel upper half-plane of genus m, Q is the matrix of an integral positive definite quadratic form in m variables, χ a Dirichlet character, and ν = 0, 1 (if ν = 0 it is understood that χ(det N)(det N)^{ν} = χ(det N) for all N). In the same paper Maass had established automorphic properties of the series and indicated a number of cases, where the series are not identically zero. He had also suggested that ”die Thetareihen (*) in der von Andrianov[1] begründeten Theorie der Dirichletreihen zu Modulformen zweiten Grades und ihrer Verallgemeinerung eine wichtige Rolle spilen werden”. At that time I was rather young and did not pay much attention on the remark of the old man, besides, the action of Hecke operators on theta-series have not been yet investigated. Now, when I am approaching the age of Maass, I can see that he was right and can provide some positive evidences. In §3 (Proposition 7) we prove that if the proper class number of Q is 1, then, for every χ, the Maass series is an eigenfunction for all regular Hecke operators. Theorem 12 of §4 establishes that for m = 2 linear combinations of the Maass series with characters of the group of proper classes of binary quadratic forms (with respect to the Gauss composition) are eigenfunctions for all regular Hecke operators with explicitely given eigenvalues. The proof is based on combination of explicit formulae for the action of Hecke opertors on harmonical theta-series and the technique relating Hecke operators on modular forms of genus 2 with Gauss composition of binary quadratic forms introduced in [1]. This result allows us to show that the (Andrianov) zeta function of each such eigenfunction is the product of two L–functions of the corresponding imaginary quadratic ring (§5, Theorem 15).

Acknowledgements. The present paper was completed and reported during the author’s stay at Max-Planck-Institut für Mathematik (Bonn) in Spring of 2002. I want to thank the Council of the institute for kind invitation and the administration for excellent research conditions. I am grateful to all my colleague for interest to my work and useful remarks.

Notation. We reserve the letters ℕ, ℤ, ℚ, ℝ, and ℂ for the set of positive rational integers, the ring of rational integers, the field of rational numbers, the field of real numbers, and the field of complex numbers, respectively. 𝔸_{n}^{m} is the set of all m × n-matrices with entries in a set 𝔸, 𝔸^{n} = 𝔸_{1}^{n}, and 𝔸_{n} = 𝔸_{n}^{1}.

If M is a matrix, ^{t}M always denotes the transpose of M, if entries of M belong ℂ, \overline{M} is the matrix with complex conjugate entries. For a symmetric matrix Q we write Q[M] = ^{t}MQM if the product on the right is defined.

We denote by

𝔼^{m} = {Q = (q_{αβ}) ∈ ℤ_{m}^{m} | ^{t}Q = Q, q_{11}, q_{22}, …, q_{mm} ∈ 2ℤ}

the set of all even matrices of order m, i.e. the set of matrices of integral quadratic forms in m variables

q(X) = \frac{1}{2} ^{t}XQX (^{t}X = (x_{1}, …, x_{m})).

We recall that the level q of an invertible matrix Q ∈ 𝔼^{m} (and of the corresponding form) is the least positive integer satisfying qQ^{−1} ∈ 𝔼^{m}, and the divisor d of a nonzero Q ∈ 𝔼^{m} (and of the corresponding form) is the greatest positive integer such that d^{−1}Q ∈ 𝔼^{m}; the matrices and forms of divisor 1 are called primitive.

§1. Harmonic theta-series and Maass theta-series

Here we shall remind basic definitions and properties of harmonic theta-series of integral positive definite quadratic forms. For details and proofs see [5], [7], or [8].

Let Q ∈ 𝔼^{m} be the matrix of an integral positive definite quadratic form q in m variables. For n ∈ ℕ, let P be a polynomial on ℂ_{n}^{m}. Suppose that the polynomial is homogeneous of exponent ν and harmonic with respect to the quadratic form q. The conditions on P mean that it satisfies

P (V A) = (det A)^{ν}P (V) (∀ V ∈ ℂ_{n}^{m} and A ∈ GL_{n}(ℂ))

and is of the form P (V) = P_{0}(SV), where S is a real n × n-matrix such that Q = ^{t}SS, and polynomial P_{0} satisfies the Laplace equation

ΔP_{0} = ∑_{α,β} \frac{∂^{2}P_{0}(V)}{(∂v_{αβ})^{2}} = 0.

The theta-series of genus n of the matrix Q with the coefficient polynomial P and characteristic

L ∈ L_{n}(Q) = {L ∈ ℤ_{n}^{m}/qℤ_{n}^{m} | QL ≡ 0 (mod q)} ,

where q is the level of Q, is defined by

(1.1)  Θ_{P} (Z, Q|L) = ∑_{N∈ℤ_{n}^{m}} P (N + q^{−1}L)e{Q[N + q^{−1}L]Z},

where Z belongs to the upper half-plane of genus n,

(1.2)  Z ∈ ℍ^{n} = \{Z = X + \sqrt{−1}Y ∈ ℂ_{n}^{n} | ^{t}Z = Z, Y > 0\} ,

and, for a square matrix A, we write

(1.3)  e\{A\} = e^{π\sqrt{−1}traceA}.

The theta-series converge absolutely and uniformly on compact subsets of ℍ^{n}. If U is an integral unimodular matrix of order m, then, since Uℤ_{n}^{m} = ℤ_{n}^{m}, by replacing N by UN in (1.1), we get the identities

(1.4)  \begin{aligned}Θ_{P} (Z, Q|L) & = ∑_{N∈ℤ_{n}^{m}} P (U(N + q^{−1}U^{−1}L))e\{Q[U(N + q^{−1}U^{−1}L)]Z\} \\ & = Θ_{P |U}(Z, Q[U]|U^{−1}L)  ∀U ∈ GL_{m}(ℤ), \end{aligned}

where (P |U)(V) = P (UV).

Examples. 1. The constant polynomial P (V) = 1 is clearly a harmonic polynomial of exponent ν = 0 with respect to every quadratic form q, and the theta-series (1.1) turns into the ordinary theta-series

(1.5)  Θ_{0}(Z, Q|L) = ∑_{N∈ℤ_{n}^{m}} e{Q[N + q^{−1}L]Z}  (Z ∈ ℍ^{n}, L ∈ L_{n}(Q))

of genus n of the form q (or matrix Q) with characteristic L.

2. If n = m, the polynomial

P (V) = det V (V = (v_{αβ}) ∈ ℂ_{m}^{m})

is clearly homogeneous of exponent 1 and is linear in each of m^{2} variables v_{αβ}. It follows that the polynomial P_{0}(T) = P (S^{−1}T) is also linear in each of the variables for every nonsingular m × m–matrix S and so it satisfies the Laplace equation. Thus, the polynomial is homogeneous of exponent 1 and harmonic with respect to every positive definite quadratic form q(X) in m variables. The corresponding theta-series of genus m is

(1.6) Θ_{1}(Z, Q|L)

= ∑_{N∈ℤ_{m}^{m}} det(N + q^{−1}L)e{Q[N + q^{−1}L]Z} (Z ∈ ℍ^{m}, L ∈ L_{m}(Q)).

3. Maass theta-series (*) are linear combinations of series (1.5) with n = m or series (1.6), depending on whether ν = 0 or 1: if χ is a character modulo g, then we have

(1.7) Θ_{ν}(Z, Q, χ)

= ∑_{R∈ℤ_{m}^{m}/gℤ_{m}^{m}} χ(det R)g^{νm} ∑_{N∈ℤ_{m}^{m}} det^{ν} (N + \frac{qg}{qg^{2}} R) e\{g^{2}Q [N + \frac{qg}{qg^{2}} R] Z\}

= g^{νm} ∑_{R∈ℤ_{m}^{m}/gℤ_{m}^{m}} χ(det R)Θ_{ν}(Z, g^{2}Q|qgR),

where q is the level of Q (note that the level of g^{2}Q is qg^{2}, and, for every R ∈ ℤ_{m}^{m}, g^{2}Q(qgR) ≡ 0 (mod qg^{2})).

The following theorem is a particular case of well-known transformation formulas for harmonic theta-functions (see, for example, [5], Theorems 4.1-4.3). In order to formulate the theorem it is convenient to consider the theta-series Θ_{P} (Z, Q|L) with various characteristics L ∈ L_{n}(Q) as entries of the theta-column

Θ Θ Θ_{P} (Z, Q) = ( ⋮ ; Θ_{P} (Z, Q|L_{α}) ; ⋮ )

of order

l_{n}(Q) = #(L_{n}(Q)) =^{L→qQ^{−1}L′} #(ℤ_{n}^{m}/Qℤ_{n}^{m}) = (det Q)^{n},

where we assume that all different residue classes of L_{n}(Q) are ordered in a fixed way.

Theorem 1. Let Q be even positive definite matrix of order m, and P a homogeneous polynomial on ℂ_{n}^{m} of an integral nonnegative exponent ν and harmonic with respect to the quadratic form with matrix Q. Then the following assertions hold.

(1). For every integral symplectic matrix

M = ( \begin{matrix}A & B \\ C & D \end{matrix} ) ∈ Γ^{n} = Sp_{n}(ℤ) = \{M ∈ ℤ_{2n}^{2n} | ^{t}MJ_{n}M = J_{n}\} ,

where

(1.8)  J_{n} = ( \begin{matrix}0_{n} & 1_{n} \\ −1_{n} & 0_{n} \end{matrix} ) ,

there is a square matrix Λ_{Q}(M) of order l, independent of Z, such that the thetacolumn satisfies the functional equation

Θ Θ Θ_{P} ((AZ + B)(CZ + D)^{−1}, Q) = det(CZ + D)^{ν+m/2}Λ_{Q}(M)Θ Θ Θ_{P} (Z, Q).

If the order m is even, then the matrices Λ_{Q}(M) satisfy the relations

(1.9)  Λ_{Q}(MM′) = Λ_{Q}(M)Λ_{Q}(M′)  (M, M′ ∈ Γ^{n}).

(2). Suppose that

(1.10)  M = ( \begin{matrix}A & B \\ C & D \end{matrix} ) ∈ Γ_{0}^{n}(q) = \{( \begin{matrix}A & B \\ C & D \end{matrix} ) ∈ Γ^{n} | C ≡ 0 (mod q)\} ,

where q is the level of Q. Then the entries Λ_{Q}(M)_{αβ} of the matrix Λ_{Q}(M) can be defined by the formulae

Λ_{Q}(M)_{αβ} = \{ \begin{matrix}μ_{Q}(M)e\{q^{−2}Q[L_{β}]^{t}DB\} & if L_{β} ≡ L_{α}A (mod q) \\ 0 & otherwise , \end{matrix}

where μ_{Q}(M) is an eight root of the unit.

(3). If the level q of Q is equal to 1, then the order m of Q satisfies m ≡ 0 (mod 8), and

μ_{Q}(M) = 1 (∀ M ∈ Γ^{n}),

but if q > 1 and m is even, then for every M = ( \begin{matrix}A & B \\ C & D \end{matrix} ) ∈ Γ_{0}^{n}(q) the scalar

μ_{Q}(M) can be computed by formula

μ_{Q}(M) = χ_{Q}(det D),

where χ_{Q} is the real Dirichlet character modulo q, satisfying

χ_{Q}(−1) = (−1)^{m/2},
χ_{Q}(p) = (\frac{(−1)^{m/2} det Q}{p}) (the Legendre symbol)

if p is an odd prime number which does not divide q, and

χ_{Q}(2) = 2^{−m/2} ∑_{r∈ℤ^{m}/2ℤ^{m}} e^{\frac{πi}{2} Q[r]}

if q is odd.

In principle, when m is even, the relations (1.9) enable one to evaluate multiplier matrices Λ_{Q}(M) for all M ∈ Γ^{n} if they are known for a system of generators of Γ^{n}. It is well known that the matrices

U(D) = ( \begin{matrix}^{t}D^{−1} & 0 \\ 0 & D \end{matrix} )  with D ∈ GL_{n}(ℤ),
T (B) = ( \begin{matrix}1_{n} & B \\ 0 & 1_{n} \end{matrix} )  with B = ^{t}B ∈ ℤ_{n}^{n}

together with the matrix J_{n} generate the group Γ^{n} (see, for example, [2], Theorem 1.3.6). According to Theorem 1(2) or by direct computation, we obtain

Θ_{P} (^{t}D^{−1}ZD^{−1}, Q|L) = det(D)^{ν+m/2}χ_{Q}(det D)Θ_{P} (Z, Q|L^{t}D^{−1})

and

Θ_{P} (Z + B, Q|L) = e{q^{−2}Q[L]B}Θ_{P} (Z, Q|L),

where L ∈ L_{n}(Q), D ∈ GL_{n}(ℤ), and B = ^{t}B ∈ ℤ_{n}^{n}. As to the generator J_{n}, the inversion formulae for harmonic theta-functions (see [5], Lemma 5.1) give the formulae

Θ_{P} (−Z^{−1}, Q|L) = det(−Z)^{ν+m/2} \frac{i^{mn/2}}{\sqrt{(det Q)^{n}}} ∑_{L′∈L_{n}(Q)} e\{2q^{−2} ^{t}LQL′\}Θ_{P} (Z, Q|L′)

for L ∈ L_{n}(Q).

§2. Modular forms

Here we shall remind basic definitions and properties of Siegel modular forms of integral weights. The general real positive symplectic group of genus n consisting of all real symplectic matrices of order 2n with positive multipliers,

𝔾^{n} = GSp_{n}^{+}(ℝ) = {M ∈ ℝ_{2n}^{2n} | ^{t}MJ_{n}M = μ(M)J_{n}, μ(M) > 0} ,

is a real Lie group acting as a group of analytic automorphisms on the n(n + 1)/2dimensional open complex variety (1.2) by the rule

𝔾^{n} ∋ M = ( \begin{matrix}A & B \\ C & D \end{matrix} ) : Z ↦ M⟨Z⟩ = (AZ + B)(CZ + D)^{−1} (Z ∈ ℍ^{n}).

Acting on the upper half-plane ℍ^{n}, the general symplectic group operates also on complex-valued functions F on ℍ^{n} by Petersson operators of integral weights k,

(2.1)  𝔾^{n} ∋ M = ( \begin{matrix}A & B \\ C & D \end{matrix} ) : F ↦ F |_{k}M = det(CZ + D)^{−k}F (M ⟨Z⟩).

The Petersson operators satisfy the rules

(2.2)  F |_{k}MM′ = (F |_{k}M)|_{k}M′  (M, M′ ∈ 𝔾^{n}).

Let S be a subgroup of 𝔾^{n} commensurable with the modular group Γ^{n}, χ a character of S, that is a multiplicative homomorphism of S into nonzero complex numbers with the kernel of finite index in S, and k an integral number. A complexvalued function F on ℍ is called a (Siegel) modular form of weight k and character χ for the group S, if the following conditions are fulfilled:

(i) F is a holomorphic function in n(n + 1)/2 complex variables on ℍ^{n}; (ii) For every matrix M ∈ S, the function F satisfies the functional equation

(2.3)  F |_{k}M = χ(M)F,

where |_{k} is the Petersson operator of weight k;

(iii) If n = 1, then every function F |_{k}M with M ∈ Γ^{1} is bounded on each subset of ℍ^{1} of the form ℍ_{ε}^{1} = {x + iy ∈ ℍ^{1}|y ≥ ε} with ε > 0.

The set 𝔐_{k}(S, χ) of all modular forms of weight k and character χ for the group S is clearly a linear space over the field ℂ. Each space 𝔐_{k}(S, χ) has finite dimension over ℂ.

For a positive integer q, we shall denote by

(2.4)  Γ^{n}(q) = {M ∈ Γ^{n} | M ≡ 1_{2n} (mod q)}

the principal congruence subgroup of level q of the modular group. Γ^{n}(q) is clearly a normal subgroup of finite index of the modular group.

A subgroup S of the (real) symplectic group

𝕊^{n} = {M ∈ 𝔾^{n} | μ(M) = 1}

is called a congruence subgroup if it contains a principal congruence subgroup as a subgroup of finite index. A character of such S is said to be a congruence character if it is trivial on a principal congruence subgroup contained in S. If S is a congruence subgroup and χ is a congruence character of S, then for each modular form F of 𝔐_{k}(S, χ) and each matrix M ∈ 𝔾^{n} ∩ ℚ_{2n}^{2n} the function F |_{k}M has a Fourier expansion of the form

(2.5)  (F |_{k}M)(Z) = ∑_{A∈𝔼^{n}, A≥0} f_{M}(A)e{AZ/r}

with a positive integer r depending on S, χ, and M, which converges absolutely on ℍ^{n} and uniformly on every subset of the form

(2.6)  ℍ_{ε}^{n} = {Z = X + iY ∈ ℍ^{n} | Y ≥ ε1_{2n}} with ε > 0.

The modular form F is called a cusp form, if coefficients f_{M}(A) of the expansion (2.5) satisfy the conditions

f_{M}(A) = 0 for all M ∈ Γ^{n} and A ∈ 𝔼^{n} with det A = 0.

The subspace of cusp forms of 𝔐_{k}(S, χ) will be denoted by 𝔑_{k}(S, χ).

Theorem 2. Let Q be an even positive definite matrix of an even order m, and P a homogeneous polynomial on ℂ_{n}^{m} of an integral nonnegative exponent ν harmonic with respect to the quadratic form with matrix Q. Then each of the theta-series of genus n of the matrix Q with the coefficient form P and characteristic L ∈ L_{n}(Q) is a modular form of weight k = ν +m/2 and the unit character for the group Γ^{n}(q), where q is the level of Q:

(2.7)  Θ_{P} (Z, Q|L) ∈ 𝔐_{ν+m/2} (Γ^{n}(q), 1) = 𝔐_{ν+m/2}^{n}(q).

If ν > 0, then the theta-series is a cusp form:

(2.8)  Θ_{P} (Z, Q|L) ∈ 𝔑_{ν+m/2} (Γ^{n}(q), 1) = 𝔑_{ν+m/2}^{n}(q) (ν > 0).

Proof. Trivial estimates imply that each of the theta-series (1.1) converges absolutely on ℍ^{n} and uniformly on its subsets of the form (2.6). It follows that each of the theta-series satisfies the condition (i) and, by Theorem 1(1), the condition (iii) of the definition of modular forms. By Theorem 1, the multiplier matrix Λ_{Q}(M) is the unit matrix of order l_{n}(Q) if M ∈ Γ^{n}(q). Thus, each of the function F (Z) = Θ_{P} (Z, Q|L) satisfies

(F |_{k}M)(Z)

= det(CZ + D)^{−k}F (M⟨Z⟩) = F (Z) for every M = ( \begin{matrix}A & B \\ C & D \end{matrix} ) ∈ Γ^{n}(q) and so satisfies the condition (ii) for S = Γ^{n}(q) with χ = 1.

In order to prove (2.8) we note first that the condition ν > 0 implies that P (T) = 0 for every matrix T ∈ ℂ_{n}^{m} with rank(T) < n. Really, in this case there is a nonzero complex n-column b such that T b = 0. Let B be a nonsingular square matrix with the first column b. Then the matrix T B has the zero first column,

whence T BA = T B for every matrix A of the form A = ( \begin{matrix}a & 0 \\ 0 & 1_{n−1} \end{matrix} ) with arbitrary

a ∈ ℂ. It follows that

P (T BA) = (det A)^{ν}P (T B) = a^{ν}P (T B) = P (T B).

Thus, P (T B) = 0 and so P (T) = (det B)^{−ν}P (T B) = 0. Now, the Fourier expansion of the series (1.1) can be obviously written in the form

(2.9) Θ_{P} (Z, Q|L) = ∑_{A∈𝔼^{n}, A≥0} (  ∑_{N∈ℤ_{n}^{m}, qQ[N+q^{−1}L]=A} P (N + q^{−1}L)) e{AZ/q}.

Since Q > 0, the condition det A = det(qQ[N + q^{−1}L]) = 0 implies the inequality rank(N + q^{−1}L) < n, whence, by the above, P (N + q^{−1}L) = 0, and so the corresponding Fourier coefficient of the series (2.9) is equal to zero. Then, by Theorem 1(1), we conclude that the function Θ_{P} (Z, Q|L)|_{k}M for every M ∈ Γ^{n} is a linear combination with constant coefficients of the series Θ_{P} (Z, Q|L_{α}) and so, together with these series, has zero Fourier coefficients corresponding to singular arguments. △

In the case of Maass theta-series (*) one can say more:

Proposition 3. Let Q be an even positive definite matrix of even order m, g a positive integer, and χ a Dirichlet character modulo g. Then the Maass theta-series (*) are modular forms of weight ν + m/2 for the group Γ_{0}^{m}(qg^{2}) with the character χχ_{Q} given by

(2.10)  Γ_{0}^{m}(qg^{2}) ∋ M = ( \begin{matrix}A & B \\ C & D \end{matrix} ) ↦ (χχ_{Q})(M) = χ(det D)χ_{Q}(det D) :

Θ_{ν}(Z, Q, χ) ∈ 𝔐_{ν+m/2}(Γ_{0}^{m}(qg^{2}), χχ_{Q}) (ν = 0, 1),

where q is the level of Q and χ_{Q} the character of Q defined in Theorem 1. If ν = 1 then the series is a cusp form:

Θ_{1}(Z, Q, χ) ∈ 𝔑_{1+m/2}(Γ_{0}^{m}(qg^{2}), χχ_{Q}).

Proof. Since the level of g^{2}Q is qg^{2}, by applying Theorem 1 for a matrix M =

( \begin{matrix}A & B \\ C & D \end{matrix} ) of Γ_{0}^{m}(qg^{2}) to a theta-series in the right hand side of the relation (1.7),

we obtain

Θ_{ν}(Z, g^{2}Q|qgR)|_{ν+m/2}M

= χ_{g^{2}Q}(det D)e{(qg^{2})^{−2}(g^{2}Q)[qgRA]^{t}DB}Θ_{ν}(Z, g^{2}Q|qgRA)

= χ_{Q}(det D)Θ_{ν}(Z, g^{2}Q|qgRA).

Hence, since A^{t}D ≡ 1_{m} (mod qg^{2}), we get

Θ_{ν}(Z, Q, χ)|_{ν+m/2}M = g^{νm} ∑_{R∈ℤ_{m}^{m}/gℤ_{m}^{m}} χ(det R)Θ_{ν}(Z, g^{2}Q|qgR)|_{ν+m/2}M

= χ_{Q}(det D)g^{νm} ∑_{R∈ℤ_{m}^{m}/gℤ_{m}^{m}} χ(det RA^{t}D)Θ_{ν}(Z, g^{2}Q|qgRA)

= χ_{Q}(det D)χ(det ^{t}D)g^{νm} ∑_{R∈ℤ_{m}^{m}/gℤ_{m}^{m}} χ(det R)Θ_{ν}(Z, g^{2}Q|qgR)

= (χχ_{Q})(M)Θ_{ν}(Z, Q, χ).

The rest follows from Theorem 2.△

§3. Hecke operators

Here we shall recall basic definitions concerning Hecke operators on the spaces 𝔐_{k}^{n}(q) and specialize the transformation formulas of general harmonic theta-series under regular Hecke operators obtained in [6] to the cases of series Θ_{P} (Z, Q|L) and Maass theta-series. For details and proofs see the book [2], Chapters 3 and 4 and [6], §5.

Let Δ be a multiplicative semigroup and S a subgroup of Δ such that every double coset SMS of Δ modulo S is a finite union of left cosets SM′. Let us consider the vector space over a field, say, the field ℂ of complex numbers consisting of all formal finite linear combinations with coefficients in ℂ of symbols (SM) with M ∈ Δ being in one-to-one correspondence with left cosets SM of the set Δ modulo S. The group S naturally acts on this space by right multiplication defined on the symbols (SM) by

(SM)γ = (SMγ), (M ∈ Δ, γ ∈ S).

We denote by

H(S, Δ) = HS_{ℂ}(S, Δ)

the subspace of all S–invariant elements. The multiplication of elements of H(S, Δ) given by the formula

(∑_{α} a_{α}(SM_{α}))(∑_{β} b_{β}(SN_{β})) = ∑_{α,β} a_{α}b_{β}(SM_{α}N_{β})

does not depend on the choice of representatives M_{α} ∈ SM_{α} and N_{β} ∈ SN_{β}, and turns H(S, Δ) into an associative algebra over ℂ with the unity element (S1_{S}), called the Hecke–Shimura ring or HS–ring of Δ relative to S (over ℂ). Elements

(3.1)  (M) = (M)_{S} = ∑_{M_{i}∈S\SMS}(SM_{i}),  (M ∈ Δ)

being in one-to-one correspondence with double cosets of Δ modulo S belong to H(S, Δ) and form a basis of the ring over ℂ. For brevity, the symbols (SM) and (M) will be refered as left and double classes (of Δ modulo S), respectively.

We consider the semigroup

Σ^{n} = 𝔾^{n} ⋂ ℤ_{2n}^{2n} = {M ∈ ℤ_{2n}^{2n} | ^{t}MJ_{n}M = μ(M)J_{n}, μ(M) > 0}

and its subsemigroups

(3.2)  \{ \begin{matrix}Σ_{q}^{n} & = \{M ∈ Σ^{n} | gcd(μ(M), q) = 1\} \\ Σ_{0}^{n}(q) & = \{M = ( \begin{matrix}A & B \\ C & D \end{matrix} ) ∈ Σ_{q}^{n} | C ≡ 0 (mod q)\} , \\ Σ^{n}(q) & = \{M ∈ Σ_{0}^{n}(q) | M ≡ ( \begin{matrix}μ(M)1_{n} & 0 \\ 0 & 1_{n} \end{matrix} ) (mod q)\} \end{matrix}

with q ∈ ℕ. It is easy to see that the semigroups satisfy the relation

(3.3)  Σ_{0}^{n}(q) = Γ_{0}^{n}(q)Σ^{n}(q) = Σ^{n}(q)Γ_{0}^{n}(q).

If S is a subgroup of finite index of the modular group Γ^{n}, then the pair S, Σ^{n} satisfies the conditions of the definition of HS–rings, as well as each pair S, Δ with S ⊂ Δ ⊂ Σ^{n}, and we can define corresponding Hecke–Shimura rings H(S, Δ). We shall say that a group S satisfying Γ^{n}(q) ⊂ S ⊂ Γ^{n} is q-symmetric if SΣ^{n}(q) = Σ^{n}(q)S. For such S the HS–ring

H_{qr}(S) = H(S, R(S)) with R(S) = SΣ^{n}(q) = Σ^{n}(q)S

is called the q-regular HS–ring of S. According to Theorem 3.3.3 of [2], all q-regular HS–rings for given genus n are isomorphic to each other. In particular, by (3.3), the group Γ_{0}^{n}(q) is q-symmetric, and the q-regular HS–rings of Γ_{0}^{n}(q) and Γ^{n}(q),

(3.4)  \begin{aligned}H_{0}^{n}(q) & = H_{qr}(Γ_{0}^{n}(q)) = H(Γ_{0}^{n}(q), Σ_{0}^{n}(q)), \\ H^{n}(q) & = H_{qr}(Γ^{n}(q)) = H(Γ^{n}(q), Σ^{n}(q)) \end{aligned}

are isomorphic. The isomorphism can be defined as follows. Let

(3.5)  T′ = ∑_{α} a_{α}(Γ_{0}^{n}(q)M_{α}) ∈ H_{0}^{n}(q),

where the left classes (Γ_{0}^{n}(q)M_{α}) with a_{α} ≠ 0 are pairwise distinct. By (3.3), without loss of generality one may assume that all the representatives M_{α} of the left cosets Γ_{0}^{n}(q)M_{α} belong to Σ^{n}(q). Then, obviously, we have

(3.6)  η : T′ ↦ T = ∑_{α} a_{α}(Γ^{n}(q)M_{α}) ∈ H^{n}(q),

and the map η is a homomorphic embedding of the ring H_{0}^{n}(q) into H^{n}(q). In fact, η is a ring isomorphism, and the inverse isomorphism ξ : H^{n}(q) ↦ H_{0}^{n}(q) is determined by the condition

(3.7)  ξ : ∑_{β} b_{β}(Γ^{n}(q)N_{β}) ↦ ∑_{β} b_{β}(Γ_{0}^{n}(q)N_{β}).

The isomorphism of the rings allows one to transfer various constructions from one of the rings to another. For example, Zharkovskaya homomorphisms Ψ^{n,r} = Ψ_{k,χ}^{n,r} of the ring H_{0}^{n}(q) to H_{0}^{r}(q), where n > r ≥ 1, k is an integer, and χ a Dirichlet character modulo q satisfying χ(−1) = (−1)^{k}, can be carried out to the rings H^{n}(q), so that we get the homomorphisms

(3.8)  Ψ^{n,r} = Ψ_{k,χ}^{n,r} : H^{n}(q) ↦ H^{r}(q)  (n > r ≥ 1)

satisfying the condition of commutativity of the diagramm

(3.9)  \begin{matrix}H_{0}^{n}(q) & ⟶^{η} & H^{n}(q) \\ _{Ψ^{n,r}} ↓ & _{Ψ^{n,r}} ↓ \\ H_{0}^{r}(q) & ⟶^{η} & H^{r}(q) \end{matrix}.

We remind that the Zharkovskaya map from genus n to genus r,

(3.10)  Ψ^{n,r} = Ψ_{k,χ}^{n,r} : H_{0}^{n}(q) ↦ H_{0}^{r}(q),

can be defined in the following way. Let T′ ∈ H_{0}^{n}(q) be an element of the form (3.5). One can assume that each representative M_{α} ∈ Γ_{0}^{n}(q)\Σ_{0}^{n}(q) is choosen in the form

M_{α} = ( \begin{matrix}A_{α} & B_{α} \\ 0 & D_{α} \end{matrix} ) with D_{α} = ( \begin{matrix}D′_{α} & ∗ \\ 0 & D′′_{α} \end{matrix} ) and D′_{α} ∈ ℤ_{r}^{r}.

If A_{α} = ( \begin{matrix}A′_{α} & ∗ \\ ∗ & ∗ \end{matrix} ) and B_{α} = ( \begin{matrix}B′_{α} & ∗ \\ ∗ & ∗ \end{matrix} ) with r × r-blocks A′_{α} and B′_{α}, then

M′_{α} = ( \begin{matrix}A′_{α} & B′_{α} \\ 0 & D′_{α} \end{matrix} ) ∈ Σ_{0}^{r}(q),

and we put

Ψ_{k,χ}^{n,r}(T′) = ∑_{α} a_{α}| det D′′_{α}|^{−k}χ^{−1}(| det D′′_{α}|)(Γ_{0}^{r}(q)M′_{α}).

Then, in view of commutativity of the diagramm (3.9), we can define the Zharkovskaya map (3.8) by

Ψ^{n,r}(T) = (ηΨ′ξ)(T) ∈ H^{r}(q) (T ∈ H^{n}(q))

with maps η and ξ defined by (3.6) and(3.7), where Ψ′ is the map (3.10).

Hecke–Shimura rings act on modular forms by means of linear representation given by Hecke operators. Let, say, F belongs to the space 𝔐_{k}(S) = 𝔐_{k}(S, 1) of modular forms of integral weight k and the trivial character χ = 1 for a subgroup S of finite index in the modular group Γ^{n}, and let

T = ∑_{i} a_{α}(SM_{α}) ∈ H(S, Σ^{n}).

Then, as it easily follows from definitions of modular forms and HS–rings, and the properties (2.2), (2.3) of the Petersson operators (2.1), the function

F ∥T = F ∥_{k}T = ∑_{α} a_{α}F |_{k}M_{α}

does not depend on the choice of representatives M_{α} ∈ SM_{α} and again belongs to the space 𝔐_{k}(S). These operators are called Hecke operators (of weight k for the group S). The Hecke operators corresponding to elements of regular HS–rings are called regular. It follows from the definition of multiplication in HS–rings and (2.2) that the map T ↦ ∥T is a linear representation of the ring H(S) on the space 𝔐_{k}(S). The subspace of cusp forms 𝔑_{k}(S) is invariant with respect to all of the Hecke operators.

The following theorem expresses the images of harmonic theta-series under regular Hecke operators in the form of linear combinations of similar theta-series with the coefficients explicitely given in terms of certain trigonometric sums.

Theorem 4. Let Q be an even positive definite matrix of even order m and level q, and let P be a homogeneous polynomial on ℂ_{n}^{m} of exponent ν and harmonic with respect to the quadratic form with matrix Q. Suppose that

T = ∑_{α} a_{α}(Γ^{n}(q)M_{α})

is an element of the Hecke–Shimura ring H^{n}(q) with n ≥ 1 such that all representatives M_{α} ∈ Σ^{n}(q) have the same multiplier μ(M_{α}) = μ > 0. Let us assume also that in the case n < m the element T belongs to the image of the ring H^{m}(q) under the Zharkovskaya map (3.8) from genus m to genus n with k = m/2 and the character χ = χ_{Q} defined in Theorem 1,

T = Ψ_{m/2,χ_{Q}}^{m,n}(T̃) with T̃ ∈ H^{m}(q) (n < m).

Then the image of a theta-series (1.1) under the Hecke operator ∥_{ν+m/2}T can be written in the form

(3.11) Θ_{P} (Z, Q|L)∥_{ν+m/2}T

= ∑_{D∈Δ(Q,μ)/Λ} I(D, Q, Ψ^{n,m}(T))Θ_{P |μ^{−1}D}(Z, μ^{−1}Q[D]|μD^{−1}L),

where

(3.12) Δ(Q, μ)

= {D ∈ ℤ_{m}^{m} | det D = μ^{m/2}, μ^{−1}Q[D] ∈ 𝔼^{m}}, Λ = Λ^{m} = SL_{m}(ℤ),

Ψ^{n,m}(T) = Ψ_{m/2,χ_{Q}}^{n,m}(T) = \{ \begin{matrix}Ψ_{m/2,χ_{Q}}^{n,m}(T) & if n > m \\ T & if n = m \\ T̃ ∈ (Ψ_{m/2,χ_{Q}}^{m,n})^{−1}(T) & if n < m, \end{matrix}

I(D, Q, T̃) with T̃ ∈ H^{m}(q) are the trigonometric sums defined by the following conditions: if

T′ = ξ(T̃) = ∑_{β} b_{β} (Γ_{0}^{m}(q) ( \begin{matrix}A_{β} & B_{β} \\ 0 & D_{β} \end{matrix} )) ∈ H_{0}^{m}(q)

with ^{t}A_{β}D_{β} = μ1_{m}, then

(3.13) I(D, Q, T̃) = I(D, Q, T′)

= ∑_{β; D^{t}D_{β}≡0 (mod μ)} b_{β}| det D_{β}|^{−m/2}χ_{Q}^{−1}(| det D_{β}|)e{μ^{−2}Q[D]^{t}D_{β}B_{β}},

e{⋯ } being the exponent (1.3), and where

(P |μ^{−1}D)(V) = P (μ^{−1}DV).

In particular, the image of the theta-series (1.5) with n = m and (1.6) under the Hecke operator ∥_{ν+m/2}T with T ∈ H^{m}(q) can be written in the form

(3.14) Θ_{ν}(Z, Q|L)∥_{ν+m/2}T

= μ^{−νm/2} ∑_{D∈Δ(Q,μ)/Λ} I(D, Q, T)Θ_{ν}(Z, μ^{−1}Q[D]|μD^{−1}L),

where ν = 0 in the first case, and ν = 1 in the second.

Proof. The formulae (3.11) were proved in [6], Theorem 5.3 with the sums taken over representatives sets

{D ∈ ℤ_{m}^{m} | | det D| = μ^{m/2}, μ^{−1}Q[D] ∈ 𝔼^{m}}/GL_{m}(ℤ).

Since, clearly, each such set can be replaced by the set {D ∈ Δ(Q, μ)/Λ}, the formulas follow.

As to the formulas (3.14), it is sufficient to note that det(μ^{−1}D) = μ^{−m/2} for all D ∈ Δ(Q, μ). △

In connection with the formulae (3.11) we note that

(3.15)  μD^{−1} ∈ Δ (μ^{−1}Q[D], μ) if D ∈ Δ(Q, μ) and gcd(μ, q) = 1.

Indeed, let us set D′ = μD^{−1} and Q′ = μ^{−1}Q[D]. Then the matrix D′ can be also written as {Q′}^{−1} ^{t}DQ. It follows that the products of D′ by det D = μ^{m/2} or det Q′ = det Q are both integral. Since the numbers are coprime, D′ is integral. The inclusion follows.

Proposition 5. Let Q be an even positive definite matrix of even order m, g a positive integer, and χ a Dirichlet character modulo g. Suppose that

T = ∑_{α} a_{α}(Γ^{n}(qg^{2})M_{α})

is an element of the regular Hecke–Shimura ring H^{m}(qg^{2}) such that all representatives M_{α} ∈ Σ^{n}(qg^{2}) (see (3.2)) have the same multiplier μ(M_{α}) = μ. Then the image of the Maass theta-series (*) under the Hecke operator ∥_{ν+m/2}T is again a linear combination of Maass theta-series of the form

(3.16) Θ_{ν}(Z, Q, χ)∥_{ν+m/2}T

= χ(μ)^{−m/2}μ^{−νm/2} ∑_{D∈Δ(Q,μ)/Λ} I(D, g^{2}Q, T)Θ_{ν}(Z, μ^{−1}Q[D], χ),

where I(⋯) are the trigonometric sums (3.13).

Proof. By (1.7) and (3.14), we have

Θ_{ν}(Z, Q, χ)∥_{ν+m/2}T = g^{νm} ∑_{R∈ℤ_{m}^{m}/gℤ_{m}^{m}} χ(det R)Θ_{ν}(Z, g^{2}Q|qgR)∥_{ν+m/2}T

= μ^{−νm/2}g^{νm} ∑_{R∈ℤ_{m}^{m}/gℤ_{m}^{m}} χ(det R)

× ∑_{D∈Δ(g^{2}Q,μ)/Λ} I(D, g^{2}Q, T)Θ_{ν}(Z, μ^{−1}g^{2}Q[D]|qgμD^{−1}R)

= μ^{−νm/2}g^{νm} ∑_{D∈Δ(g^{2}Q,μ)/Λ} I(D, g^{2}Q, T)

× ∑_{R∈ℤ_{m}^{m}/gℤ_{m}^{m}} χ(det μD^{−1}R)χ(det μD^{−1})^{−1}Θ_{ν}(Z, μ^{−1}g^{2}Q[D]|qgμD^{−1}R)

= χ(μ)^{−m/2}μ^{−νm/2} ∑_{D∈Δ(Q,μ)/Λ} I(D, g^{2}Q, T)

× g^{νm} ∑_{R′∈ℤ_{m}^{m}/gℤ_{m}^{m}} χ(det R′)Θ_{ν}(Z, μ^{−1}g^{2}Q[D]|qgR′)

= χ(μ)^{−m/2}μ^{−νm/2} ∑_{D∈Δ(Q,μ)/Λ} I(D, g^{2}Q, T)Θ_{ν}(Z, μ^{−1}Q[D], χ),

since clearly Δ(g^{2}Q, μ) = Δ(Q, μ) if μ is prime to qg^{2}, the matrix R′ = μD^{−1}R runs together with R the set ℤ_{m}^{m}/gℤ_{m}^{m}, and the level of μ^{−1}Q[D] with D ∈ Δ(Q, μ) is again q. △

It is sometimes convenient to rewrite the above transformation formulae in somewhat different form. For this we have to do several preliminary remarks. We shall say that an even matrix Q′ is properly similar to a nonsingular matrix Q ∈ 𝔼^{m}, Q′ ∼_{p} Q, if it can be written in the form Q′ = μ^{−1}Q[D] with D ∈ Δ(Q, μ), where μ is coprime with the level q of Q. It follows from (3.15) that the relation of proper similarity is symmetric. Besides, it is clearly reflexive and transitive. Thus, the set of all nonsingular matrices of 𝔼^{m} is disjoint union of ps-classes

ps{Q} = {Q′ ∈ 𝔼^{m} | Q′ ∼_{p} Q} (Q ∈ 𝔼^{m}, det Q ≠ 0).

It is easy to see that all matrices of the ps-class of a matrix Q have the same signature, determinant, level, and divisor as Q (see Notation). Further, we recall that two matrices Q, Q′ of 𝔼^{m} are said to be properly equivalent, Q ≃_{p} Q′, if Q′ = Q[U] with U ∈ Λ = SL_{m}(ℤ). All matrices properly equivalent to a matrix Q form the pe-class of Q,

(3.17)  pe{Q} = {Q′ = Q[U] | U ∈ Λ} .

According to the reduction theory of integral quadratic forms, the ps-class of every nonsingular matrix Q ∈ 𝔼^{m} is a finite union of the pe-classes:

ps{Q} = ⋃_{j=1}^{h_{p}(Q)} pe{Q_{j}}.

The quantity h_{p}(Q) we shall call the proper class number of Q.

Proposition 6. With the notation and assumptions of Theorem 4 formula (3.11) can be rewritten in the form

(3.18) Θ_{P} (Z, Q|L)∥_{ν+m/2}T

= ∑_{j=1}^{h_{p}(Q)} ∑_{D∈R_{p}(Q,μQ_{j})/E_{p}(Q_{j})} I(D, Q, Ψ^{n,m}(T))Θ_{P |μ^{−1}D}(Z, Q_{j}|μD^{−1}L),

where Q_{1}, …, Q_{h_{p}(Q)} is a system of representatives of all different pe-classes contained in the ps-class of Q,

R_{p}(Q, Q′) = {M ∈ ℤ_{m}^{m} | Q[M] = Q′, det M > 0} (Q, Q′ ∈ 𝔼^{m})

is the set of all proper integral representations of Q′ by Q, and

E_{p}(Q′) = R_{p}(Q′, Q′) (Q′ ∈ 𝔼^{m}, det Q′ ≠ 0)

denotes the group of all proper units of Q′.

In particular, the formula (3.14) can be rewritten in the form

(3.19) Θ_{ν}(Z, Q|L)∥_{ν+m/2}T

= μ^{−νm/2} ∑_{j=1}^{h_{p}(Q)} ∑_{D∈R_{p}(Q,μQ_{j})/E_{p}(Q_{j})} I(D, Q, T)Θ_{ν}(Z, Q_{j}|μD^{−1}L).

Proof. It is an easy consequense of definitions that the sums (3.13) are independent of the choice of representatives in the decomposition of T′ and satisfy

(3.20)  I(UDU′, Q, T′) = I(D, Q[U], T′) for all U, U′ ∈ GL_{m}(ℤ)

(see [6], Lemma 3.1). From (1.4) and (3.20) we conclude that the term of the sum on the right in (3.11) corresponding to a matrix D ∈ Δ(Q, μ) depends only on the coset DΛ. On the other hand, each of the matrices μ^{−1}Q[D] with D ∈ Δ(Q, μ) is properly similar to Q and so is properly equivalent to one of the matrices Q_{1}, …, Q_{h_{p}(Q)}, say, μ^{−1}Q[D] ≃_{p} Q_{j}. This means that μ^{−1}Q[D][U] = μ^{−1}Q[DU] = Q_{j} with U ∈ Λ. On replacing D by DU, we can assume that D ∈ R_{p}(Q, Q_{j}). If D′ = DU′ is another such matrix, then, clearly, Q_{j}[U′] = Q_{j}, whence U′ ∈ E_{p}(Q_{j}). This proves the formulae. △

Similarly, in the same notations we have the following formulae for the action of regular Hecke operators on Maass theta-series:

Proposition 7. In the notation and assumption of Proposition 5 the formulae (3.16) can be rewritten in the form

(3-21) Θ_{ν}(Z, Q, χ)∥_{ν+m/2}T

= χ(μ)^{−m/2}μ^{−νm/2} ∑_{j=1}^{h_{p}(Q)} ∑_{D∈R_{p}(Q,μQ_{j})/E_{p}(Q_{j})} I(D, g^{2}Q, T)Θ_{ν}(Z, Q_{j}, χ),

where Q_{1}, …, Q_{h_{p}(Q)} is a system of representatives of all different pe-classes contained in the ps-class of Q, and where sets R_{p}(Q, Q′), E_{p}(Q′) were defined in Proposition 6.

In particular, if h_{p}(Q) = 1, then, for each ν = 0, 1 and every Dirichlet character χ, the Maass series Θ_{ν}(Z, Q, χ) is an eigenfunction of the operator ∥_{ν+m/2}T with the eigenvalue

χ(μ)^{−m/2}μ^{−νm/2} ∑_{D∈R_{p}(Q,μQ)/E_{p}(Q)} I(D, g^{2}Q, T).

We conclude this section with summary of known at results of computation of the sums (3.13). We remind that the HS–ring H_{0}^{n}(q) is generated over ℂ by the following double classes of the form (3.1):

(3.22)  T^{n}(p) = (diag(\underbrace{1, …, 1}_{n}, \underbrace{p, …, p}_{n}))_{Γ_{0}^{n}(q)},

and

(3.23) T_{j}^{n}(p^{2}) = (diag(\underbrace{1, …, 1}_{n−j}, \underbrace{p, …, p}_{j}, \underbrace{p^{2}, …, p^{2}}_{n−j}, \underbrace{p, …, p}_{j}))_{Γ_{0}^{n}(q)} (j = 1, …, n),

where p runs over all prime numbers not dividing q (see [2], Theorem 3.3.23). It follows that the ring H^{n}(q) is generated over ℂ by the classes

(3.24)  T̃^{n}(p) = η(T^{n}(p)) and T̃_{j}^{n}(p^{2}) = η(T_{j}^{n}(p^{2})) (j = 1, …, n),

where η is the map (3.6) and p again runs over all prime numbers not dividing q.

Proposition 8. Let Q be an even positive definite matrix of even order m, q the level of Q, and χ_{Q} the Dirichlet character corresponding to Q. Then the sums (3.13) on elements (3.22)–(3-24) for each prime number p not dividing q can be computed by the following formulae:

I(D, Q, T̃^{m}(p)) = I(D, Q, T^{m}(p)) = p^{m/2} ∏_{j=1}^{m/2}(1 + χ_{Q}(p)p^{−j})

for all D ∈ Δ(Q, p);

I(D, Q, T̃_{m−1}^{m}(p)) = I(D, Q, T_{m−1}^{m}(p))

= \{ \begin{matrix}χ_{Q}(p)p^{(2+m−m^{2})/2} \frac{(p^{m} − 1)}{p − 1} + p^{−m^{2}/2}(χ_{Q}(p)p^{m/2} − 1) if D ∈ Λ(p1_{m}), \\ χ_{Q}(p)p^{(2+m−m^{2})/2} if D ∈ Λdiag(1, \underbrace{p, …, p}_{m−2}, p^{2})Λ, \\ 0 otherwise; \end{matrix}

I(D, Q, T̃_{m}^{m}(p^{2})) = I(D, Q, T_{m}^{m}(p^{2})) = \{ \begin{matrix}p^{−m^{2}/2} & if D ∈ Λ(p1_{m}), \\ 0 & otherwise . \end{matrix}

Proof. In [3] (formula (2.19) and Lemma 5.1) the sums γ(Q, D, T) similar to the sums (3.11) were defined and computed for T = T^{m}(p). In §2 of [4] the sums γ(Q, D, T) were, in fact, computed for T = T_{m}^{m}(p^{2}) = (p1_{2m})_{Γ_{0}^{m}(q)} and T = T_{m−1}^{m}(p^{2}) (see also Lemma 3.3.32 of [2] for the presentation of T_{m−1}^{m}(p^{2}) used in [4] ). It directly follows from definitions of the sums that I(D, Q, T) = μ^{m/2}γ(Q, μD^{−1}, T). The rest is clear. △

Note that the formulae of Proposition 7 determine sums I(D, Q, T) for all generators of the rings H_{0}^{2}(q) and H^{2}(q). We do not know any other results on computation of the sums I(D, Q, T).

§4 Theta-series of binary forms with characters of class groups

In this section we shall construct eigenfunctions of regular Hecke operators in the form of linear combinations of Maass theta-series of binary quadratic forms with characters of class groups of modules in the corresponding quadratic fields.

At first we shall recall basic definitions and facts concerning to modules in algebraic number fields and relations quadratic modules with binary quadratic forms. For more details and proofs see, for example, [2], Appendix 3.

Let K be a (finite) algebraic number field, i.e. a finite extension of the field ℚ. Any finitely generated ℤ–submodule M of K is called a module in K. Two modules M and M′ are said to be similar if M′ = αM for some α ≠ 0 of K. All modules in K are divided into classes of similar modules. A module M in K is said to be full if ℚM = K. Each full module in K has a ℤ–basis of n = [K : ℚ] elements and vice versa.

A full module in K that contains 1 and is itself a ring is called an order in K. Every order in K is contained in the maximal order O of all integeral numbers of K over ℚ. If M is a module in K, then the ring

O(M) = {α ∈ K | αM ⊂ M}

is called the ring of multipliers of M. Similar modules have equal rings of multipliers. The ring of multipliers of a full module is an order in K. For each full module, there is a similar module contained in its ring of multipliers.

Let O′ be an order in the field K and ω_{1}, …, ω_{n} a basis of O′. Then the number d(O′) = det(tr(ω_{i}ω_{j})), where tr means the trace from K to ℚ, is independent of the choice of the basis and is called the discriminant of O′. The discriminant of the maximal order O,

d = d_{K} = d(O),

is called the (absolute)discriminant of K.

For a full module M with the ring of multipliers O′ = O(M), let α_{1}, …, α_{n} and ω_{1}, …, ω_{n} be bases of M and O′, respectively. Then the absolute value of the determinant of the transition matrix from the first basis to the second,

N(M) = | det(a_{ij})|, where α_{i} = ∑ a_{ij}ω_{j},

is independent of the choices of bases, and is called the norm of M. If M ⊂ O′, then N(M) = [O′ : M] (the index of M in O′).

Let M and M′ be two full modules in K. Then the set MM′ is again a full module in K, which is called the product of M and M′. The norm of the product is equal to the product of norms:

(4.1)  N(MM′) = N(M)N(M′).

All full modules in K with a fixed ring of multipliers O′ form a commutative group under multiplication of modules. The quotient group of this group by the subgroup of modules similar to O′ is a finite group H(O′) called the class group of O′.

Let O′ be an order in an algebraic number field. Full modules in K with the ring of multipliers O′ contained in O′ are called regular ideals of O′. Each regular ideal of an order in a finite extension K of ℚ is uniquely (up to order) decomposable into product of regular prime ideals.

Now let K be a quadratic extension of ℚ. Every such field K is of the form K = ℚ(\sqrt{d_{0}}), where d_{0} ≠ 0, 1 is a square-free rational integer. As a basis of the maximal order O of K one can take the numbers 1 and ω, where ω = (1 + \sqrt{d_{0}})/2 if d_{0} ≡ 1 (mod 4), and ω = \sqrt{d_{0}} if d_{0} ≡ 2 or 3 (mod 4). The discriminant d = d_{K} = d(O) of the field K and the ring O is equal to d_{0} in the first case, and to 4d_{0} in the second, and in both cases one can write K = ℚ(\sqrt{d}). Any order O′ of K has the form

O_{l} = ℤ + lωℤ,

where l is the index [O : O′]. The discrininant of O_{l} is equal to dl^{2}.

For a full module M in K we denote by \overline{M} the module consisting of the conjugates ᾱ over ℚ of the elements α ∈ M. The conjugate module \overline{M} is again a full module with the same ring of multipliers as that of M, and the following formula holds

(4.2)  M\overline{M} = N(M)O(M).

Let K be a quadratic extension of ℚ. For α, β, … ∈ K we shall denote by {α, β, … } the full module in K generated over ℤ by α, β, …:

{α, β, … } = ℤα + ℤβ + · · · .

The following lemma, which is proved, for example, in [2], Lemma A.3.2, is often useful in consideration of quadratic modules.

Lemma 9. Let K be a quadratic field, γ ∈ K, γ ∉ ℚ, and let aγ^{2} + bγ + c = 0, where a, b, and c are the rational integers satisfying gcd(a, b, c) = 1 and a > 0. Then the module M = {1, γ} satisfies

N(M) = 1/a, O(M) = {1, aγ}, d(O(M)) = b^{2} − 4ac.

The relation between quadratic modules and prime numbers is described by the following proposition (the proof can be found, for example, in [1], Proposition 2.3.1 or [2], Proposition A.3.5).

Proposition 10. Let K = ℚ(\sqrt{d}) be the quadratic field with discriminant d, and O_{l} the order of K with discriminant dl^{2}. Suppose that p is a rational prime number not dividing l. Then the existence of a full module M in K satisfying the conditions

(4.3)  O(M) = O_{l}, M ⊂ O_{l}, N(M) = p,

is equivalent to the solvability of the congruence x^{2} ≡ d (mod 4p). If this congruence is solvable, and p does not divide d, then there are precisely two modules 𝔭 and 𝔭′ with the properties (4.3), and 𝔭′ = \overline{𝔭}. But if the congruence has a solution and p divides d, then there is exactly one module 𝔭 satisfying (4.3), and \overline{𝔭} = 𝔭. Finally, if the congruence has no solutions, then there is the unique complete module 𝔭 satisfying

O(𝔭) = O_{l}, 𝔭 ⊂ O_{l}, N(𝔭) = p^{2},

and 𝔭 = pO_{l}. All of the listed modules M = 𝔭 are regular prime ideals of the ring O_{l}.

We conclude the survey part of this section with a description of the correspondence between modules in quadratic fields and integral binary quadratic forms. We restrict ourselves to imaginary quadratic field and positive definite forms, since for applications we shall not need any others.

Let

q(X) = \frac{1}{2}Q[X] = ax^{2} + bxy + cy^{2} (X = ( \begin{matrix}x \\ y \end{matrix} ))

be a binary quadratic form with matrix Q = ( \begin{matrix}2a & b \\ b & 2c \end{matrix} ). The number d(q) =

d(Q) = b^{2} − 4ac is called the discriminant of q and Q. The form is called integral if Q ∈ 𝔼^{2}, i.e. if a, b, and c are rational integers. In this case gcd(a, b, c) is the divisor of q, and the form is primitive if the divisor equals 1 (see Notation). We remind that two forms q and q′ with matrices Q and Q′ are properly equivalent if

q(UX) = q′(X) or ^{t}UQU = Q′ with U ∈ Λ^{2} = SL_{2}(ℤ).

According to the reduction theory of integral quadratic forms (see, for example, [2], Theorem 2.1.12), the set of all integral binary forms with a fixed nonzero discriminant d is a finite union of classes (3.17) of properly equivalent forms. All forms from a fixed class have the same signatur, level, discriminant, and divisor.

Let K = ℚ(\sqrt{d}) be an imaginary quadratic field with discriminant d < 0, and let M be a full module in K with the ring of multipliers O(M) = O_{l}. With every ℤ basis α, β of M ordered by the condition ℑ(β/α) > 0 we associate the binary quadratic form

q = q(M)(X) = \frac{1}{N(M)}(αx + βy)(\overline{α}x + \overline{β}y) = ax^{2} + bxy + cy^{2}.

The form q is, clearly, positive definite. It follows easily from Lemma 9 that q is integral and primitive with the discriminant d(q) = b^{2} − 4ac = dl^{2} equal to the discriminant of the ring of multipliers O(M). Conversely, if q(X) = ax^{2} +bxy +cy^{2} is a positive definite integral primitive quadratic form of discriminant b^{2}−4ac = dl^{2}, then the module

(4.4)  M = M(q) = \{a, \frac{−b + l\sqrt{d}}{2}\}

is a full module in K with the ring of multipliers O_{l} contained in the ring. It is not difficult to check that the indicated correspondences define a bijection between the set of all classes of equivalent full modules in K with the ring of multipliers O_{l} and the set of all classes of properly equivalent positive definite integral primitive binary quadratic forms of discriminant dl^{2}.

Let q and q′ be two integral primitive positive definite binary quadratic forms of the same negative discriminant −δ, and let Q = Q(q) and Q′ = Q(q′) be the matrices of the forms. Since the modules M = M(q) and M′ = M(q′) are full modules in the field K = ℚ(\sqrt{−δ}) with the same ring of multipliers O_{l} of discriminant −δ = dl^{2}, where d is the discriminant of the field K, the same is true for their product M · M′. We define the product q × q′ of the two forms q and q′ by

(4.5)  q × q′ = q(M · M′)

and denote by Q × Q′ the matrix of q × q′:

(4.6)  Q × Q′ = Q(q × q′).

Further, for any full module N in K with the ring of multipliers O_{l}, we set

(4.7)  q × N = N × q = q(M(q) · N),

and denote by Q × N the matrix of q × N:

(4.8)  Q × N = N × Q = Q(q × N).

The multiplication (4.5) is called the Gauss composition of quadratic forms. The proper equivalence class of the product q × q′ depends only on proper classes of q and q′, and the proper equivalence class of q×N depends only on proper class of q and class of N. Thus, the formulae (4.5)–(4.8) define the products of corresponding classes.

Since the set of equivalence classes of full modules with the ring of multipliers of discriminant −δ form an Abelian group under multiplication of the classes, the set of proper equivalence classes of integral primitive positive definite binary quadratic forms of the discriminant −δ, can also be endowed with a group structure. We shall denote by H(−δ) both of the class groups.

The following useful lemma is an easy consequence of definitions.

Lemma 11. The level q and the discriminant −δ of an integral primitive positive definite binary quadratic form with matrix Q satisfy the relation q = det Q = δ.

We turn now to the action of regular Hecke operators on Maass theta-series (*) of genus 2.

Theorem 12. Let q be a positive integer, and let Q_{1}, …, Q_{h(−q)} be a system of representatives of all different proper equivalence classes (3.17) of even primitive positive definite matrices of order 2 and level q. Finally, let ψ be a character of the group H(−q) of the proper equivalence classes of matrices of integral primitive positive definite binary quadratic forms of discriminant −q under the composition (4.6), and χ a Dirichlet character modulo a rational integer g. Then the linear combination of Maass theta-series of the form

F = Θ_{ν}(Z, q, ψ, χ) = ∑_{j=1}^{h(−q)} ψ(Q_{j})Θ_{ν}(Z, Q_{j}, χ) (ν = 0, 1)

has the following properties:

(1). The function F is independent of the choice of representatives Q_{j} in the proper equivalence classes; it is a modular form of weight ν +1 for the group Γ_{0}^{2}(qg^{2}) with the character χχ_{Q}, where χ_{Q} is the character of an even primitive positive definite matrix Q of order 2 and level q defined in Theorem 1 ; if ν = 1, then F is a cusp form.

(2). The modular form F is an eigenfunction for all Hecke operators of the regular Hecke–Shimura ring H^{2}(qg^{2}):

F ∥_{ν+1}T = λ_{F} (T)F (∀T ∈ H^{2}(qg^{2})).

(3). For a prime number p not dividing qg, the eigenvalues of the operators corresponding to the generators (3.24) of the ring H^{2}(qg^{2}) let T be one of the images can be given by the formulae:

(4.9)  λ_{F} (T̃^{2}(p)) = \{ \begin{matrix}\frac{(p+1)(ψ(𝔭)+ψ(\overline{𝔭}))}{χ(p)p^{ν}} if χ_{Q}(p) = 1, \\ 0 if χ_{Q}(p) = −1; \end{matrix}

(4.10)  λ_{F} (T̃_{1}^{2}(p^{2})) = \{ \begin{matrix}\frac{(p^{3}+p^{2}+p−1+(pψ(𝔭))^{2}+(pψ(\overline{𝔭}))^{2})}{χ(p)^{2}p^{2(ν+1)}} if χ_{Q}(p) = 1, \\ −\frac{(p^{3}+p^{2}+p+1)}{χ(p)^{2}p^{2(ν+1)}} if χ_{Q}(p) = −1, \end{matrix}

where, for the case χ_{Q}(p) = 1, in both formulae 𝔭 and \overline{𝔭} are the conjugate regular prime ideals of the order O′ with discriminant −q in the field K = ℚ(\sqrt{−q}) satisfying 𝔭\overline{𝔭} = O′;

(4.11)  λ_{F} (T̃_{2}^{2}(p^{2})) = \frac{1}{χ(p)^{2}p^{2(ν+1)}} .

At first we shall establish some preliminary results.

Lemma 13. Each of the Maass series (*) satisfies the relations

Θ_{ν}(Z, Q[U], χ) = χ(det U)^{−1}(det U)^{−ν}Θ_{ν}(Z, Q, χ) (∀U ∈ GL_{m}(ℤ)).

In particular, it depends only on the proper equivalence class (3.17) of Q.

Proof. The relation follows directly from the definition by the change N ↦ U^{−1}N.△

Lemma 14. Let Q be an even primitive positive definite matrix of order 2, q the level of Q, and χ_{Q} the corresponding Diriclet character modulo q. Let d be the discriminant of the imaginary quadratic field K = ℚ(\sqrt{−q}) so that −q = dl^{2}, and O_{l} is the order in K with discriminant −q. For a prime number p not dividing q and a positive integer β, we denote by Π(p^{β}) the linear operator on functions Φ of proper equivalence classes of matrices Q′ properly similar to Q given by

(4.12)

(Φ ∘ Π(p^{β})) (Q) = ∑ ; D∈\{Λ( \begin{matrix}1 & 0 \\ 0 & p^{β} \end{matrix} )Λ∩Δ(Q, p^{β})\}/Λ Φ(p^{−β}Q[D]),

(Φ ∘ Π(p^{β})) (Q) = 0 if Λ ( \begin{matrix}1 & 0 \\ 0 & p^{β} \end{matrix} ) Λ ∩ Δ(Q, p^{β}) = ∅,

where Δ(Q, μ) are the sets (3.12), and Λ = Λ^{2} = SL_{2}(ℤ). Then the following formulae hold:

(1). If χ_{Q}(p) = 1, then there are precisely two regular prime ideals 𝔭 and 𝔭′ of norm p in the ring O_{l} conjugate with each other, 𝔭′ = \overline{𝔭}, and the formula is true

(Φ ∘ Π(p^{β})) (Q) = Φ(Q × 𝔭^{β}) + Φ(Q × \overline{𝔭}^{β}),

where × means the multiplication (4.8).

(2). If χ_{Q}(p) = −1 , then the ring O_{l} contains no regular prime ideals of norm p, and the formula is true

(Φ ∘ Π(p^{β})) (Q) = 0.

Proof. First, we introduce on the set of all pairs (u, v) of coprime integers the equivalence relation ∼ (mod p^{β}) as on the ”projective line modulo p^{β}”:

(u′, v′) ∼ (u, v) (mod p^{β}) ⇔ u′ ≡ tu, v′ ≡ tv (mod p^{β})

and denote by R(p^{β}) a set of integral matrices U of order 2, where the first column runs a system of reptesentatives for the classes ∼ (mod p^{β}) of all coprime pairs and the second row is fixed by the condition U ∈ Λ. Then it is easy to see that the following decomposition into different right cosets modulo Λ holds

(4.13)  Λ ( \begin{matrix}1 & 0 \\ 0 & p^{β} \end{matrix} ) Λ = ⋃_{U∈R(p^{β})} U ( \begin{matrix}1 & 0 \\ 0 & p^{β} \end{matrix} ) Λ.

Now, let q(x, y) = ax^{2} + bxy + cy^{2} be the quadratic form with matrix Q. We consider the congruence

(4.14)  q(x, y) = ax^{2} + bxy + cy^{2} ≡ 0 (mod p^{β}).

Let us call a solution (u, v) of the congruence primitive if the integers u, v are coprime. An easy consideration of the corresponding quadratic space over the field of p elements shows that if χ_{Q}(p) = 1, then the congruence x^{2} ≡ d (mod 4p) is solvable, and the congruence (4.14) has precisely two nonequivalent modulo p^{β} primitive solutions, and these solutions already are not equivalent modulo p, but if χ_{Q}(p) = −1 the congruence x^{2} ≡ d (mod 4p) is unsolvable, and the congruence (4.14) has no primitive solutions (see, for example, [2], Proof of Proposition 4.3.14).

On the other hand, note that a matrix of the form

U ( \begin{matrix}1 & 0 \\ 0 & p^{β} \end{matrix} ) with U = ( \begin{matrix}u & u_{1} \\ v & v_{1} \end{matrix} ) ∈ Λ

belongs to the set Δ(Q, p^{β}) if and only if the pair u, v is a primitive solution of (4.14). Hence, the part (2) follows from Proposition 10.

Let χ_{Q}(p) = 1, and (u, v), (u′, v′) be two nonequivalent primitive solutions of (4.14). Since p does not divide q = det Q, we may assume that q(u, v) = p^{β}a_{1} and q(u′, v′) = p^{β}a_{2}, where a_{1} and a_{2} are not divisible by p. We can choose the set R(p^{β}) so that

U_{1} = ( \begin{matrix}u & u_{1} \\ v & v_{1} \end{matrix} ) ∈ R(p^{β}) and U_{2} = ( \begin{matrix}u′ & u′_{1} \\ v′ & v′_{1} \end{matrix} ) ∈ R(p^{β}).

Let us set for i = 1, 2

Q_{i} = Q[U_{i}] = ( \begin{matrix}2p^{β}a_{i} & b_{i} \\ b_{i} & 2c_{i} \end{matrix} ) , Q′_{i} = p^{−β}Q [U_{i} ( \begin{matrix}1 & 0 \\ 0 & p^{β} \end{matrix} )] = ( \begin{matrix}2a_{i} & b_{i} \\ b_{i} & 2p^{β}c_{i} \end{matrix} ) .

Then according to (4.22), (4.25), and the above considerations, we can write

Φ(Q) ∘ Π(p^{β}) = Φ(Q′_{1}) + Φ(Q′_{2}).

Now, let us consider the modules in K given by

M_{1} = \{p^{β}, \frac{−b_{1} + l\sqrt{d}}{2}\} , M_{2} = \{p^{β}, \frac{−b_{2} + l\sqrt{d}}{2}\}

and show at first that the modules are regular ideals of O_{l} of norm p^{β}. Let us consider, for example, the module M_{1}. The number γ_{1} = (−b_{1} + l\sqrt{d})/2p^{β} is a root of the polynomial p^{β}x^{2} + b_{1}x + a_{1}c_{1}. Since q = 4ac − b^{2} = 4p^{β}a_{1}c_{1} − b_{1}^{2} is not divisible by p, it follows that b_{1} is not divisible by p, and so the coefficients of the polynomial are coprime. Then, by Lemma 9, the ring of multipliers of the module M_{1} = p^{β}{1, γ_{1}} is O_{l}, and its norm N(M_{1}) = N(p^{β})/p^{β} = p^{β}. It is also clear that M_{1} ⊂ O_{l}. Next, we shall show that the ideals M_{1} and M_{2} are coprime, i.e. that M_{1} + M_{2} = O_{l}. The numbers (b_{2} − b_{1})/2 and p^{β} are obviously both contained in M_{1} + M_{2}. Hence, every integral linear combination of the numbers is contained in the sum; in particular, s = gcd((b_{2} − b_{1})/2, p^{β}) is contained in M_{1} + M_{2}. Thus, we only have to prove that s = 1. Let us set

U_{1}^{−1}U_{2} = ( \begin{matrix}v_{1} & −u_{1} \\ −v & u \end{matrix} ) ( \begin{matrix}u′ & u′_{1} \\ v′ & v′_{1} \end{matrix} ) = ( \begin{matrix}t_{1} & t_{2} \\ t_{3} & t_{4} \end{matrix} ) = T ∈ Λ.

Since the pairs (u, v) and (u′, v′) are not equivalent modulo p, it follows that t_{3} = uv′ − vu′ is not divisible by p. By comparing the corresponding entries of matrices in the obvious relation Q_{1}[T] = Q_{1}[U_{1}^{−1}U_{2}] = Q_{2}, we obtain the relations

p^{β}a_{1}t_{1}^{2} + b_{1}t_{1}t_{3} + c_{1}t_{3}^{2} = p^{β}a_{2} and 2p^{β}a_{1}t_{1}t_{2} + b_{1}(t_{1}t_{4} + t_{2}t_{3}) + 2c_{1}t_{3}t_{4} = b_{2}.

Since t_{3} is not divisible by p, the first relation implies that b_{1}t_{1} + c_{1}t_{3} ≡ 0 (mod p). Since t_{1}t_{4} − t_{2}t_{3} = 1, the second relation implies that (b_{2} − b_{1})/2 ≡ t_{3}(b_{1}t_{2} + c_{1}t_{4}) (mod p^{β}). Thus, if (b_{2} − b_{1})/2 is divisible by p, then we have congruences

b_{1}t_{1} + c_{1}t_{3} ≡ b_{1}t_{2} + c_{1}t_{4} ≡ 0 (mod p),

which would imply the congruences b_{1} ≡ c_{1} ≡ 0 (mod p), contrary to the assumption that q = 4p^{β}a_{1}c_{1} − b_{1}^{2} is not divisible by p. From what we have proved, by the theorem on prime ideals factorization of regular ideals of the ring O_{l} and Proposition 10, it follows that M_{1} = 𝔭^{β} and M_{2} = \overline{𝔭}^{β}, where 𝔭 and \overline{𝔭} are the only regular prime ideals of norm p in O_{l}. Let q_{1}, q_{2}, q′_{1}, and q′_{2} be the quadratic forms with the matrices Q_{1}, Q_{2}, Q′_{1}, and Q′_{2}, respectively, and let M(q_{1}), M(q_{2}), M(q′_{1}), and M(q′_{2}) be the modules (4.4) corresponding to the forms. We set δ_{1} = (−b_{1}+l\sqrt{d})/2. Since clearly δ_{1}^{2} = −b_{1}δ_{1} − p^{β}a_{1}c_{1} and, as it was noted above, b_{1} is prime to p, it follows that

M(q′_{1})𝔭^{β} = {a_{1}, δ_{1}}{p^{β}, δ_{1}} = {a_{1}p^{β}, a_{1}δ_{1}, p^{β}δ_{1}, −b_{1}δ_{1} − p^{β}a_{1}c_{1}}
= {a_{1}p^{β}, δ_{1}} = M(q_{1}).

Hence, by (4.2), we obtain

M(q′_{1}) = \frac{1}{p^{β}} M(q′_{1})𝔭^{β}\overline{𝔭}^{β} = \frac{1}{p^{β}} M(q_{1})\overline{𝔭}^{β}.

Since the form q_{1} is properly equivalent to q, it follows that the last module is equivalent to M(q)\overline{𝔭}^{β}. Thus, the matrix Q′_{1} of q′_{1} is properly equivalent to the matrix Q × \overline{𝔭}^{β}. Similarly, Q′_{2} is properly equivalent to Q × 𝔭^{β}. △

Proof of the theorem. The part (1) follows from the definition of the function F and Proposition 3.

Since the elements (3.24) generate the ring H^{2}(qg^{2}) over ℂ, in order to prove the parts (2) and (3) it will be sufficient to show that the function F is an eigenfunction for all Hecke operators corresponding to the elements T of the form (3.24) with eigenvalues given by the formulae (4.9)–(4.11).

By Proposition 5, we can write

F ∥_{ν+1}T

= \frac{1}{χ(μ)μ^{ν}} ∑_{j=1}^{h(−q)} ψ(Q_{j}) ∑_{D∈Δ(Q_{j},μ)/Λ} I(D, g^{2}Q_{j}, T)Θ_{ν}(Z, μ^{−1}Q_{j}[D], χ),

where μ = μ(T) equals p for the first of the elements (3.24) and p^{2} for the second and the third of the elements.

Let T = η(T^{2}(p)). By the definition (3.13) of the sums I(D, Q, T) and Proposition 8, for all D ∈ Δ(Q_{j}, p) we have

I(D, Q_{j}, η(T^{2}(p))) = I(D, Q_{j}, T^{2}(p)) = p(1 + χ_{Q}(p)p^{−1}) = p + χ_{Q}(p),

since all of the matrices Q_{j} have the same character as the matrix Q of part (1). If the set Δ(Q_{j}, p) is not empty, it is clearly contained in the double coset

Λ ( \begin{matrix}1 & 0 \\ 0 & p \end{matrix} ) Λ. Therefore, using the notation (4.12), we can rewrite the above for-

mula for T = η(T^{2}(p)) in the form

F ∥_{ν+1}T = \frac{p + χ_{Q}(p)}{χ(p)p^{ν}} ∑_{j=1}^{h(−q)} ψ(Q_{j})Θ_{ν}(Z, Q_{j}, χ) ∘ Π(p),

which, by Lemma 14, is equal to

\frac{p + χ_{Q}(p)}{χ(p)p^{ν}} ∑_{j=1}^{h(−q)} ψ(Q_{j})(Θ_{ν}(Z, Q_{j} × 𝔭, χ) + Θ_{ν}(Z, Q_{j} × \overline{𝔭}, χ))

= \frac{p + χ_{Q}(p)}{χ(p)p^{ν}}(ψ(\overline{𝔭}) ∑_{j=1}^{h(−q)} ψ(Q_{j} × 𝔭)Θ_{ν}(Z, Q_{j} × 𝔭, χ)

+ ψ(𝔭) ∑_{j=1}^{h(−q)} ψ(Q_{j} × \overline{𝔭})Θ_{ν}(Z, Q_{j} × \overline{𝔭}, χ))

= \frac{p + 1}{χ(p)p^{ν}} (ψ(𝔭) + ψ(\overline{𝔭})) ∑_{j=1}^{h(−q)} ψ(Q_{j})Θ_{ν}(Z, Q_{j}, χ)

if χ_{Q}(p) = 1, and is 0 if χ_{Q}(p) = −1. This proves the formulae (4.9).

Now, let T = η(T_{1}^{2}(p^{2})). Then, by Proposition 8, with the notation (4.12), we obtain

F ∥_{ν+1}T = \frac{1}{χ(p^{2})p^{2ν}} ((χ_{Q}(p)(p + 1) + p^{−2}(χ_{Q}(p)p − 1)) ∑_{j=1}^{h(−q)} ψ(Q_{j})Θ_{ν}(Z, Q_{j}, χ)

+ χ_{Q}(p) ∑_{j=1}^{h(−q)} ψ(Q_{j})Θ_{ν}(Z, Q_{j}, χ) ∘ Π(p^{2})).

If χ_{Q}(p) = 1, then, by Lemma 14, the last expression is equal to

\frac{1}{χ(p^{2})p^{2ν}} ((p + 1 + p^{−1} − p^{−2}) ∑_{j=1}^{h(−q)} ψ(Q_{j})Θ_{ν}(Z, Q_{j}, χ)

+ ∑_{j=1}^{h(−q)} ψ(Q_{j})(Θ_{ν}(Z, Q_{j} × 𝔭^{2}, χ) + Θ_{ν}(Z, Q_{j} × \overline{𝔭}^{2}, χ)))

= \frac{1}{χ(p)^{2}p^{2(ν+1)}} (p^{3} + p^{2} + p − 1 + (pψ(𝔭))^{2} + (pψ(\overline{𝔭}))^{2})F.

But if χ_{Q}(p) = −1 we have just

F ∥_{ν+1}T = \frac{−(p^{3} + p^{2} + p + 1)}{χ(p)^{2}p^{2(ν+1)}}F.

It proves the formula (4.10).

Finally, let T = η(T_{2}^{2}(p^{2})). By Proposition 8, or directly, by the definition of Hecke operators, we obtain

F ∥_{ν+1}T = \frac{1}{χ(p^{2})p^{2ν}} ∑_{j=1}^{h(−q)} ψ(Q_{j})p^{−2}Θ_{ν}(Z, Q_{j}, χ) = \frac{1}{χ(p^{2})p^{2(ν+1)}} F,

which proves the formula (4.11) and the theorem.△

§5 Zeta functions of Maass theta-series of binary forms

Let F ∈ 𝔐_{k}(Γ_{0}^{2}(q)) be a nonzero eigenfunction for all Hecke operators of the regular Hecke–Shimura ring H^{2}(q):

F ∥_{k}T = λ_{F} (T)F (∀T ∈ H^{2}(q)).

As it was noted in §3, the ring H^{2}(q) is generated over ℂ by the elements (3.14) for all prime number p not dividing q. For any such p we introduce the p-polynomial of the eigenfunction F by

(5.1) Q_{p,F} (t) = 1 − p^{2k−3}λ_{F} (T̃(p))t + p^{4k−6}λ_{F} (pT̃_{1}(p^{2}) + p(p^{2} + 1)T̃_{2}(p^{2}))t^{2}

− p^{6k−6}λ_{F} (T̃(p)T̃_{2}(p^{2}))t^{3} + p^{8k−6}λ_{F} (T̃_{2}(p^{2})^{2})t^{4}

and define the zeta function of F as the Euler product

(5.2)  Z_{F} (s) = ∏_{p∤q} Q_{p,F} (p^{−s})^{−1}

(compare with [1], §1.3 and [2], §4.3.2). One can show that the product converges absolutely and uniformly in a right half-plane of the complex variable s.

Theorem 15. Let −q = dl^{2} be the discriminant of an even primitive positive definite matrix Q of order 2, where d is the discriminant of the field K = ℚ(\sqrt{−q}), and let Q_{1}, …, Q_{h(−q)} be a system of representatives of all different proper equivalence classes (3.17) of even primitive positive definite matrices of order 2 and discriminant −q. Finally, let ψ be a character of the group H(−q) of the proper equivalence classes under the composition (4.6), g a rational integer, and χ a Dirichlet character modulo g. Suppose that the linear combination of Maass theta-series

F = Θ_{ν}(Z, q, ψ, χ) = ∑_{j=1}^{h(−q)} ψ(Q_{j})Θ_{ν}(Z, Q_{j}, χ) (ν = 0, 1)

is not identically zero. Then the zeta function (5.2) of the eigenfunction F satisfies the identity

(5.3)  Z_{F} (s) = L_{O_{l}}(s − ν, ψ, \overline{χ})L_{O_{l}}(s − ν + 1, ψ, \overline{χ}),

where O_{l} is the subring of index l in the maximal order of K, and

(5.4)  L_{O_{l}}(s, ψ, \overline{χ}) = ∏_{𝔭} (1 − \frac{ψ(𝔭)\overline{χ}(N(𝔭))}{N(𝔭)^{s}})^{−1} ,

with 𝔭 running over all regular prime ideals of the ring O_{l} not dividing gq is a L-function of O_{l} .

Proof. According to Theorem 12, the function F is a modular form of weight k = ν + 1 and character χχ_{Q} for the group Γ_{0}^{2}(q′) of the level q′ = qg^{2}, and it is an eigenfunction for all Hecke operators corresponding to elements (3.24) for all prime numbers p ∤ q′ with eigenvalues given by formulae (4.9)–(4.11). By substituting the eigenvalues to formulae for coefficients of polynomials (5.1), we obtain that if χ_{Q}(p) = 1, then

Q_{p,F} (t) = 1 − p^{ν−1}(p + 1)(ψ(𝔭) + ψ(\overline{𝔭})\overline{χ}(p)t

+ p^{2ν−3}(p^{3} + p^{2} + p − 1 + p^{2}ψ(𝔭)^{2} + p^{2}ψ(\overline{𝔭})^{2} + p^{2} + 1)\overline{χ}(p)^{2}t^{2}
− p^{3ν−2}(p + 1)(ψ(𝔭) + ψ(\overline{𝔭}))\overline{χ}(p)^{3}t^{3} + p^{4ν−2}\overline{χ}(p)^{4}t^{4}

= (1 − p^{ν}ψ(𝔭)\overline{χ}(p)t)(1 − p^{ν}ψ(\overline{𝔭})\overline{χ}(p)t)(1 − p^{ν−1}ψ(𝔭)\overline{χ}(p)t)(1 − p^{ν−1}ψ(\overline{𝔭})\overline{χ}(p)t),

but if χ_{Q}(p) = −1, then

Q_{p,F} (t) = 1 + p^{2ν−3}(−p^{3} − p^{2} − p − 1 + p^{2} + 1)\overline{χ}(p)^{2}t^{2} + p^{4ν−2}\overline{χ}(p)^{4}t^{4}
= (1 − p^{2ν} \overline{χ}(p^{2})t^{2})(1 − p^{2ν−2}\overline{χ}(p^{2})t^{2}).

Since, by Proposition 10, in the first case 𝔭 and \overline{𝔭} are the only regular prime ideals of the ring O_{l} of the norm N(𝔭) = N(\overline{𝔭}) = p, and in the second case the principal ideal 𝔭 = pO_{l} is the only regular prime ideal of the norm N(𝔭) = p^{2}, the above calculations and factorization (5.4) show that the infinite products in both sides of relation (5.3) have equal p-factors for every prime number p not dividing q′. △

References

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7. E. Freitag, Die Wirkung von Heckeoperatoren auf Thetareihen mit harmonischen Koefficienten, Math. Ann. 258 (1981/82), 419-440.

8. E. Freitag, Singular modular forms and theta relations, Lecture Notes in Math.,1487, Springer-Verlag, Berlin Heidelberg..., 1991.

9. H. Maass, Konstruktion von Spitzenformen beliebigen Grades mit Hilfe von Thetareihen, Math. Ann. 226 (1977), 275-284.

10. R. Salvati-Manni and J. Top, Cusp forms of weight 2 for the group Γ_{2}(4, 8), Amer. J. Math. 115 (1993), no. 2, 455-486.

11. H. Yoshida, Siegel’s Modular Forms and the Arithmetic of Quadratic Forms, Invent. Math. 60 (1980), no. 3, 193-248.

Steklov Mathematical Institute, St.Petersburg Branch, Fontanka 27, 191011 St.Petersburg, Russia

Current address: Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn

E-mail address: anandr@pdmi.ras.ru