SINGULAR HECKE–SHIMURA RINGS AND HECKE OPERATORS ON SIEGEL MODULAR FORMS

ANATOLI ANDRIANOV Max-Planck-Institut für Mathematik, Bonn Institut Fourier, Grenoble St.Petersburg Steklov Mathematical Institute, St.Petersburg

Abstract

Theory of singular Hecke–Shimura rings and Hecke operators on Siegel modular forms for the groups Γ_{0}^{n}(q) is developed. It is proved, in particular, that, if n = 2, the spaces of new cusp forms of integral weights are spanned by common eigenfunctions of all regular Hecke operators, Hecke operators corresponding to elements of singular unramified rings, and to all of the singular Frobenius elements. The classical case of one variable, considered by Atkin and Lehner, is included.

Table of contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Chapter 1. The background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 §1.1. Symplectic matrices, groups, and operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 §1.2. Integral matrices and discrete subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 §1.3. Modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 §1.4. Petersson scalar product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 §1.5. Hecke–Shimura rings and Hecke operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 §1.6. Reducible, regular, and singular subrings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Chapter 2. Regular reducible rings and operators for Γ_{0}^{n}(q) . . . . . . . 20 §2.1. Global rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 §2.2. Local rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 §2.3. Parabolic embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 §2.4. Regular reducible Hecke operators and triangular operators . . . . . . . . . . 30 Chapter 3. Singular Hecke–Shimura rings for Γ_{0}^{n}(q) . . . . . . . . . . . . . . . . 33 §3.1. Centralizers of Frobenius elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 §3.2. Neutral rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 §3.3. Unramified embeddings of local rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Chapter 4. Old forms, new forms, eigenforms . . . . . . . . . . . . . . . . . . . . . . 62 §4.1. Trace-idempotents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 §4.2. Definition of old and new forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 §4.3. Invariant subspaces and eigenforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Introduction

The most impressive applications of modular forms to Diophantine number theory during the last decade, the proof of the Fermat’s last theorem by Willes and Teylor, would be literally impossible without a reasonable complete theory of Hecke p-operators on modular forms in one variable for all prime numbers p , both for primes which do not divide the level of modular forms (regular p-operators due to Hecke [He(37)] and Petersson [Pet(39)]), and for prime divisors of the level (singular p-operators studied by Atkin and Lehner [A-L(70)], Li [Li(75)], and others). Without this one could not even define all of the p-factors of Euler products corresponding to modular forms and, therefore, formulate precisely the Shimura– Taniyama conjecture on zeta functions of elliptic curves, which lies in the heart of the Willes–Teylor proof.

One can hardly doubt that the relation between zeta functions of elliptic curves and zeta functions of modular forms in one variable known as Shimura–Taniyama conjecture is only a particular case of some general links of global zeta functions of algebraic varieties and zeta functions of automorphic forms in several variables. Speaking on Abelian varieties in place of elliptic curves, one can expect that modular forms in one variable should be replaced by Siegel modular forms for congruence subgroups of the Siegel modular group Γ^{n} = Sp_{n}(ℤ), which expectation is supported by numerical evidences (see, for example, [SM-T(93)] ). One of the main obstacle in this direction is complete absence of theory of singular Hecke–Shimura rings and Hecke operators on Siegel modular forms. Here we start to fill this gap.

We develop the theory of singular(i.e. related to divisors of levels) Hecke-Shimura rings and Hecke operators on Siegel modular forms for the congruence subgroups of Γ^{n} of the form Γ_{0}^{n}(q) (see (2-1)). In particular, it is proved that, if n = 2, the spaces of new cusp forms of integral weights are spanned by common eigenfunctions of all regular Hecke operators, Hecke operators corresponding to elements of singular nonramified rings, and to all of the singular Frobenius elements.

Contents. Chapter 1 contains all necessary definitions and facts on Siegel modular forms, general Hecke–Shimura rings, and Hecke operators. Chapter 2 treats specifically regular reducible rings and operators for Γ_{0}^{n}(q). The proofs in Chapters 1 and 2 are, as a rule, omitted and referred to ours books [An(87)] or [An-Z(90)]. The Chapter 3 is the key chapter. It deals with singular Hecke–Shimura rings for the groups Γ_{0}^{n}(q). Especially important is §3.3, where, for prime divisors p of q not dividing q/p (nonramified primes),canonical embedding of regular reducible p-rings of Γ_{0}^{n}(q/p) into (singular) p-rings of Γ_{0}^{n}(q) are defined and studied. The embeddings replace parabolic embeddings of regular rings. Chapter 4 contains definitions of old and new cusp forms and considers the questions of their invariance under various Hecke operators and existence of eigenforms. The exposition is essentially based on properties of trace-idempotents, considered in §4.1. Conclusion contains some remarks on possible generalization of the results and their applications.

Acknowledgment. First of all, I wish to thank my colleagues of Institut Fourier (Universite Joseph Fourier, Grenoble, France) and Max-Planck-Institut für Mathematik (Bonn, Germany), where big parts of the present research were carried out and reported, and the paper was prepared, for kind invitations, substantial discussions, and various help. Especially I should like to thank Professor Roland Gillard and Professor Alexei Panchishkin of Institut Fourier, and Professor Yuri Manin of Max-Planck-Institut, who has also taught me to AMS-TEX this paper. I am grateful to Fedor Andrianov, who has expertly read the files and initiated a number of improvements.

Last, but foremost, my thanks to my wife, as a small acknowledgment of her patience and moral support.

The research was partly supported by Russian Fund of Fundamental Researches (Grant 96-01-00663).

NOTATION. We fix 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.We set

ℕ_{(q)}={n ∈ ℕ; gcd(n, q) = 1},

and

ℕ_{q} = {n ∈ ℕ; n|q^{∞}},

where the expression n|q^{∞} means that n divides a power of q.

𝔸_{n}^{m} is the set of all m × n–matrices with entries in a set 𝔸. We denote by

𝕊_{n} = {M ∈ ℤ_{n}^{n}; ^{t}M = M},

where ^{t}M always denotes the transposed matrix, the set of all integral symmetric matrices of order n, and by

𝔼_{n} = {M = (m_{ij}) ∈ 𝕊_{n}; m_{ii} ∈ 2ℤ, i = 1, …, n}

the set of all even matrices of order n.

E_{n} will denote the unit matrix of order n , and

J_{n} = ( \begin{matrix}0 & −E_{n} \\ E_{n} & 0 \end{matrix} ) ,

where 0 = 0_{n} is the zero matrix of order n.

For brevity we set

a(a + 1)/2 = ⟨a⟩.

Chapter 1

The Background

In this chapter we shall briefly recall basic definitions and properties of symplectic matrices, Siegel modular forms, Hecke-Shimura rings, and Hecke operators for arithmetical discrete subgroups of the real symplectic group Sp_{n}(ℝ). Majority of the proofs can be found in the books [An(87)], [An-Z(90)], or can be restored by the reader in the course of easy exercises. We make an exception only for Theorem 1.9, because it is difficult to point out at explicit reference.

§1.1. Symplectic matrices, groups, and operators

A matrix M ∈ ℂ_{2n}^{2n} is said to be symplectic, if it satisfies the relation

tM · J_{n} · M = μJ_{n},

with a nonzero scalar μ = μ(M) called the multiplier of M (see Notation).

Lemma 1.1. Let M = ( \begin{matrix}A & B \\ C & D \end{matrix} ) , where A, B, C, D are complex matrices of

dimension n×n, and μ be a nonzero complex number. Then the following conditions are equivalent:

(1) M is symplectic of the multiplier μ(M) = μ;

(2) ^{t}M is symplectic of the multiplier μ(^{t}M) = μ;

(3) M is invertible, and

μM^{−1} = ( \begin{matrix}^{t}D & −^{t}B \\ −^{t}C & ^{t}A \end{matrix} );

(4) ^{t}A · C = ^{t}C · A, ^{t}B · D = ^{t}D · B, ^{t}A · D −^{t}C · B = μE_{n};

(5) A ·^{t}B = B ·^{t}A, C ·^{t}D = D ·^{t}C, A ·^{t}D − B ·^{t}C = μE_{n}.

In the course of our arithmetical considerations we shall be interested in discrete subgroups and subsemigroups of the general real positive symplectic group of a genus n ∈ ℕ consisting of all real symplectic matrices of order 2n with positive multipliers,

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

The group G^{n} is a real Lie group acting as a group of analytical automorphisms on the n(n + 1)/2 – dimensional complex variety

ℍ_{n} = {Z = X + iY ∈ ℂ_{n}^{n}; ^{t}Z = Z, Y > 0},

the Siegel upper half-plane of genus n, 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}).

The action satisfies

(M · M′)⟨Z⟩ = M⟨M′⟨Z⟩⟩, (M, M′ ∈ G^{n}, Z ∈ ℍ_{n}).

Acting on ℍ_{n}, the group G^{n} acts also on functions F : ℍ_{n} → ℂ by means of the Petersson operators of integral weights k,

(1-1)  G^{n} ∋ M = ( \begin{matrix}A & B \\ C & D \end{matrix} ) : F → F |_{k}M = det(CZ + D)^{−k}F (M⟨Z⟩),

satisfying

(1-2)  F |_{k}(M · M′) = F |_{k}M|_{k}M′, F |_{k}rM = r^{−nk}F |_{k}M

for every F : ℍ_{n} → ℂ, M, M′ ∈ G^{n}, and r ∈ ℝ − {0}.

§1.2. Integral matrices and discrete subgroups

We consider the semigroup

Σ = Σ^{n} = G^{n} ⋂ ℤ_{2n}^{2n}

of integral symplectic matrices of order 2n with positive multipliers. The inverse M^{−1} of a matrix M of Σ not necessarily belongs to Σ, but the matrix

(1-3)  M^{∗} = μ(M)M^{−1} = J_{n}^{−1} ·^{t}M · J_{n}

does, and the map M → M^{∗} is an antiautomorphism of the semigroup Σ preserving multipliers:μ(M^{∗}) = μ(M). Each subgroup of Σ is contained in the maximal subgroup

Γ = Γ^{n} = Sp_{n}(ℤ) = {M ∈ Σ^{n}; μ(M) = 1},

called the Siegel modular group of genus n.

In order to build matrices ( \begin{matrix}A & B \\ C & D \end{matrix} ) belonging to Γ, it will be useful to charac-

terize the ”second rows” (C, D) of such matrices. A pair (C, D) of square matrices of order n will be called symmetric, if it satisfies C ·^{t}D = D ·^{t}C; it will be called coprime, if the matrices C and D are integral and the products RC, RD by a matrix R of order n are both integral only if R is integral.

Lemma 1.2. Let (C, D) be a pair of integral matrices of order n. Then the following conditions are equivalent:

(1) There are matrices A, B such that ( \begin{matrix}A & B \\ C & D \end{matrix} ) ∈ Γ;

(2) The pair (C, D) is symmetric and coprime.

The modular group Γ = Γ^{n} is clearly discrete in G^{n}.This is also true for every of subgroups of G^{n} commensurable with Γ. We shall remind that two subgroups K, K′ of a group G is said to be commensurable, if their intersection K∩K′ has finite indexes both in K and K′. The relation of commensurability is reflexive, symmetric, and transitive, and so the set of all subgroups of G is a disjoint union of classes of commensurable subgroups. The principal congruence subgroup of level q,

Γ(q) = Γ^{n}(q) = {M ∈ Γ^{n}; M ≡ E_{2n} (mod q)},

and each of intermediate subgroups K,

Γ(q) ⊂ K ⊂ Γ,

called congruence subgroups of level q, have finite indexes in Γ and therefore are commensurable with Γ. Also, if K is commensurable with Γ, it is also true for every conjugated subgroup M^{−1} ·K ·M with M ∈ Σ. The following easy lemma relates decompositions of double cosets with decompositions of corresponding groups, which is often useful in the theory of Hecke–Shimura rings and Hecke operators.

Lemma 1.3. Let G be a multiplicative group, K and K′ subgroups of G, and g ∈ G. Then the following disjoint decomposition holds

KgK′ = ⋃_{γ_{i}∈g^{−1}Kg∩K′\K′} Kgγ_{i}.

The last lemma implies

Lemma 1.4. If K is commensurable with Γ^{n} and M ∈ Σ^{n},then the double coset KMK is a finite union of left cosets modulo K

Similar lemmas are also true for right cosets.

§1.3. Modular forms

We shall consider complex-valued functions F on the upper half-plane ℍ_{n}. Such a function is called K-automorphic of weight k, where K is a subgroup of G^{n} and k ∈ ℤ, if it is invariant under each of the Petersson operators |_{k}γ of weight k with γ ∈ K. We shall denote by A_{k}(K) the ℂ–linear space of all such functions:

(1-4)  A_{k}(K) = {F : ℍ_{n} → ℂ; F |_{k}γ = F, for all γ ∈ K}.

If M ∈ G^{n}, it follows from (1-4) that the condition F |_{k}γ = F is equivalent with the condition (F |_{k}M)|_{k}M^{−1}γM = F |_{k}M. Therefore we have

A_{k}(M^{−1}KM) = A_{k}(K)|_{k}M, (M ∈ G^{n}).

If a subgroup K ⊂ G^{n} is commensurable with the modular group Γ = Γ^{n}, then it contains subgroups of finite index in Γ, and so there are numbers r ∈ ℕ, called exponents of K, such that each of the matrices of the form

T (rB) = ( \begin{matrix}E_{n} & rB \\ 0 & E_{n} \end{matrix} ) , with B ∈ 𝕊_{n},

belongs to K. If F ∈ A_{k}(K), where K is a group of exponent r, then, in particular,

F |_{k}T (rB) = F (Z + rB) = F (Z)

for every integral symmetric B of order n, and so the function F = F ((z_{ij})) is periodic of period r in each of the complex variables z_{ij} = z_{ji}.

Lemma 1.5. Let F = F ((z_{ij})) be a function on ℍ_{n} which is periodic of period r in each of the variables z_{ij}. Suppose that F is holomorphic on ℍ_{n}. Then F has an expansion of the form

(1-5)  F = ∑_{A∈r^{−1}𝔼_{n}} f(A)e(AZ),  (Z ∈ ℍ_{n})

with constant coefficients f(A), where, for a complex square matrix S,

(1-6)  e(S) = exp(πi · Trace S).

The series (1-5) converges absolutely on ℍ_{n} and uniformly on each of the subsets

ℍ_{n}(ε) = {Z = X + iY ∈ ℍ_{n}; Y ≥ εE_{n}}

with ε > 0.

The series (1-5) and its coefficients f(A) will be called the Fourier expansion and the Fourier coefficients of F, respectively.

A function F : ℍ_{n} → ℂ is called a modular form (resp.,a cusp form) for a subgroup K of G^{n} commensurable with Γ^{n}, if it satisfies the following three conditions:

(1) F ∈ A_{k}(K),i.e. F |_{k}γ = F for all γ ∈ K;

(2) F is holomorphic on ℍ_{n};

(3) for each matrix M ∈ Σ^{n}, the coefficients of the Fourier expansion

F |_{k}M = ∑_{A∈r^{−1}𝔼_{n}} f_{M}(A)e(AZ)

of the function F |_{k}M ∈ A_{k}(M^{−1}KM), where r is an exponent of M^{−1}KM, satisfy

f_{M}(A) = 0, unless A ≥ 0 (resp., unless A > 0).

If n ≥ 2, the condition (3) of the definition of modular forms is a consequence of conditions (1) and (2) (the Koecher effect).We let denote by 𝔐_{k}(K) and 𝔑_{k}(K) the space of all modular forms and all cusp forms of weight k for the group K , respectively. Each of the spaces of modular forms for a group K commensurable with Γ is finite dimensional over ℂ.

§1.4. Petersson scalar product

Every space of cusp forms can be equipped with the structure of a Hilbert space by means of Petersson scalar product, whose definition and properties we shall now recall. By the definition of K–automorphic functions for subgroups K of G^{n}, the values of such functions on ℍ_{n} are determined by their values on a set of representatives of orbits of K on ℍ_{n}. We shall say that a subset D = D(K) of ℍ_{n} is a fundamental domain of K on ℍ_{n}, if it is closed in ℍ_{n}, contains a representative of each K–orbit on ℍ_{n}, and has no inner points belonging to the same K–orbit. In particular, the upper half-plane is the union of images of the fundamental domain under different transformations by elements of K:

ℍ_{n} = ⋃_{γ′∈K′} γ′⟨D(K)⟩,

where K′ is the group of transformations of ℍ_{n} corresponding to K, and the images γ′⟨D(K)⟩ have pairwise no common inner points.Fundamental domains not necessarily exist, but if a fundamental domain D(K) exists for a group K, it also exists for every group of the form M^{−1}KM with M ∈ G^{n} and for every subgroup K_{1} of finite index in K, and one can take

(1-7)  D(M^{−1}KM) = M^{−1}⟨D(K)⟩,

and

(1-8)  D(K_{1}) = ⋃_{γ′∈K′_{1}\K′} γ′⟨D(K)⟩,

where K′ and K′_{1} are the corresponding groups of transformations of the upper half-plain.

We turn now to invariant integration on fundamental domains. Recall first that the measure on ℍ_{n} given by

d^{∗} Z = (detY)^{−(n+1)} ∏_{1≤α≤β≤n} d x_{αβ}d y_{αβ},

where Z = X + iY = (x_{αβ}) + i(y_{αβ}) ∈ ℍ_{n}, and d x_{αβ}, d y_{αβ} are the Euclidean measure on ℝ, is G^{n}–invariant, i.e. it satisfies

(1-9)  d^{∗} M⟨Z⟩ = d^{∗} Z  for every M ∈ G^{n},

as it follows by a routine check from the formula:

(1-10)  Y′ = μ(M)^{t}(C\overline{Z} + D)^{−1}Y (CZ + D)^{−1},

where M = ( \begin{matrix}A & B \\ C & D \end{matrix} ) ∈ G^{n}, Z = X + iY ∈ ℍ_{n}, M⟨Z⟩ = X′ + iY′, and where

the bar means the complex conjugation. It follows from (1-9) and (1-10) that every differential form on ℍ_{n} given by

(1-11)  ω_{k}(F, F′)(Z) = F (Z)\overline{F′(Z)}(det Y)^{k} d^{∗}Z,

for arbitrary functions F and F′ on ℍ_{n}, and any integer k, satisfies

ω_{k}(F, F′)(M⟨Z⟩) = μ(M)^{nk}ω_{k}(F |_{k}M, F′|_{k}M)(Z),

for each matrix M of G^{n}, where |_{k} is the Petersson operator (1-1).

The invariance of the measure d^{∗}Z implies that the volume

(1-12)  v(D(K)) = ∫_{D(K)} d^{∗}Z

of a fundamental domain for a subgroup K of G^{n} does not depend on the choice of fundamental domain and can be refereed as the volume v(K) of K. More generally, if K is a subgroup of G^{n} with a fundamental domain D(K), and F, F′ are two K–automorphic forms of an integral weight k , such that the integral

∫_{D(K)} ω_{k}(F, F′)(Z)

converges absolutely, then it does not depend of the choice of fundamental domain.

Turning to subgroups of G^{n} commensurable with the modular group, we shall remind first the following lemma, which essentially rests on Minkowski reduction theory of quadratic forms.

Theorem 1.6. The modular group Γ = Γ^{n} has a fundamental domain D_{n} = D(Γ^{n}) on ℍ_{n}. The invariant volume v(Γ^{n}) = v(D_{n}) is finite.

The lemma and decompositions (1-8) imply that the volume of every subgroup of G^{n} commensurable with Γ^{n} is finite.

Theorem 1.7. Let K be a subgroup of G = G^{n} commensurable with the modular group Γ = Γ^{n}, k an integer, and F , F′ two modular forms of weigh k for K. Suppose that at least one of the forms is a cusp form. Then the integral

(1-13)  (F, F′) = ν(K′_{1})^{−1} ∫_{D(K_{1})} ω_{k}(F, F′)(Z),

where K_{1} is a subgroup of finite index of K ∩ Γ , K′_{1} = K_{1} ∪ (−E_{n})K_{1}/(±E_{n}) is the transformation group corresponding to K_{1}, ν(K′_{1}) = [Γ′ : K′_{1}] is its index in Γ′ = Γ/(±E_{n}), D(K_{1}) is a fundamental domain of K_{1} on ℍ_{n}, and ω_{k} is the differential form (1-11), satisfies the following rules:

(1) (F, F′) converges absolutely and is independent of the choice of fundamental domain;

(2) (F, F′) is independent neither of the choice of the group K in the commensurability class of Γ with F, F′ ∈ 𝔐_{k}(K), nor of the choice of K_{1}; (3) (F, F′) is linear in F and conjugate linear in F′;

(4) \overline{(F, F′)} = (F′, F);

(5) (F, F) ≥ 0, and (F, F) = 0 only if F = 0;

(6) If M is a matrix of G with rational entries, then

(1-14)  (F |_{k}M, F′|_{k}M) = μ(M)^{−nk}(F, F′),

where |_{k} is the Petersson operator (1-1) , and the functions F |_{k}M, F′|_{k}M are considered as elements of 𝔐_{k}(M^{−1}KM).

The number (F, F′) is called the (Petersson) scalar product of the forms F, F′. The product was originally introduced by Petersson in [Pet(39)] for modular forms in one variable and then extended on Siegel modular forms by Maass in [Ma(51)].

The space 𝔑_{k}(K) of cusp forms of weight k for a subgroup K of G commensurable with Γ is a finite-dimensional Hilbert space with respect to the Petersson scalar product.

§1.5. Hecke–Shimura rings and Hecke operators

Suppose that G is a multiplicative group, K is a subgroup,and Δ is a subsemigroup of G, containing K. We shall say that the pair (K, Δ) is finite (more precisely, left–finite), if for every g ∈ Δ the double coset KgK consists of a finite number of left cosets Kg′. For such a pair (K, Δ), and a commutative and associative ring 𝔸 with the identity we let denote by

W = Lin_{𝔸}(K\Δ)

the free 𝔸–module consisting of all formal finite linear combinations

t = ∑_{i} a_{i}(Kg_{i})

with coefficients a_{i} ∈ 𝔸 of symbols (Kg_{i}) with g_{i} ∈ Δ, which are in one-to-one correspondence with the left cosets Kg of the set Δ modulo the group K. For brevity, we shall refer to these symbols as left cosets of Δ modulo K.

Next, let us denote by

H = HS_{𝔸}(K, Δ)

the submodule of W consisting of all elements which are invariant relative to the natural (right) action of the group K on W:

K ∋ γ : t → tγ = ∑_{i} a_{i}(Kg_{i}γ),

i.e.,

H = {t ∈ W; tγ = t for all γ ∈ K}

The multiplication of elements of H given by the formula

(∑_{i} a_{i}(Kg_{i})) · (∑_{j} b_{j}(Kh_{j})) = ∑_{i,j} a_{i}b_{j}(Kg_{i}h_{j})

does not depend on the choice of representatives g_{i} ∈ Kg_{i} and h_{j} ∈ Kh_{j}, and it turns H into an associative 𝔸–algebra with the identity element (K) = (K · 1_{G}), called the Hecke–Shimura ring (HS–ring) of the pair (K, Δ) over 𝔸. The elements

(1-15)  (g) = (g)_{K} = ∑_{g_{i}∈K\KgK}(Kg_{i}),

which are in one-to-one correspondence with the two-sided cosets KgK of Δ with respect to K, belong to the ring H and form a free 𝔸–basis of H. For brevity, we shall call these elements the double cosets of Δ modulo K.

We suppose next that the semigroup Δ acts (from the right) on a (left) 𝔸–module V by linear operators

|g : v → v|g, (v ∈ V, g ∈ Δ),

satisfying

v|g|g′ = v|gg′, (v ∈ V, g, g′ ∈ Δ),

and we let denote by

𝔐 = {v ∈ V; v|γ = v for all γ ∈ K}

the submodule of K–invariant elements of V. Then for every v ∈ 𝔐 and every t = ∑_{i} a_{i}(Kg_{i}) of H = HS_{𝔸}(K, Δ) the element

v|t = ∑_{i} a_{i}v|g_{i}

does not depend on the choice of representatives g_{i} ∈ Kg_{i}, it again belong to the submodule 𝔐, and we have

v|t|t′ = v|tt′,

for every v ∈ 𝔐, t, t′ ∈ H. Thus, the correspondence t → |t is a linear representation of the HS–ring H on the module 𝔐.The operators |t are called Hecke operators.

The Hecke operators were first studied by Hecke in [He(37)] for spaces of modular forms in one variable,although the idea of the operators was originally used by Mordell in [Mo(17)] in order to prove the conjectures of Ramanujan on multiplicative properties of the function τ(n) and similar functions. General Hecke–Shimura rings over ℤ were originally introduced by Shimura at the end of fifties in a formally different, but equivalent, form as an abstraction of rings of Hecke operators.

In the situation which interests us in this paper—the action of Hecke operators on modular forms—the role of G is played by the general real positive symplectic group of genus n, G^{n} = GSp_{n}^{+}(ℝ), introduced in §1.1.; the role of Δ is played by the semigroup Σ^{n} of integral matrices of G^{n}; and the role of K is played by a subgroup of G^{n} commensurable with the modular group Γ^{n} of genus n. Then, according to Lemma 1.4., the pair (K, Σ) is finite, and so we can define the corresponding HS–ring over arbitrary commutative ring 𝔸. In this paper we take 𝔸 = ℚ, the field of rational numbers, for all HS–rings of symplectic groups, and therefore the indication of the ground ring is, as a rule, omitted. In particular, the HS–ring of K,

(1-16)  H(K) = HS(K, Σ)

is the HS–ring of the pair (K, Σ) over ℚ. In order to define Hecke operators, we shall consider the action of Σ on complex-valued functions F on the upper halfplane ℍ = ℍ^{n}, given by the Petersson operators (1-1). Then, by the above, the subspace (1-4) of all K–invariant with weight k functions is invariant under all of the Hecke operators of weight k,

|_{k}T : F → F |_{k}T = ∑_{i} a_{i}F |_{k}M_{i}, (T = ∑_{i} a_{i}(KM_{i}) ∈ H(K), F ∈ A_{k}(K)),

and the linear operators

(1-17)  |_{k}T : A_{k}(K) → A_{k}(K)  (T ∈ H(K))

satisfy

|_{k}T |_{k}T′ = |_{k}T T′.

Proposition 1.8. The subspaces 𝔐_{k}(K) and 𝔑_{k}(K) of modular forms and cusp forms of A_{k}(K) are invariant under all of the Hecke operators (1-17).

Generally speaking, the whole ring H(K) is noncommutative. However, it turns out that various questions of the theory of Hecke operators can often be reduced to consideration of suitable commutative subrings.Essential part in the reduction is played by the star map

(1-18)  T → T^{∗}  (T ∈ H(K))

of the ring H(K) , which is defined on the elements of the form (1-15) by

(M)_{K}^{∗} = (M^{∗})_{K}, (M ∈ Σ)

where M → M^{∗} is the antiautomorphism (1-3), and then extended on whole of H(K) by linearity.

Theorem 1.9. (1) The star map is a linear antiautomorphism of degree two of the ring H(K); in particular, it satisfies the rules

(T^{∗})^{∗} = T, (T · T_{1})^{∗} = T_{1}^{∗} · T^{∗} (T, T_{1} ∈ H(K)).

(2) The Hecke operators |_{k}T and |_{k}T^{∗} are conjugate with respect to the Petterson scalar product (1-13). More precisely, if F and F′ are two modular forms of weight k for K , and at least one of the forms is a cusp form, then

(F |_{k}T, F′) = (F, F′|_{k}T^{∗}). (T ∈ H(K))

Proof. (1).The linear map

H(K) → HS(K, Σ^{−1}) = H′

given on double cosets by

(M)_{K} → (M^{−1})_{K} (M ∈ Σ)

is an antiisomorphism of ℚ–algebras (see, for example, [An(87), Proposition 3.1.7]). On the other hand , the linear map given on double cosets by

(M^{−1})_{K} → (μ(M)M^{−1})_{K} (M ∈ Σ)

is clearly an isomorphism of H′ onto H(K).Then the assertion follows, since the star map is the composition of the two maps.

(2). It suffices to consider the case T = (M)_{K} with M ∈ Σ. By Lemma 1.3, the number #(K \ KMK) of the left cosets of KMK modulo K is equal to the index ν = [K : K ∩ M^{−1}KM]. The number of the right cosets of KMK modulo K is clearly equal to the number of the left cosets in KM^{−1}K and so equals ν′ = [K : K ∩ MKM^{−1}]. Let D be a fundamental domain for the group K ∩ M^{−1}KM on ℍ_{n}. Since

K ∩ MKM^{−1} = M(K ∩ M^{−1}KM)M^{−1},

it follows, by (1-7), that we can take the set M⟨D⟩ as a fundamental domain for the group K ∩ MKM^{−1}.Then, by (1-8), we get the following relation for the invariant volumes (1-12):

v(D) = νv(D(K)), v(M⟨D⟩) = ν′v(D(K)),

where D(K) is a fundamental domain for K. By (1-9), we conclude that v(D) = v(M⟨D⟩), and so ν = ν′. It follows that the numbers of left and right cosets of KMK modulo K are equal:

(1-19)  #(K \ KMK) = #(KMK/K)  (M ∈ Σ).

Since each of the left cosets clearly meets each of the right cosets,it follows, by (1-19), that there is a set M_{1}, …, M_{ν} of common representatives for left and right cosets of KMK modulo K. Then the set M_{1}^{∗}, …, M_{ν}^{∗} is a set of common representatives for left and right cosets of KM^{∗}K modulo K. Finally, by (1-14) and (1-2), we get

(F |_{k}(M), F′) = (∑_{i} F |_{k}M_{i}, F′) = μ(M)^{−nk}(F, ∑_{i} F′|_{k}M_{i}^{−1})

= (F, ∑_{i} F′|_{k}M_{i}^{∗}) = (F, F′|_{k}(M)^{∗}).

□

In this paper we practically do not touch, except for §2.4, the important question on action of Hecke operators on Fourier coefficients of modular forms, leaving it to our forthcoming papers on zeta functions. It turns out to be convenient for the applications to normalize the Hecke operators on modular forms for groups K commensurable with Γ^{n} by certain constant factors. For T = ∑_{i} a_{i}(KM_{i}) ∈ H(K) we define the normalized Hecke operator ∥_{k} T on 𝔐_{k}(K) by

(1-20)  F ∥_{k} T = ∑_{i} a_{i}μ(M_{i})^{nk−⟨n⟩}F |_{k}M_{i},  (F ∈ 𝔐_{k}(K))

Since, clearly,

F ∥_{k} (M)_{K} = μ(M)^{nk−⟨n⟩}F |_{k}(M)_{K}, (F ∈ 𝔐_{k}(K), M ∈ Σ^{n}),

it follows that the normalized operators map the spaces 𝔐_{k}(K) and 𝔑_{k}(K) into themselves, the map T →∥_{k} T is a linear representation of the ring H(K), and it satisfies

(1-21)  (F ∥_{k} T, F′) = (F, F′ ∥_{k} T^{∗}),

if T ∈ H(K), and F, F′ ∈ 𝔑_{k}(K).We note that Hecke operators considered in the books [An(87)] and [An-Zhu(90),(95)] are the normalized operators.

§1.6. Reducible, regular, and singular subrings

We close this chapter with several simple but useful definitions concerning Hecke– Shimura rings of discrete symplectic groups K commensurable with the modular group Γ^{n}.

Standard applications of Hecke operators are usually based on relations between eigenvalues of the operators on spaces of modular forms and Fourier coefficients of eigenfunctions. To do this one have first to compute Fourier coefficients of images under Hecke operators in terms of Fourier coefficient of originals. Since a Hecke operator |_{k}T with T ∈ H(K) is a linear combination of the Petersson operators |_{k}M with M ∈ Σ, the problem reduces to that for the Petersson operators. If F ∈ 𝔐_{k}(K), then the only known general situation when the Fourier coefficients of F |_{k}M can be directly calculated in terms of Fourier coefficients of F is the case when the operator |_{k}M can be reduced to a ”triangular operator” |_{k}M′ with a matrix M′ belonging to the semigroup

(1-22)  Σ_{0} = Σ_{0}^{n} = \{( \begin{matrix}A & B \\ 0 & D \end{matrix} ) ∈ Σ\}

of ”triangular” matrices of Σ. It happens if, say, M ∈ KΣ_{0}, since then

F |_{k}M = F |_{k} ( \begin{matrix}A & B \\ 0 & D \end{matrix} ) = (detD)^{−k}F (AZD^{−1} + BD^{−1}),

and the Fourier coefficients of the last function can be easily expressed through those of F. These arguments justify the following definition. A nonzero element

(1-23)  T = ∑_{i} a_{i}(KM_{i}) ∈ H(K),

where all of the coefficients are nonzero and all of the left cosets are pairwise distinct (the shortest form) is said to be reducible, if each of the left cosets KM_{i} contains a ”triangular” representative, i. e.

KM_{i} ⋂ Σ_{0} ≠ ∅.

If T is reducible, one can assume that all of M_{i} belong to Σ_{0}, in which case we shall say that T is written in the triangular form, or is triangular. Since the product of two triangular elements is clearly triangular, it follows that the set

R(D) = {T ∈ D; T is reducible }

of all reducible elements contained in a subring D of H(K) is a subring, which will be called the reducible subring of D.

It follows from definitions that an element (M)_{K} of H(K) of the form (1-15) is reducible if and only if KMK ⊂ KΣ_{0}. In this case we shall say that the element M of Σ is K-reducible. The set

R(K) = {M ∈ Σ; KMK ⊂ KΣ_{0}}

of all K–reducible elements of Σ is clearly a subsemigroup of Σ, the K-reducible semigroup. If

D = HS(K, Σ′) ⊂ H(K)

is the HS–ring of a subsemigroup Σ′ of Σ, then the reducible subring R(D) of D is nothing else but the HS–ring of the pair (K, Σ′ ∩ R(K)).

In applications one can usually limit oneself to consideration of appropriate reducible subrings or their simple extensions. We shall see in the following chapter how it looks like in the case of the groups K = Γ_{0}^{n}(q).

Properties of left and double cosets of matrices M of Σ = Σ^{n} for a congruence subgroup of a level q > 1 depend very much on common divisors of their multiplier μ(M) and the level. The two extreme cases, the regular case, when

(1-24)  gcd(μ(M), q) = 1,

and the singular case, when

(1-25)  μ(M)|q^{∞},

that is each of prime factors of μ(M) divides q, are of especial importance, since the general case is usually reducible to the particular cases. The reduction is based on the following consequence of the theory of elementary divisors for symplectic groups.

Lemma 1.10. Let M ∈ Σ = Σ^{n}. Suppose that the multiplier μ(M) is a product of two coprime factors: μ(M) = ab, gcd(a, b) = 1. Then there are matrices M_{1}, M_{2} ∈ Σ such that M = M_{1}M_{2}, and μ(M_{1}) = a, μ(M_{2}) = b. If M = M′_{1}M′_{2} is another such factorization, then M′_{1} = M_{1}γ, M′_{2} = γ^{−1}M_{2} with γ ∈ Γ^{n}.

We shall say that a matrix M of Σ = Σ^{n} is q-regular (resp., q-singular), if it satisfies (1-24) (resp., (1-25)) and denote by

(1-26)  Σ_{(q)} = Σ_{(q)}^{n} and Σ_{q} = Σ_{q}^{n}

the sets of all q-regular and q-singular matrices of Σ, respectively. An element of a HS–ring H(K) is called q-regular (resp., q-singular) if each of the the cosets entering in the shortest form (1-23) consists of q–regular (resp., q–singular) matrices.The subrings

(1-27)  H_{(q)}(K) = HS(K, Σ_{(q)}) and H_{q}(K) = HS(K, Σ_{q})

of all q–regular and q–singular elements of H(K) are called the q-regular and the q-singular HS–ring of the group K, respectively. A subring D of H(K) is said to be q-regular (resp., q-singular), if it is contained in H_{(q)}(K) (resp., H_{q}(K)). If q is fixed, the indication of q , will be, as a rule, omitted.

Chapter 2

Regular reducible rings and operators for Γ_{0}^{n}(q)

In this chapter we shall remind the regular theory for the groups

(2-1)  K = Γ_{0}^{n}(q) = \{( \begin{matrix}A & B \\ C & D \end{matrix} ) ∈ Γ^{n}; C ≡ 0 (mod q)\}

with n, q ∈ ℕ. The numbers n and q are fixed through out the chapter and often are not indicated. For details and omitted proofs see [An(87)] or [An-Z(90)]. The reader may also find there the regular theory for more general congruence subgroups.

§2.1. Global rings

Lemma 2.1. The set R^{∗}(K) of all q-regular and K-reducible elements of Σ^{n} for the group (2-1) has the form

(2-2)  R^{∗}(K) = \{( \begin{matrix}A & B \\ C & D \end{matrix} ) ∈ Σ_{(q)}^{n}; C ≡ 0 (mod q)\};

in particular, R^{∗}(Γ^{n}) = Σ^{n}.

Proof . For the inclusion ⊃, see, for example, [An(87), Lemma 3.3.4]. The inverse inclusion follows from the definition of K-reducible elements.

□

It follows from the lemma that the (maximal) regular reducible subring of the ring H(K) is the ring

(2-3)  H_{rr} = H_{rr}^{n}(q) = HS(K, R^{∗}(K)).

According to general properties of HS–rings, the elements (M)_{K} of the form (1-15) associated to different double cosets of R^{∗}(K) modulo K form a free basis of the ring H_{rr} over ℚ. In order to describe the basis we shall recall the following lemma on elementary divisors for the symplectic group over ℤ.

Lemma 2.2. Each of the double cosets KMK with M ∈ R^{∗}(K) for the group (2-1) contains the unique representative of the form

(2-4)  sd(M) = diag(d_{1}, …, d_{n}; e_{1}, …, e_{n}),

where d_{i}, e_{j} ∈ ℕ with d_{i}|d_{i+1}, d_{n}|e_{n}, e_{i+1}|e_{i}, d_{i}e_{i} = μ(M). If M, M′ ∈ R^{∗}(K), and the multipliers μ(M) and μ(M′) are coprime, then

(2-5)  sd(MM′) = sd(M)sd(M′).

The diagonal matrix sd(M) is called the matrix of symplectic divisors of M , and the positive integers d_{1} = d_{1}(M), …, d_{n} = d_{n}(M), e_{1} = e_{1}(M), …, e_{n} = e_{n}(M) are the symplectic divisors of M .

Theorem 2.3. The star map (1-18) is identical on the double cosets (M)_{K} contained in the regular reducible ring H_{rr} of the group (2-1):

(2-6)  (M)_{K}^{∗} = (μ(M)M^{−1})_{K} = (M)_{K},  (M ∈ R^{∗}(K));

in partiqular, the ring H_{rr} is commutative.

A problem of explicit multiplication rules for specific elements of H_{rr} can be quite complicated, except for certain special cases, which, in partiqular, allow one to bring about an impotant reduction of the problem.

Note, first of all, that, for every M ∈ Σ and r ∈ ℕ, one has the rules

(2-7)  [r](M)_{K} = (M)_{K}[r] = (rM)_{K},

where

(2-8)  [r] = [r]_{K} = (rE_{2n})_{K} = (K(rE_{2n}));

in particular, the element [1] is the unity element of H(K). Next we have the following

Proposition 2.4. Let us suppose that the quotients of the divisors e_{1}(M)/d_{1}(M) and e_{1}(M′)/d_{1}(M′) of two matrises M, M′ ∈ R^{∗}(K) are coprime. Then the following relation holds:

(2-9)  (M)_{K}(M′)_{K} = (MM′)_{K}.

If a diagonal matrix diag(d_{1}, …, d_{n}; e_{1}, …, e_{n}) is the matrix of symplectic divisors of a matrix M ∈ R^{∗}(K), we shall write

(2-10)  T (d_{1}, …, d_{n}; e_{1}, …, e_{n}) = (sd(M))_{K} = (M)_{K},

and for a number a ∈ ℕ prime to q we denote by T (a) = T_{K}(a) the sum of all different double cosets (M)_{K} with M ∈ R^{∗}(K) and μ(M) = a:

(2-11)  T (a) = ∑_{KMK⊂R^{∗}(K),μ(M)=a}(M)_{K}.

Then, by (2-5), (2-7), and (2-9), we get the relations

(2-12)  T (d_{1}, …, e_{n})T (d′_{1}, …, e′_{n}) = T (d_{1}d′_{1}, …, e_{n}e′_{n}),

provided that the numbers e_{1}/d_{1} and e′_{1}/d′_{1} are coprime, and the relations

(2-13)  T (a)T (b) = T (ab), if a, b are coprime and prime to q.

For a prime number p not dividing q ,we define the matrix of symplectic p-divisors of M by

(2-14)  sd_{p}(M) = diag(p^{ν_{p}(d_{1})}, …, p^{ν_{p}(e_{n})}),

where ν_{p}(r) is the p-order of r. The matrix (2-14) is clearly the matrix of symplectic divisors of a matrix M_{p} ∈ R^{∗}(K) (for example, M_{p} = sd_{p}(M)):

(2-15)  sd_{p}(M) = sd(M_{p}),  (M, M_{p} ∈ R^{∗}(K), gcd(p, q) = 1).

The relations (2-9) imply the factorization

(2-16)  (M)_{K} = ∏_{p}(M_{p})_{K},

where p runs over all prime divisors of μ(M).

The p-components (M_{p})_{K} of double cosets (M)_{K} ∈ H_{rr} span clearly a subring H_{p}, the (reducible) p-ring, which can be interpreted by itself as the HS–ring of the pair

(K, R_{p}(K) = {M ∈ R(K); μ|p^{∞}}) :
(2-17)  H_{p} = H_{p}^{n}(q) = HS(K, R_{p}(K)).

The factorization (2-16) imply the following

Theorem 2.5. The ring H_{rr} is generated by the p-rings H_{p} for all primes p not dividing q.

The factorization of elements of H_{rr} into p-factors can be naturally reformulated in terms of Euler decomposition of formal symplectic zeta functions. For example, it follows from (2-13) that

T (∏_{i} p_{i}^{ν_{i}}) = ∏_{i} T (p_{i}^{ν_{i}}),

if p_{i} runs over a finite set of different prime numbers not dividing q, which implies the following (formal) Euler decomposition of the formal Dirichlet series over H_{rr} with the coefficients T (a) for a prime to q:

(2-18)  ∑_{a∈ℕ, gcd(a,q)=1} \frac{T (a)}{a^{s}} = ∏_{\begin{matrix}p prime, \\ p∤q \end{matrix}}(∑_{δ=0}^{∞} T (p^{δ}) · p^{−δs}).

We shall see in the next section that each of the formal power series

(2-19)  Z_{p}(v) = Z_{p}^{n}(v) = ∑_{δ=0}^{∞} T (p^{δ})v^{δ}

is formally a rational fraction over the ring H_{p}.

§2.2. Local rings

Theorem 2.6. Let p be a prime number not dividing q. Then the following assertions holds:

(1) If T = ∑_{i} a_{i}(KM_{i}) is an element of H_{p}^{n}(q) with triangular representatives M_{i} , then the map T → i(T) = ∑_{i} a_{i}(Γ^{n}M_{i}) is an isomorphism of H_{p}^{n}(q) onto H_{p}^{n}(1);

(2) The ring H_{p}^{n}(q) is generated over ℚ by the n + 1 elements

(2-20) \{ \begin{matrix}T^{n}(p) & = T (\underbrace{1, …, 1}_{n}; \underbrace{p, …, p}_{n}), \\ T_{i}^{n}(p^{2}) & = T (\underbrace{1, …, 1}_{n−i}, \underbrace{p, …, p}_{i}; \underbrace{p^{2}, …, p^{2}}_{n−i}, \underbrace{p, …, p}_{i}),  (i = 1, …, n); \end{matrix}

(3) The elements (2-20) are algebraically independent over ℚ.

The problem to express elements of the ring H_{p} = H_{p}^{n}(q) in terms of generators (2-20) is not easy one. But it can be considerably simplified, if one uses another realization of the rings given by spherical polynomials.

Since the ring H_{p} is reducible, each of the left cosets KM with K = Γ_{0}^{n}(q) and M ∈ R_{p}(K) contains a triangular representative

(2-21)  ( \begin{matrix}A & B \\ 0 & D \end{matrix} ) , with ^{t}AD = p^{δ}E_{n},  (p^{δ} = μ(M)).

Since the group K contains the subgroup

(2-22)  V^{n} = \{v(U) = ( \begin{matrix}^{t}U^{−1} & 0 \\ 0 & U \end{matrix} ); U ∈ GL_{n}(ℤ)\} ,

one can replace the matrix D in (2-21) by any of the matrices UD with U ∈ GL_{n}(ℤ) and so assume that D is an upper triangular matrix of the form

(2-23) D = (d_{ij}), where d_{ij} = 0 , if i > j, and d_{ii} = p^{d_{i}},  (i, j = 1, …, n).

We set then

ω(KM) = x_{0}^{δ} ∏_{i=1}^{n}(x_{i}p^{−i})^{d_{i}},

where x_{0}, x_{1}, …, x_{n} are algebraically independent over ℚ. By extending the map ω by linearity on the whole ring H_{p}, we obtain a linear map

(2-24)  ω = ω_{p} : H_{p} → ℚ[x_{0}, …, x_{n}]

of the ring H_{p} into the ring of polynomials in n + 1 independent variables over ℚ, the spherical map. The polynomials ω(T) with T ∈ H_{p} are called the spherical polynomials.

Theorem 2.7. The spherical mapping ω_{p} is an isomorphism of the ring H_{p} = H_{p}^{n}(q) with the subring ℚ^{W} [x_{0}, …, x_{n}] of ℚ[x_{0}, …, x_{n}] consisting of all polynomials invariant with respect to the group W = W_{n} of ℚ–automorphisms of the rational function field ℚ(x_{0}, …, x_{n}) generated by all permutations in the variables x_{1}, …, x_{n} and by the automorphisms τ_{1}, …, τ_{n} satisfying

τ_{i}(x_{0}) = x_{0}x_{i}, τ_{i}(x_{i}) = x_{i}^{−1}, τ_{i}(x_{j}) = x_{j} (j ≠ 0, i).

The ring ℚ^{W} [x_{0}, …, x_{n}] is generated over ℚ by n + 1 polynomials

t = t^{n}(x_{0}, …, x_{n}) = x_{0} ∏_{i=1}^{n}(1 + x_{i}),

ρ_{a} = ρ_{a}^{n}(x_{0}, …, x_{n}) = x_{0}^{2}x_{1} … x_{n}r_{a}^{n}(x_{1}, …, x_{n}) (a = 0, 1, …, n − 1),

where

∑_{a=0}^{2n}(−1)^{a}r_{a}^{n}(x_{1}, …, x_{n})v^{a} = ∏_{i=1}^{n}(1 − x_{i}v)(1 − x_{i}^{−1}v).

With the help of spherical map one can prove the following formulae for the formal power series (2-19):

(2-25)  Z_{p}^{1}(v) = Q_{p}^{1}(v)^{−1}, where Q_{p}^{1}(v) = [1] − T^{1}(p)v + p[p]v^{2};

Z_{p}^{2}(v) = Q_{p}^{2}(v)^{−1}([1] − p^{2}[p]v^{2}),

where

(2-26)  Q_{p}^{2}(v) = [1] − T^{2}(p)v + q_{2}(p)v^{2} − p^{3}[p]T^{2}(p)v^{3} + p^{6}[p^{2}]v^{4}

with q_{2}(p) = pT (1, p, p^{2}, p) + p(p^{2} + 1)[p]; and for general n,

Z_{p}^{n}(v) = Q_{p}^{n}(v)^{−1}N_{p}^{n}(v),

where

(2-27)  Q_{p}^{n}(v) = ∑_{i=0}^{m}(−1)^{i}q_{i}^{n}(p)v^{i}  (m = 2^{n})

is the polynomial of degree m = 2^{n} over H_{p}^{n}(q) satisfying

ω(Q_{p}^{n})(v) = ∑_{i=0}^{m}(−1)^{i}ω(q_{i}^{n}(p))v^{i}

= (1 − x_{0}v) ∏_{r=1}^{n} ∏_{1≤i_{1}<···<i_{r}≤n}(1 − x_{0}x_{i_{1}} ⋯ x_{i_{r}}v),

ω = ω_{p} being the spherical mapping (2-24), and where N_{p}^{n}(v) is a polynomial over H_{p}^{n}(q) of degree at most 2^{n} − 2 . The coefficients q_{i}^{n}(p) satisfy the rules

q_{m−i}^{n}(p) = (p^{⟨n⟩}[p])^{(m/2)−i}q_{i}^{n}(p), (0 ≤ i ≤ m = 2^{n}),
q_{0}^{n}(p) = [1], and q_{m}^{n}(p) = (p^{⟨n⟩}[p])^{m/2}.

The formula (2-25) goes back to Hecke and was formally proved along with the formula for Z_{p}^{2}(v) in [Sh(63)] and [Sa(63)]. The general case was considered in [An(69)] and [An(70)].

§2.3. Parabolic embeddings

It is often useful to split elements of a symplectic HS–ring on certain elementary components. Howerever, in general, the components do not belong to the original ring. According to our approach, they live in HS–rings of the ”triangular” subgroup

Γ_{0} = Γ_{0}^{n} = \{( \begin{matrix}A & B \\ 0 & D \end{matrix} ) ∈ Γ^{n}\}

of the modular group Γ = Γ^{n}.

It is easy to see that the pair (Γ_{0}, Σ_{0}), where Σ_{0} = Σ_{0}^{n} is defined by (1-22), is finite, and therefore one can introduce the corresponding Hecke–Shimura ring over ℚ,

(2-28)  L = L^{n} = HS(Γ_{0}^{n}, Σ_{0}^{n}),

along with its q-regular and q-singular subrings

(2-29)  L_{(q)} = L_{(q)}^{n} = HS(Γ_{0}^{n}, Σ_{0,(q)}^{n}),  L_{q} = L_{q}^{n} = HS(Γ_{0}^{n}, Σ_{0,q}^{n}),

for q ∈ ℕ, where

Σ_{0,(q)} = Σ_{0,(q)}^{n} = Σ_{0}^{n} ⋂ Σ_{(q)}^{n}, Σ_{0,q} = Σ_{0,q}^{n} = Σ_{0}^{n} ⋂ Σ_{q}^{n}

(see (1-22) and (1-26)).

Let D be a reducible subring of the ring H(K) (see (1-16)) for K = Γ_{0}^{n}(q), and let T = ∑_{i} a_{i}(KM_{i}) be an element of D written in the triangular form. Since K ∩ Σ_{0} = Γ_{0}, the map

j : T → j(T) = T̂ = ∑_{i} a_{i}(Γ_{0}M_{i})

does not depend on the choice of triangular representatives M_{i} ∈ KM_{i}, the image T̂ belongs to L, and T̂ = 0, if and only if T = 0. It follows from the definition of multiplication in HS–rings, that the map T → T̂ is a linear ring homomorphism. Thus, we get the homomorphic inclusion

(2-31)  j : D ↪ D̂ ⊂ L

of the ring D into L, which will be referred as parabolic embedding.

We now turn to multiplicative properties of the ring L . Contrary to reduced rings we have considered earlier, the ring L is not commutative and contains zero divisors. Howerever, some important properties remain valid and, as a matter of fact, we shall need only certain subrings and submodules of L with good properties.

Note, first of all, that the star map

(2-32)  X → X^{∗}  (X ∈ L)

of the ring L, which is defined on double cosets by

(2-33)  (M)_{Γ_{0}}^{∗} = (μ(M)M^{−1})_{Γ_{0}}

and then extended on whole of L by linearity, is again a linear antiautomorphism of the second order of the ring L. The same is obviously true for the subrings L_{(q)} and L_{q}. The images of double cosets (M)_{K} ∈ H_{rr}^{n}(q) under parabolic embedding (2-30) are invariant under the star map:

(2-34)  j((M)_{K})^{∗} = j((M)_{K})  (M ∈ R^{∗}(K)).

Note that the image j((M)_{K}) is not necessarily a double coset modulo Γ_{0}, but a sum of such cosets.

Next, by analogy with (2-7), we have the relations

(2-35)  ˆ[r](M)_{Γ_{0}} = (M)_{Γ_{0}} ˆ[r] = (rM)_{Γ_{0}},  (r ∈ ℕ, M ∈ Σ_{0}),

where

(2-36)  ˆ[r] = [r]_{Γ_{0}} = (Γ_{0}(rE_{2n}));

in particular, the element ˆ[1] is the unity element of the ring L.

It turns out that the ring L contains not only HS–rings of the symplectic group, but also subrings which are naturally isomorphic to certain subrings of HS–rings of the general linear group. This allows one to reduce certain questions of the theory of Hecke–Shimura rings and Hecke operators for symplectic group to similar questions for general linear group.

Let us consider the elements of L = L^{n} defined by

(2-37)  Π̂(a) = Π̂^{n}(a) = (( \begin{matrix}E_{n} & 0 \\ 0 & aE_{n} \end{matrix} ))_{Γ_{0}}

with a ∈ ℕ. The elements have the following decomposition into left cosets modulo Γ_{0}^{n}:

(2-38)  Π̂(a) = ∑_{B∈𝕊/a𝕊} (Γ_{0} ( \begin{matrix}E & B \\ 0 & aE \end{matrix} )) ,

where E = E_{n} and 𝕊 = 𝕊_{n} (see Notation).It follows that

(2-39)  Π̂(a)Π̂(b) = Π̂(ab)  (a, b ∈ ℕ)

The elements Π̂(a) will be called the Frobenius elements of the ring L. We now consider centralizers of Frobenius elements in the rings L, L_{(q)}, and L_{q}:

(2-40)

Ĉ^{n} = {X ∈ L; Π̂(a)X = XΠ̂(a), a ∈ ℕ} ,
Ĉ_{(q)}^{n} = {X ∈ L_{(q)}; Π̂(a)X = XΠ̂(a), a ∈ ℕ, gcd(a, q) = 1} ,
Ĉ_{q}^{n} = {X ∈ L_{q}; Π̂(a)X = XΠ̂(a), a|q^{∞}} .

The following theorem is proved in [An(87)] and [An-Z(90)] only for rings L_{(q)}, but the same proof with obvious modifications goes through also for the rings L and L_{q}.

Theorem 2.8. The ring Ĉ (resp.,Ĉ_{(q)}, Ĉ_{q}) is spanned over ℚ by all of the double cosets modulo Γ_{0} of the matrices of Σ_{0} (resp., Σ_{0,(q)}, Σ_{0,q}) of the form

(2-41)  M = U(μ, D) = ( \begin{matrix}μ ·^{t} D^{−1} & 0 \\ 0 & D \end{matrix} ) ,

satisfying the condition

(2-42)  μ|d(D)^{2},

where d(D) is the greatest common divisor of entries of D. For every such M, the double coset (M)_{Γ_{0}} has the following decomposition into left cosets:

(2-43)  (M)_{Γ_{0}} = ∑_{D′,B}(Γ_{0}U(μ, D′) ( \begin{matrix}E & B \\ 0 & E \end{matrix} )),

where D′ runs a set of representatives of the left cosets modulo the group

(2-44)  Λ = Λ^{n} = GL_{n}(ℤ),

contained in the double coset ΛDΛ , and B runs over a set of representatives for the additive group 𝕊 = 𝕊_{n} modulo the subgroup μ^{−1} ·^{t}D′𝕊D′.

Let now

X = ∑_{i} a_{i} (Γ_{0} ( \begin{matrix}μ_{i} ·^{t} D_{i}^{−1} & B_{i} \\ 0 & D_{i} \end{matrix} ))

be an element of the ring L. We set

(2-45)  Ω(X) = ∑_{i} a_{i}{μ_{i}}(ΛD_{i}),

where the coefficients a_{i}{μ_{i}} are considered as elements of the semigroup ring ℚ[ℕ] of the semigroup of positive integers over the field ℚ. It is easy to see that in fact Ω(X) belongs to the HS–ring of the pair (Λ, Δ), where

(2-46)  Δ = Δ^{n} = {D ∈ ℤ_{n}^{n}; det D ≠ 0},

over ℚ[ℕ], and the map

(2-47)  Ω : L → HS_{ℚ[ℕ]}(Λ, Δ)

is a ℚ–linear ring homomorphism.

Proposition 2.9. The restriction Ω|Ĉ of the homomorphism Ω to the centralizer Ĉ of Frobenius elements in L is a monomorphism; in partiqular, Ĉ is a commutative integral domain.

Note that HS–ring of the pair (Λ^{n}, Δ^{n}) over any commutative ring without zero divisors is commutative and contains no zero divisors.

Let us turn now to local triangular rings. For each prime number p, the local subring L_{p} = L_{p}^{n} of the ring L consists of those linear combinations of left (or double) cosets of L whose multipliers are powers of p.

According to Proposition 2.9, the local centalizers

(2-48)  Ĉ_{p} = {X ∈ L_{p}; Π̂(p)X = XΠ̂(p)} = Ĉ ∩ L_{p}

are commutative integral domains.

Theorem 2.10. The ring Ĉ (resp., Ĉ_{(q)}, Ĉ_{q}) is generated by the subrings Ĉ_{p}, where p runs over all prime numbers (resp.,all prime numbers not dividing q, all prime numbers dividing q ).

Although the ring L_{p} is noncommutative and contains zero divisors, it turns out that each element of L_{p} can be ”projected” into commutative integral domain Ĉ_{p} and, moreover, each element of Ĥ_{p} = H_{p}(1) can be restored by its projections:

Proposition 2.11. Let X = ∑_{i} a_{i}(Γ_{0}M_{i}) be an element of L_{p} with μ(M_{i}) = p^{ν_{i}} , and let d be an integer satisfying d ≥ ν_{i} for all i. Then

(2-49)  XΠ̂(p)^{d} ∈ Ĉ_{p}.

If, moreover, X ∈ Ĥ_{p} , then

(2-50)  Π_{+}^{−d} · XΠ̂(p)^{d} = X,

where Π_{+}^{−d} = Π̂(p)^{(−d)} are independent of X elements of the extension Ĉ′_{p} = Ĉ[[p]^{−1}] of the ring Ĉ_{p}.

The elements Π_{+}^{−d} are so-called ”negative powers” of Frobenius elements. For their definition and properties see, for example, [An(87), §3.5.1].

It is clear from the definitions that the image Ĥ_{p}(q) of a p-ring of the group Γ_{0}^{n}(q) is contained in L_{p}. Moreover, it follows from the first part of Theorem 2.6, that the parabolic embedding of the ring H_{p}(q) into the ring L_{p} can be presented as a composition of the isomorphism of H_{p}^{n}(q) onto H_{p}(1) defined in the theorem and the parabolic embedding of the ring H_{p}(1), in particular the image

(2-51)  Ĥ_{p}(q) = Ĥ_{p} = Ĥ_{p}(1)

is independent of the choice of q not divisible by p.

As to the relations between introduced local rings for different primes, we have the following Proposition 2.12. Let p and p′ be two different prime numbers. Then each element of Ĥ_{p} commutes with each element of Ĉ_{p′}.

We close this section with the following theorem which is crucial for our forthcoming constructions.

Theorem 2.13. The Frobenius element Π̂(p) = Π̂^{n}(p) , where p is a prime number, satisfies the equation

(2-52)  ∑_{j=0}^{m}(−1)^{j}q̂_{j}^{n}(p)Π̂(p)^{m−j} = 0,  (m = 2^{n}),

where q̂_{j}^{n}(p) ∈ Ĥ_{p} are images under parabolic embedding of coefficients of the polynomial (2-27) over the ring H_{p}(q) for a q not divisible by p.

§2.4. Regular reducible Hecke operators and triangular operators

We consider first the representation of the regular reducible ring H_{rr}(q) = H_{rr}^{n}(q) on the space 𝔑_{k}(K) of cusp forms of weight k for the group K = Γ_{0}^{n}(q) given by Hecke operators, as they were defined in §1.5.

According to Theorem 2.3 and Theorem 1.9(2), the Hecke operators on 𝔑_{k}(K) corresponding to elements of H_{rr}(q) commute with each other and are selfadjoint with respect to the Petersson scalar product. Since the space of cusp forms is finite-dimensional, this implies the following theorem.

Theorem 2.14. The space 𝔑_{k}(K) has an orthogonal basis consisting of common eigenfunctions for all of the Hecke operators |_{k}T and ∥_{k}T with T ∈ H_{rr}(q).

Hecke operators for the group Γ_{0} = Γ_{0}^{n} not necessarily map the spaces 𝔐_{k}(K) into themselves. Therefore, in order to apply the relations between elements of HS–rings of the groups K and Γ_{0} cited in §2.3, one has first to extend the spaces.

Every function F of 𝔐_{k}(K) satisfies, in particular, the relations

(2.53)  F |_{k}M = (det D)^{−k}F (M⟨Z⟩) = F for each M = ( \begin{matrix}A & B \\ 0 & D \end{matrix} ) ∈ Γ_{0}.

Besides, it can be decomposed into an absolutely convergent on ℍ_{n} Fourier series of the form

(2.54)  F (Z) = ∑_{A∈𝔼_{n}, A≥0} f(A)e(AZ)

(see Notation and (1-6)) with constant Fourier coefficients f(A). These remarks allows one to define the space T_{k} = T_{k}^{n} of modular forms of weight k for the group Γ_{0} as the space of all absolutely convergent on ℍ_{n} Fourier series of the form (2-54), which satisfy the relations (2-53). Note that the space T_{k} depends only on parity of k. It is clear that

(2-55)  𝔐_{k}(K) ⊂ T_{k}.

The HS–ring L of the group Γ_{0} defined by (2-28) naturally acts on T_{k} ,say, by normalized Hecke operators:

(2-56)  L ∋ T = ∑_{i} a_{i}(Γ_{0}M_{i}) : F → F ∥_{k}T = ∑_{i} a_{i}μ(M_{i})^{nk−⟨n⟩}F |_{k}M_{i}.

The mapping T → ∥_{k}T is a linear representation of L on T_{k}, which is compatible with the parabolic embedding

H_{rr}(q) → Ĥ_{rr}(q) ⊂ L,

defined by (2-30), in the sense that

(2-57)  F ∥_{k}T = F ∥_{k}T̂  (F ∈ 𝔐_{k}(K), T ∈ H_{rr}(q)).

The following lemma shows that the normalized Hecke operators corresponding to elements of the centralizer C of Frobenius elements Π(a) (see (2-38) and (2-40)) have an especially simple action on Fourier coefficients of functions in T_{k}.

Lemma 2.15. Let F be a function of T_{k} with Fourier coefficients f(A), and let M = U(μ, D) be a matrix of Σ_{0} of the form (2-41) satisfying (2-42). Then the Fourier coefficients (f∥_{k}(M)_{Γ_{0}})(A) of the function F ∥_{k}(M)_{Γ_{0}} can be given by the formula

(2-58) (f∥_{k}(M)_{Γ_{0}})(A)

= μ^{nk−n(n+1)}| det D|^{n+1} ∑_{D′∈Λ\ΛDΛ}(det D)^{−k}f(μ^{−1}D′A ·^{t}D′).

In particular,

(2-59)  (f∥_{k}Π̂(a))(A) = f(aA)  (a ∈ ℕ, A ∈ 𝔼_{n}).

As an illustration of the technique of parabolic embeddings we shall prove here a rationality theorem originally proved for the group Γ^{n} by Zharkovskaya in [Zha(75)].

Theorem 2.16. Let

F = ∑_{A∈𝔼_{n}, A≥0} f(A)e(AZ) ∈ 𝔐_{k}(K)

be an eigenfunction for all Hecke operators ∥_{k}T with T ∈ H_{p}^{n}(q), where p is a prime number not dividing q:

F ∥_{k}T = λ(T)F (T ∈ H_{p}^{n}(q)).

Then each of the formal power series

∑_{δ=0}^{∞} f(p^{δ}A)v^{δ} with A ∈ 𝔼_{n}

is formally a rational fraction with the denominator

(λQ_{p}^{n})(v) = ∑_{i=0}^{m}(−1)^{i}λ(q_{i}^{n}(p))v^{i} (m = 2^{n}),

where Q_{p}^{n}(v) is the polynomial (2-27), and with a numerator of degree at most 2^{n}−1. Proof .We have

\begin{aligned}(∑_{i=0}^{m}(−1)^{i}λ(q_{i}^{n}(p))v^{i})(∑_{δ=0}^{∞} f(p^{δ}A)v^{δ}) & = ∑_{i,δ}(−1)^{i}(f∥_{k}q_{i}^{n}(p))(p^{δ}A)v^{i+δ} \\ & = ∑_{i,δ}(−1)^{i}(f∥_{k}q̂_{i}^{n}(p)Π̂(p)^{δ})(A)v^{i+δ}, \end{aligned}

where we have used (2-57) and (2-59). If i + δ = r ≥ m, then

∑_{i+δ=r}(−1)^{i}q̂_{i}^{n}(p)Π̂(p)^{δ} = ∑_{i=0}^{m}((−1)^{i}q̂_{i}^{n}(p)Π̂(p)^{m−i})Π̂(p)^{r−m} = 0,

by (2-52).

□

Chapter 3

Singular Hecke–Shimura rings for Γ_{0}^{n}(q)

Our theory of singular HS–rings for the groups Γ_{0}^{n}(q) rests on two fundamental facts: the first is that the centralizer of the Frobenius element Π(p) in the reducible p-subring for Γ_{0}^{n}(q), where p is a prime divisor of q, is naturally isomorphic to the centralizer of Π̂(p) in the triangular ring L_{p}, which follows from Theorem 3.3 and Theorem 2.8; the second is that the elements Π(p) with primes p dividing q but not dividing q/p are invertible, which is based on properties of neutral HS– rings considered in §3.2. These circumstances will allow us to construct in §3.3 natural embeddings of regular reducible p-rings for Γ_{0}^{n}(q/p) into p-subring for Γ_{0}^{n}(q), provided that p|q and p^{2} ∤ q, which replace parabolic embeddings of regular rings considered in §2.3.

§3.1. Centralizers of Frobenius elements

We let denote by

(3-1)  \begin{aligned}S(q) & = S^{n}(q) ⊂ H_{q}(K)  and \\ S_{p}(q) & = S_{p}^{n}(q) = \{∑_{i} a_{i}(KM_{i}) ∈ S(q); μ(M_{i})|p^{∞}\}, \end{aligned}

where p is a prime divisor of q, the (maximal) reducible subring of the singular HS–ring H_{q}(K) of the group K = Γ_{0}^{n}(q) and the singular reducible p-ring. In this section we shall show that the singular Frobenius elements

(3-2)  Π(a) = Π_{K}^{n}(a) = (( \begin{matrix}E_{n} & 0 \\ 0 & aE_{n} \end{matrix} ))_{K} with a|q^{∞}

belong to the ring S(q) and determine their centralizer in the ring.

It turns out that the centralizer is naturally isomorphic to the centralizer Ĉ_{q}^{n} of the corresponding elements Π̂(a) in the q-singular subring L_{q}^{n} of the triangular ring L^{n} (see Theorem 3.3). This is crucial for our approach to singular HS–rings and operators for the group K.

We shall start with two simple lemmas on rational matrices (see [Si(63)]).

Lemma 3.1. Let R be a rational matrix of order n. Then there is an integral matrix D of order n with nonzero determinant, which satisfies the following conditions:

(1) the matrix DR is integral;

(2) if a product D′R is integral, then D′ = T D with an integral matrix T.

Proof . If R = 0, one can take D = E_{n}. If R ≠ 0, then it can be diagonalized by left and right multiplications by matrices of the group Λ = GL_{n}(ℤ):

URV = R′ = diag(a_{1}/b_{1}, …, a_{r}/b_{r}, 0, …, 0), (U, V ∈ Λ),

where a_{i}, b_{j} are positive integers satisfying gcd(a_{i}, b_{i}) = 1 for i = 1, …, r. Note that when S = (s_{i}, …, s_{n}) is an integral matrix with columns s_{i}, then the product SR′ is again an integral matrix if and only if the columns s_{i} are divisible by b_{i} for i = 1, …, r , which is equivalent to the condition that S is divisible from the right by the matrix B = diag(b_{1}, …, b_{r}, 1, …, 1). Let us set D = BU .Then the matrix DR = BR′V^{−1} is integral, and if a product D′R is integral, then the matrix D′U^{−1}R′ is integral, which implies that D′U^{−1} = T B with an integral matrix T, that is D′ = T BU = T D.

□

A matrix D satisfying the conditions (1) and (2) of the lemma is called a (left) denominator of the matrix R. It follows from the definition that all denominators of R form the single left coset ΛD modulo the group Λ = Λ^{n}, which can be called the denominator of R.

Lemma 3.2. If R is a rational matrix of order n and D is a denominator of R, then the pair (DR, D) is coprime.

Proof . If DR = SC_{1} and D = SD_{1} with integral matrices S, C_{1}, D_{1}, then R = D_{1}^{−1}C_{1}, and so the matrix D_{1}R is integral. It follows that D_{1} = T D with an integral matrix T. Therefore, ST = E , and so S ∈ Λ.

□

Theorem 3.3. Let n, q be positive integers and q > 1. Then the following assertions hold:

(1) Each double coset (M)_{K} of a matrix M = U(μ, D) of the form (2-41), which is contained in Σ_{0,q}^{n} and satisfies the condition (2-42), has the following decomposition into left cosets modulo the group K = Γ_{0}^{n}(q):

(3-3)  (M)_{K} = ∑_{D′,B}(KU(μ, D′) ( \begin{matrix}E & B \\ 0 & E \end{matrix} )),

where D′ runs a set of representatives of the left coset modulo the group Λ = Λ^{n} contained in the double coset ΛDΛ, and B runs over a set of representatives for the additive group 𝕊 = 𝕊_{n} modulo the subgroup μ^{−1} · ^{t}D′𝕊D′. In particular, the singular Frobenius elements (3-2) have the decomposition

(3-4)  Π(a) = Π_{K}^{n}(a) = (U(a, aE))_{K} = ∑_{B∈𝕊/a𝕊}(K ( \begin{matrix}E & B \\ 0 & aE \end{matrix} ));

(2) All double cosets (M)_{K} of matrices M = U(μ, D) in Σ_{0,q}^{n} satisfying (2-42) span over ℚ a subring C_{q} = C_{q}^{n} of the ring S(q) = S^{n}(q);

(3) Restriction of the parabolic embedding

(3-5)  j : S → L_{q}

on the subring C_{q} is an isomorphism of C_{q} onto the centralizer Ĉ_{q} of the Frobenius elements contained in L_{q}. In particular , C_{q} is a commutative integral domain, which is generated by the local subrings

(3-6)  C_{p} = {T = ∑_{i} a_{i}(KM_{i}) ∈ C_{q}; μ(M_{i})|p^{∞}}

for all prime numbers p dividing q;

(4) The ring C_{q} (resp., C_{p}) can be characterized as the centralizer in S(q) (resp., S_{p}(q)) of the Frobenius elements (3-2) (resp., Π(p)).

We prove first two lemmas.

Lemma 3.4. Each element of the form U(μ, D) ∈ Σ_{0}^{n} satisfying the condition (2-42) is K–reducible.

Proof. By Lemma 2.1, one can assume that q > 1. It suffices to prove that , for

every matrix γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ K, there is a matrix γ′ = ( \begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix} ) ∈ K such that

(3-7)  γ′U(μ, D)γ = ( \begin{matrix}∗ & ∗ \\ c′Aa + d′Dc & ∗ \end{matrix} ) = ( \begin{matrix}∗ & ∗ \\ 0 & ∗ \end{matrix} ) ,

where A = μ ·^{t}D^{−1}. Since a ·^{t}d ≡ 1 (mod q), it follows that det a is prime to q, and so the matrix a is invertible. Set

R = −Dca^{−1}A^{−1} = −μ^{−1}Dca^{−1} ·^{t}D.

By Lemma 1.1(4), the matrix R is symmetric. Since μ|d(D)^{2}, it follows that the matrix det a · R is integral. Since det a is prime to q and c ≡ 0 (mod q), we conclude that det a · R ≡ 0 (mod q). Let d′ be a denominator of R and c′ = d′R. Then d′| det a · E, whence det d′|(det a)^{n}, and so det d′ is prime to q. Further, since (det a)c′ = d′ · (det a)R ≡ 0 (mod q), we have c′ ≡ 0 (mod q). The pair (c′, d′) is coprime, by Lemma 3.2, and is symmetric, since R is symmetric. By Lemma 1.2,

there are matrices a′,b′ such that the matrix γ′ = ( \begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix} ) ∈ Γ^{n}. Then γ′ ∈ K,

and γ′ satisfies (3-7).

□

Lemma 3.5. Each of left cosets modulo K contained in the double coset KMK of a matrix M = U(μ, D) ∈ Σ_{0,q} of the form (2-41) satisfying (2-42) contains a representative of the form

(3-8)  ( \begin{matrix}A′ & B′ \\ 0 & D′ \end{matrix} )  with D′ ∈ ΛDΛ.

Proof . Let us take a representative

( \begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix} ) M ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ KM ( \begin{matrix}a & b \\ c & d \end{matrix} ) of the form ( \begin{matrix}A′ & B′ \\ 0 & D′ \end{matrix} ) ,

where ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ K, which exists, by previous lemma. Then c′Aa + d′Dc = 0,

where A = μ ·^{t}D^{−1}, and so (d′)^{−1}c′ = −μ^{−1}Dca^{−1} ·^{t}D. On the other hand, we have

D′ = c′Ab + d′Dd = d′((d′)^{−1}c′Ab + Dd)
= d′(−μ^{−1}Dca^{−1} ^{t}DAb + Dd) = d′D(−ca^{−1}b + d)
= d′D ·^{t}a^{−1}(^{t}ad −^{t}a ·^{t}a^{−1} ·^{t}cb) = d′D ·^{t}a^{−1}.

Since ( \begin{matrix}a & b \\ c & d \end{matrix} ), ( \begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix} ) ∈ K, it follows that det d′ and det a are coprime with q,

and so are coprime with μ, because μ|q^{∞}. On the other hand, the matrix μ ·^{t}D^{−1} is integral, and so det D|μ^{n}|q^{∞}. Since the integral matrices d′ and D have coprime determinants, it is an easy consequence of the theory of elementary divisors for the general linear group over ℤ (see, for example, [An(87),Chapter 3]), that one can write d′D = D_{1}d_{1}, where D_{1} and d_{1} are integral matrices having the same elementary divisors as D and d′, respectively. Then we have the relation

D′ = d′D ·^{t}a^{−1} = D_{1}d_{1} ·^{t}a^{−1}.

Since all of the elementary divisors of matrices D′ and D are prime to those of d_{1} and a, the last relation implies that all of the elementary divisors of the matrix d_{1} ·^{t}a^{−1} are equal to 1. It follows that the matrices D′,D_{1}, and D have the same elementary divisors, and so D′ ∈ ΛDΛ.

□

Proof of Theorem 3.3. (1). Let us take a representative of the form (3-8) in a left

coset KM ( \begin{matrix}a & b \\ c & d \end{matrix} ) with ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ K. then we have

( \begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix} ) M ( \begin{matrix}a & b \\ c & d \end{matrix} ) = ( \begin{matrix}A′ & B′ \\ 0 & D′ \end{matrix} ) = ( \begin{matrix}^{t}η^{−1} & 0 \\ 0 & η \end{matrix} ) ( \begin{matrix}A & B \\ 0 & D \end{matrix} ) ( \begin{matrix}^{t}ν^{−1} & 0 \\ 0 & ν \end{matrix} ) ,

where ( \begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix} ) ∈ K, η, ν ∈ Λ. It follows that

M = ( \begin{matrix}a_{1} & b_{1} \\ c_{1} & d_{1} \end{matrix} ) ( \begin{matrix}A & B \\ 0 & D \end{matrix} ) ( \begin{matrix}a_{2} & b_{2} \\ c_{2} & d_{2} \end{matrix} ) = ( \begin{matrix}∗ & a_{1}Ab_{2} + a_{1}Bd_{2} + b_{1}Dd_{2} \\ ∗ & ∗ \end{matrix} ) ,

where the matrices (3-9)

( \begin{matrix}a_{1} & b_{1} \\ c_{1} & d_{1} \end{matrix} ) = ( \begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix} )^{−1} ( \begin{matrix}^{t}η^{−1} & 0 \\ 0 & η \end{matrix} ) , ( \begin{matrix}a_{2} & b_{2} \\ c_{2} & d_{2} \end{matrix} ) = ( \begin{matrix}^{t}ν^{−1} & 0 \\ 0 & ν \end{matrix} ) ( \begin{matrix}a & b \\ c & d \end{matrix} )^{−1}

belong to K. This implies the relation a_{1}Ab_{2} + a_{1}Bd_{2} + b_{1}Dd_{2} = 0, whence

B = −(Ab_{2}d_{2}^{−1} + a_{1}^{−1}b_{1}D) = AS,

where S = −(b_{2}d_{2}^{−1}+A^{−1}a_{1}^{−1}b_{1}D). Note that, since the matrices (3-9) belong to K, we have ^{t}a_{1}d_{1} ≡ 1 (mod q) and ^{t}a_{2}d_{2} ≡ 1 (mod q), and so det a_{1}, det d_{2} are prime to q. In particular, the matrices a_{1} and d_{2} are invertible, because q > 1. Also, if p is a prime number and p| det a_{1} det d_{2}, then p ∤ q and so p ∤ det A. It follows that the matrix S = A^{−1}B is p-integral, because the matrix B is integral. But if p| det A, then p|q and so p ∤ det a_{1} det d_{2}. Also the matrix S = −(b_{2}d_{2}^{−1} + μ^{−1} ·^{t}Da_{1}^{−1}bD) is p-integral, by the condition (2-42). We conclude that S is an integral matrix, and so

B′ = ^{t}η^{−1}Bν = ^{t}η^{−1}ASν = A′ ·^{t}νSν = A′S′

with an integral symmetric matrix S′.

Finally, it is clear that two matrices ( \begin{matrix}A′ & B′ \\ 0 & D′ \end{matrix} ) and ( \begin{matrix}A_{1} & B_{1} \\ 0 & D_{1} \end{matrix} ) of the form

(3-8), where B′ = A′S′ and B_{1} = A_{1}S_{1} with S, S_{1} ∈ 𝕊, belong to the same left coset modulo K, if and only if they belong to the same left coset modulo Γ_{0}^{n}:

( \begin{matrix}α & αβ \\ 0 & δ \end{matrix} ) ( \begin{matrix}A_{1} & A_{1}S_{1} \\ 0 & D_{1} \end{matrix} ) = ( \begin{matrix}A′ & A′S′ \\ 0 & D′ \end{matrix} ) ,

where α, δ ∈ Λ, ^{t}αδ = E, β ∈ 𝕊, which is equivalent to the relations

δD_{1} = D′ and αA_{1}S_{1} + αβD_{1} = A′S′.

The first relation means that ΛD_{1} = ΛD′, and so one can assume that D_{1} = D′ and δ = E. Then the second turns into

A′S_{1} + βD′ = A′S′, i.e. S′ − S_{1} = (A′)^{−1}βD′ = μ^{−1} ·^{t}D′βD′.

(2). If (M)_{K} = (U(μ, D))_{K} and (M_{1})_{K} = (U(μ_{1}, D_{1}))_{K} are two double cosets of C_{q}, written in the form (3-3), then their product is a sum of the left cosets of the products

\begin{matrix}U( & μ, D′) ( \begin{matrix}E & B \\ 0 & E \end{matrix} ) U(μ_{1}, D′_{1}) ( \begin{matrix}E & B_{1} \\ 0 & E \end{matrix} ) \\ = U(μμ_{1}, D′D′_{1}) ( \begin{matrix}E & μ_{1}^{−1} ·^{t}D_{1}BD_{1} + B_{1} \\ 0 & E \end{matrix} ) , \end{matrix}

which belong to the double coset of the matrix U(μμ_{1}, D′D′_{1}), since the matrix μ^{−1} ·^{t}D_{1}BD_{1} is integral, by the condition (2-42) for U(μ_{1}, D_{1}). Also,

μμ_{1}|(det D det D_{1})^{2} = (det D′ det D′_{1})^{2} = (det D′D′_{1})^{2}.

(3). The statements follow from parts (1), (2), Theorem 2.8, Proposition 2.9, and Theorem 2.10.

(4). Since the ring C_{q} is commutative and contains the elements (3-2), it suffices to prove that every element T of S(q) which commutes with the elements (3-2) belongs to C_{q}. The image under the parabolic embedding T̂ of T in L_{q} commutes with each of Frobenius elements of the ring L_{q} and therefore belongs to Ĉ_{q}. Then, by Theorem 2.8, T̂ is a linear combination of the left cosets modulo Γ_{0} of matrices of the form

U(μ, D′) ( \begin{matrix}E & B \\ 0 & E \end{matrix} )

with integral symmetric B and integral D′ satisfying the condition μ|(det D′)^{2}|q^{∞}. By the definition of parabolic embeddings, the element T is a linear combination of left cosets of the same matrices modulo K. Then T belongs to C_{q}, by the part (1). The case of C_{p} with a prime p dividing q is considered similarly.

□

§3.2. Neutral rings

The neutral ring

(3-10)  N (q) = N^{n}(q) = HS(K, Γ^{n})

of the group K = Γ_{0}^{n}(q) is contained both in the regular ring H_{(q)}(K) and in the singular ring H_{q}(K) ( see (1-27)), since

Γ^{n} = Σ_{(q)}^{n} ⋂ Σ_{q}^{n}.

Although the neutral ring does not play essential part in the regular theory, it is important in the singular case. In this section we shall prove several properties of the neutral ring and its subrings of the form

(3-11)  N (q_{1}, q) = N^{n}(q_{1}, q) = HS(K, Γ_{0}^{n}(q_{1})),

where q_{1} divides q.

Lemma 3.6. Two matrices M = ( \begin{matrix}A & B \\ C & D \end{matrix} ) and M′ = ( \begin{matrix}A′ & B′ \\ C′ & D′ \end{matrix} ) of the group

Γ = Γ^{n} belong to the same left coset KM = KM′ modulo the group K, if and only if their ”second rows” are ”proportional” modulo q,

(3-12)  (C′, D′) ≃ (C, D) (mod q),

in the sense that

(3-13)  (C′, D′) ≡ d(C, D) (mod q)

with an integral matrix d of order n invertible modulo q.

Proof . If KM′ = KM, then M′ = γM with γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ K, which implies

the relation

(C′, D′) = (cA + dC, cB + dD) ≡ d(C, D) (mod q).

But if (3-13) is satisfied, then the matrix

γ = ( \begin{matrix}A′ & B′ \\ C′ & D′ \end{matrix} ) ( \begin{matrix}A & B \\ C & D \end{matrix} )^{−1} = ( \begin{matrix}A′ & B′ \\ C′ & D′ \end{matrix} ) ( \begin{matrix}^{t}D & −^{t}B \\ −^{t}C & ^{t}A \end{matrix} )

= ( \begin{matrix}∗ & ∗ \\ C′ ·^{t}D − D′ ·^{t}C & ∗ \end{matrix} )

belongs to K, since C′ ·^{t}D − D′ ·^{t}C ≡ d(C ·^{t}D − D ·^{t}C) = 0 (mod q).

□

Lemma 3.7. The indices

(3-14)  ν(q) = ν_{n}(q) = [Γ^{n} : Γ_{0}^{n}(q)]

satisfy the relations

(3-15)  ν(q_{1}q_{2}) = ν(q_{1})ν(q_{2}), if gcd(q_{1}, q_{2}) = 1.

Proof . It is well known that the natural map Γ^{n} → Sp_{n}(ℤ/qℤ) is an epimorphism (see, for example, [An(87), Lemma 3.3.2]). It follows that the index ν(q) can be interpreted as the index in the group Sp_{n}(ℤ/qℤ) of the subgroup

(3-16)  \{( \begin{matrix}α & β \\ 0 & δ \end{matrix} ) ∈ Sp_{n}(ℤ/qℤ)\} .

The group and the subgroup are sets of solutions of finite systems of polynomial congruences modulo q, and the relations (3-15) follow from the Chinese remainder theorem.

□

In order to formulate next lemma, we need certain notation. We set

(3-17)  I_{r,n} = {(i_{1}, …, i_{r}) ∈ ℕ^{r}; 1 ≤ i_{1} < ⋯ < i_{r} ≤ n}.

For i = (i_{1}, …, i_{r}) ∈ I_{r,n} we denote by î the set (j_{1}, …, j_{n−r}) ∈ I_{n−r,n} which is complementary to (i_{1}, …, i_{r}) in (1, …, n). Further, to each i = (i_{1}, …, i_{r}) ∈ I_{r,n} we associate the permutation

(3-18)  σ(i) : (i_{1}, …, i_{r}, j_{1}, …, j_{n−r}) → (1, …, r, r + 1, …, n)

of the set (1, …, n), where (j_{1}, …, j_{n−r}) = î, and the matrix

(3-19)  M(i) = M(σ(i))

of the permutation, where the matrix of a permutation σ of the set (1, …, n) is the matrix M(σ) = (δ_{σ^{−1}(i),j}) with δ_{i,i} = 1 and δ_{i,j} = 0, if i ≠ j, for i, j = 1, …, n. Finally, for i ∈ I_{r,n} and a positive integer p , we set

(3-20) V_{p}(i) = {V = (v_{αβ)} ∈ ℤ_{n}^{r}; 0 ≤ v_{αβ} < p, if i_{α} < j_{β}, and v_{αβ} = 0, if i_{α} > j_{β}} ,

where (j_{1}, …, j_{n−r}) = î.

Lemma 3.8. Let p be a prime divisor of q, which does not divide q/p, and let a,b be two integers satisfying ap − bq_{1} = 1, where q_{1} = q/p. Then the set

(3-21)  ⋃_{r=0}^{n} W_{n}^{r}(p, q),

where W_{n}^{0}(p, q) = {E_{2n}}, and, for r > 1,

(3-22)  W_{n}^{r}(p, q) = {M_{r} · U_{i}(V, D); i ∈ I_{r,n}, V ∈ V_{p}(i), D ∈ 𝕊_{r}/p𝕊_{r}}

with

M_{r} = ( \begin{matrix}aE_{r} & 0 & bE_{r} & 0 \\ 0 & E_{n−r} & 0 & 0 \\ q_{1}E_{r} & 0 & pE_{r} & 0 \\ 0 & 0 & 0 & E_{n−r} \end{matrix} ) ,

and

U_{i}(V, D) = ( \begin{matrix}E_{r} & V & D & 0 \\ 0 & E_{n−r} & 0 & 0 \\ 0 & 0 & E_{r} & 0 \\ 0 & 0 & −^{t}V & E_{n−r} \end{matrix} ) ( \begin{matrix}M(i) & 0 \\ 0 & ^{t}M(i)^{−1} \end{matrix} )

is a complete set of representatives for different left cosets Γ_{0}^{n}(q) \ Γ_{0}^{n}(q_{1}).

Proof . We shall show first that matrices of the set (3-21) belong to different left

cosets modulo K = Γ_{0}^{n}(q). Let M = ( \begin{matrix}A & B \\ C & D \end{matrix} ) and M′ = ( \begin{matrix}A′ & B′ \\ C′ & D′ \end{matrix} ) be two of

the matrices. Suppose that KM = KM′. Then, by Lemma 3.6, their second rows are proportional modulo p, that is

(3-23)  C′ ≡ dC (mod q), D′ ≡ dD (mod q)

with an integral matrix d of order n invertible modulo q. It follows that the matrices C, C′ have the same rank r modulo p, and so M, M′ both belong to the same set W_{n}^{r}(p, q). If

C = ( \begin{matrix}q_{1}E_{r} & q_{1}V \\ 0 & 0 \end{matrix} ) M(i), C′ = ( \begin{matrix}q_{1}E_{r} & q_{1}V′ \\ 0 & 0 \end{matrix} ) M(i′), and d = ( \begin{matrix}d_{1} & d_{2} \\ d_{3} & d_{4} \end{matrix} ) ,

where d_{1} is the block of dimension r×r, then the congruences imply the congruence

(3-24)  (q_{1}E_{r}, q_{1}V′)M(i′) ≡ d_{1}(q_{1}E_{r}, q_{1}V )M(i) (mod q).

It follows that the matrix d_{1} is invertible modulo p. If q_{1}(E_{r}, V)M(i) = (t_{1}, …, t_{n}), then

q_{1}(E_{r}, V) = (t_{1}, …, t_{n})M(i)^{−1} = (t_{i_{1}}, …, t_{i_{r}}, t_{j_{1}}, …, t_{j_{n−r}}),

where (i_{1}, …, i_{r}) = i and (j_{1}, …, j_{n−r}) = î, by the definition of the matrix M(i), and so, by the definition of the set V_{p}(i) (see (3-20)), the system i = (i_{1}, …, i_{r}) is in the sense of lexicographic order the least system of indices of columns of the matrix (t_{1}, …, t_{n}) which are linearly independent modulo p. Thus, the congruence (3-24) implies that i′ = i, and so d_{1} ≡ E_{r} (mod p), and V′ ≡ V (mod p). By (3-30), the last congruence means that V′ = V. Then the second congruence (3-23) implies the congruence

( \begin{matrix}q_{1}D′ & 0 \\ −^{t}V & E_{n−r} \end{matrix} )^{t}M(i)^{−1} ≡ ( \begin{matrix}E_{r} & d_{2} \\ d_{3} & d_{4} \end{matrix} ) ( \begin{matrix}q_{1}D & 0 \\ −^{t}V & E_{n−r} \end{matrix} )^{t}M(i)^{−1} (mod p).

by comparing corresponding blocks, we conclude that d_{2} ≡ 0 (mod p), and q_{1}D′ ≡ q_{1}D − d_{2}^{t}V ≡ q_{1}D (mod p). It follows that D′ = D.

In order to conclude the proof of the lemma, it is sufficient to show that the number of elements of the set (3-21) is equal to the index [Γ_{0}^{n}(q_{1}) : Γ_{0}^{n}(q)]. Note first that the number of elements of the set (3-20) for i = (i_{1}, …, i_{r}) ∈ I_{r,n} is p^{j_{1}+···+j_{n−r}−⟨n−r⟩} (see Notation), where (j_{1}, …, j_{n−r}) = î, since the number of i_{α} satisfying i_{α} < j_{β} with a fixed β is equal to j_{β} − β. It follows that the number of elements of the set (3-21) is equal to (3-25)

∑_{r=0}^{n}(p^{−⟨n−r⟩} ∑_{1≤j_{1}<···<j_{n−r}≤n} p^{j_{1}+···+j_{n−r}})p^{⟨r⟩} = ∑_{r=0}^{n} p^{⟨r⟩} \frac{φ_{n}(p)}{φ_{r}(p)φ_{n−r}(p)} = ∏_{i=1}^{n}(1 + p^{i}),

where

(3-26)  φ_{0}(x) = 1, φ_{r}(x) = (x − 1)(x^{2} − 1) … (x^{r} − 1), (r ∈ ℕ)

and we have used the identity in two variables x, t:

∏_{i=1}^{n}(1 + tx^{i}) = ∑_{r=0}^{n} t^{r}x^{⟨r⟩} \frac{φ_{n}(x)}{φ_{r}(x)φ_{n−r}(x)},

and its consequences

(3-27)  ∑_{1≤i_{1}<···<i_{r}≤n} x^{i_{1}+···+i_{r}} = x^{⟨r⟩} \frac{φ_{n}(x)}{φ_{r}(x)φ_{n−r}(x)},  (1 ≤ r ≤ n).

On the other hand, according to Lemma 3.7, we can write

(3-28)  [Γ_{0}^{n}(q_{1}) : Γ_{0}^{n}(q)] = ν_{n}(q)/ν_{n}(q_{1}) = ν_{n}(p),

and

(3-29)  \begin{aligned}ν_{n}(p) & = [Γ^{n} : Γ_{0}^{n}(p)] = #(Sp_{n}(ℤ/pℤ))/p^{⟨n⟩}#(GL_{n}(ℤ/pℤ)) \\ & = p^{n^{2}} ∏_{i=1}^{n}(p^{2i} − 1)/p^{⟨n⟩}p^{⟨n−1⟩} ∏_{i=1}^{n}(p^{i} − 1) = ∏_{i=1}^{n}(p^{i} + 1). \end{aligned}

(see, for example, [Ar(57), Chapters 3,4]). The comparison of formulas (3-28) and (3-29) with (3-25) concludes the proof of the lemma.

□

Proposition 3.9. Let p be a prime number dividing q, but not dividing q_{1} = q/p. Then the following assertions hold:

(1) The ring

(3-30)  N_{p}(q) = N_{p}^{n}(q) = N^{n}(q_{1}, q)

consists of linear combinations of n + 1 different double cosets

(3-31)  ξ_{r}(p) = ξ_{r}^{n}(p) = (M_{r})_{K},

where M_{0} = E_{2n}, and M_{1}, …, M_{n} were introduced in Lemma 3.8. Each double

coset KM_{r}K can be characterized as the set of all matrices M = ( \begin{matrix}A & B \\ C & D \end{matrix} ) of

Γ_{0}^{n}(q_{1}) with the block C of rank r modulo p.

(2) Each double coset (3-31) has the following decomposition into left cosets:

(3-32)  ξ_{r}(p) = ∑_{M∈W_{n}^{r}(p,q)}(KM),

where W_{n}^{r}(p, q) is the set (3-22). In particular, the number of different left cosets in ξ_{r}(p) is

(3-33)  #(W_{n}^{r}(p, q)) = p^{⟨r⟩} \frac{φ_{n}(p)}{φ_{r}(p)φ_{n−r}(p)},

where φ_{r}(x) are the polynomials (3-26).

(3) The double cosets (3-31) are invariant with respect to the star antiautomorphism (1-18):

(3-34)  ξ_{r}^{∗}(p) = ξ_{r}(p),  (r = 0, 1, …, n).

In particular, the ring N_{p}(q) is commutative.

(4) If p and p′ are two different prime divisors of q and p^{2} does not divide q, then each element of N_{p}(q) commutes with the Frobenius element Π(p′).

Proof . First of all, note that, since K ⊂ Γ_{0}^{n}(p), the function

(3-35)  M = ( \begin{matrix}A & B \\ C & D \end{matrix} ) → r_{p}(C),  (M ∈ Σ^{n})

where r_{p}(C) is the rank of C modulo p, takes a constant value on every left and double coset of the group K; in particular, its value on W_{n}^{r}(p, q) and on KM_{r}K is r. Thus, the double cosets ξ_{r} are pairwise distinct. Since

W_{n}^{r}(p, q) ⊂ M_{r}Γ_{0}^{n} ⊂ KM_{r}K,

it follows from Lemma 3.8, that each of ξ_{r} has the decomposition (3-32), and there are no other double cosets in N_{p}(q), but ξ_{0}, ξ_{1}, …, ξ_{n}.

The formula (3-33) follows from the identities (3-27), since the number of elements of a set V_{p}(i) with i ∈ I_{r,n} is clearly p^{j_{1}+···+j_{n−r}−⟨n−r⟩}, where (j_{1}, …, j_{n−r}) = î.

Further, by the above,

M_{r}^{∗} = M_{r}^{−1} = ( \begin{matrix}∗ & ∗ & ∗ & ∗ \\ ∗ & ∗ & ∗ & ∗ \\ −q_{1}E_{r} & 0 & ∗ & ∗ \\ 0 & 0 & ∗ & ∗ \end{matrix} ) ∈ KM_{r}K,

which implies (3-34).

Finally, if M = ( \begin{matrix}A & B \\ C & D \end{matrix} ) ∈ Γ_{0}^{n}(q_{1}), S and S′ are matrices of 𝕊_{n} satisfying the

congruence AS′ ≡ B + SD (mod p′), then the matrix

\begin{aligned}M′ & = ( \begin{matrix}A′ & B′ \\ C′ & D′ \end{matrix} ) = ( \begin{matrix}E & S \\ 0 & p′E \end{matrix} ) ( \begin{matrix}A & B \\ C & D \end{matrix} ) ( \begin{matrix}E & S′ \\ 0 & p′E \end{matrix} )^{−1} \\ & = ( \begin{matrix}A + SC & (−(A + SC)S′ + B + SD)/p \\ p′C & −CS′ + D \end{matrix} ) \end{aligned}

belongs to Γ_{0}^{n}(q_{1}), r_{p}(C′) = r_{p}(C), and it satisfies the relation

( \begin{matrix}E & S \\ 0 & p′E \end{matrix} ) M = M′ ( \begin{matrix}E & S′ \\ 0 & p′E \end{matrix} ) .

It is easy to see that, for every two matrices M, M_{1} ∈ Γ_{0}^{n}(q_{1}) with different left cosets KM, KM_{1}, the left cosets KM′, KM′_{1} are also different. It follows that

\begin{aligned}Π(p′)ξ_{r}(p) & = ∑_{\begin{matrix}M∈K\Γ_{0}^{n}(q_{1}),r_{p}(C(M))=r, \\ S∈𝕊_{n}/p𝕊_{n} \end{matrix}} (K ( \begin{matrix}E & S \\ 0 & p′E \end{matrix} ) M) \\ & = ∑_{\begin{matrix}M′∈K\Γ_{0}^{n}(q_{1}),r_{p}(C(M′))=r, \\ S′∈𝕊_{n}/p𝕊_{n} \end{matrix}} (KM′ ( \begin{matrix}E & S′ \\ 0 & p′E \end{matrix} )) = ξ_{r}(p)Π(p′), \end{aligned}

where C(M), and C(M′) denote the lower left n × n–blocks of the corresponding matrices.

□

Lemma 3.10. With the notations and assumptions of Proposition 2.9, products of elements (3.31) of the ring N_{p}(q) by the element ξ_{n}(p) can be given by the following formulas

(3-36)  ξ_{n}(p)ξ_{r}(p) = ∑_{s=0}^{r} p^{⟨n−r⟩+(r−s)(n−r+s)}sm_{p}(s, n − r + s)ξ_{n−r+s}(p),

where r = 0, 1, …, n and sm_{p}(a, b) stands for the number of symmetric matrices of order b and rank a over the field of p elements.

Proof. By Proposition 3.9 and the definition of multiplication in HS–rings, the product ξ_{n}(p)ξ_{r}(p) is the sum of left cosets of all products

( \begin{matrix}aE_{n} & aD + bE_{n} \\ q_{1}E_{n} & q_{1}D + pE_{n} \end{matrix} ) M_{r} ( \begin{matrix}E_{r} & V & D′ & 0 \\ 0 & E_{n−r} & 0 & 0 \\ 0 & 0 & E_{r} & 0 \\ 0 & 0 & −^{t}V & E_{n−r} \end{matrix} ) ( \begin{matrix}M(i) & 0 \\ 0 & ^{t}M(i)^{−1} \end{matrix} )

= ( \begin{matrix}∗ & ∗ & ∗ & ∗ \\ ∗ & ∗ & ∗ & ∗ \\ Ď_{1} & 0 & ∗ & ∗ \\ q_{1}^{2}D_{3} & q_{1}E_{n−r} & ∗ & ∗ \end{matrix} ) ( \begin{matrix}E_{r} & V & D′ & 0 \\ 0 & E_{n−r} & 0 & 0 \\ 0 & 0 & E_{r} & 0 \\ 0 & 0 & −^{t}V & E_{n−r} \end{matrix} ) ( \begin{matrix}M(i) & 0 \\ 0 & ^{t}M(i)^{−1} \end{matrix} ) ,

where D = ( \begin{matrix}D_{1} & D_{2} \\ D_{3} & D_{4} \end{matrix} ), D′ ∈ 𝕊_{n}/p𝕊_{n}, V ∈ V (i) with i ∈ I_{r,n}, and where Ď_{1} =

q_{1}^{2}D_{1} + q_{1}(a + p)E_{r}. Such a product belongs to a double coset KM_{r′}K if and only if the rank of the matrix

( \begin{matrix}Ď_{1} & 0 \\ q_{1}^{2}D_{3} & q_{1}E_{n−r} \end{matrix} )

modulo p is r′ (see (3-35) and further). On the other hand, the rank is clearly equal to n − r + s, where s stands for the rank r_{p}(Ď_{1}). Since the matrix Ď_{1} runs together with D_{1} over all symmetric matrices of order r modulo p, we conclude that the number of the products belonging to KM_{n−r+s}K is

sm_{p}(s, r)p^{⟨n−r⟩+r(n−r)}#(W_{n}^{r}(p, q)) = sm_{p}(s, r)p^{⟨n⟩} \frac{φ_{n}(p)}{φ_{r}(p)φ_{n−r}(p)}

(see (3-33)). On dividing the number by the number #(W_{n}^{n−r+s}(p, q)) of left cosets in ξ_{n−r+s}(p), we see that the coefficient at ξ_{n−r+s}(p) in the product ξ_{n}(p)ξ_{r}(p) is equal to

sm_{p}(s, r)p^{⟨n⟩−⟨n−r+s⟩} \frac{φ_{n−r+s}(p)φ_{r−s}(p)}{φ_{r}(p)φ_{n−r}(p)}

\begin{matrix}= p^{⟨r−s⟩+(r−s)(n−r+s)}sm_{p}(s, s)\frac{φ_{r}(p)φ_{n−r+s}(p)φ_{r−s}(p)}{φ_{s}(p)φ_{r−s}(p)φ_{r}(p)φ_{n−r}(p)} \\ = p^{⟨r−s⟩+(r−s)(n−r+s)}sm_{p}(s, n − r + s), \end{matrix}

where we have twice used the formula

(3-37)  sm_{p}(a, b) = sm_{p}(a, a)\frac{φ_{b}(p)}{φ_{a}(p)φ_{b−a}(p)}  (0 ≤ a ≤ b)

(see, for example, [An(87), Lemma 3.5.20]) and the obvious relation ⟨a + b⟩ = ⟨a⟩ + ⟨b⟩ + ab.

□

Note that the numbers sm_{p}(a, b) can be explicitly calculated with the help of formulas (3-37) and the formula

(3-38)  sm_{p}(a, a) = ∑_{i=0}^{a}(−1)^{i}p^{⟨i−1⟩+⟨a−i⟩} \frac{φ_{a}(p)}{φ_{i}(p)φ_{a−i}(p)},

which is equivalent to the first identity of Lemma 3.5.21 of [An(87)].

Lemma 3.11. With the notation and assumptions of Proposition 2.9, the element ξ_{n}^{n} = ξ_{n}^{n}(p) is invertible in the ring N_{p}^{n}(q). In particular,

(3-39)  (ξ_{1}^{1})^{−1} = ((1 − p)ξ_{0}^{1} + ξ_{1}^{1})/p;
(3-40)  (ξ_{2}^{2})^{−1} = ((1 − p)ξ_{0}^{2} + (1 − p)ξ_{1}^{2} + ξ_{2}^{2})/p^{3}.

Proof. The formulas (3-36) show that the matrix of the operator of multiplication by ξ_{n}^{n} in the basis ξ_{0}^{n}, ξ_{1}^{n}, …, ξ_{n}^{n} of the space N_{p}^{n}(q) has zeros above the second diagonal and nonzero elements p^{⟨r⟩+r(n−r)}sm_{p}(0, n − r) = p^{⟨r⟩+r(n−r)} for r = 0, 1, …, n on this diagonal. For example, for n = 1 and n = 2, we get the matrices

( \begin{matrix}0 & 1 \\ p & p − 1 \end{matrix} ) and ( \begin{matrix}0 & 0 & 1 \\ 0 & p^{2} & p^{2} − 1 \\ p^{3} & p^{2}(p − 1) & p^{2}(p − 1) \end{matrix} ) ,

respectively. Such a matrix is clearly invertible, which proves that ξ_{n}^{n} is invertible. The formulas (3-39) and (3-40) follow by inversion of the matrices.

□

In contrast to Lemma 3.8, where q/q_{1} and q_{1} are coprime, in the case when q and its divisor q_{1} have the same prime factors, the left cosets Γ_{0}^{n}(q) \ Γ_{0}^{n}(q_{1}) admit a very simple description, which we shall need in the case when q/q_{1} is a prime number.

Proposition 3.12. Let p be a prime number such that p^{2} divides q. Then the following assertions hold:

(1) The set

\{( \begin{matrix}E_{n} & 0 \\ q_{1}C & E_{n} \end{matrix} ) ; C ∈ 𝕊_{n}/p𝕊_{n}\} ,

where q_{1} = q/p, is a complete set of representatives for different left cosets of the group K_{1} = Γ_{0}^{n}(q_{1}) modulo the subgroup K = Γ_{0}^{n}(q);

(2) The element

(3-41)  \begin{aligned}τ(p) & = τ^{n}(p, q) = p^{−⟨n⟩} ∑_{M_{i}∈K\K_{1}}(KM_{i}) \\ & = p^{−⟨n⟩} ∑_{C∈𝕊_{n}/p𝕊_{n}} (K ( \begin{matrix}E_{n} & 0 \\ q_{1}C & E_{n} \end{matrix} )) ∈ N (q_{1}, q) \end{aligned}

satisfies the relation

Π(p)τ(p) = Π(p),

where Π(p) is the Frobenius element of the multiplier p ;

(3) The element τ(p) commutes with every of Frobenius elements Π(p′) with primes p′ ≠ p dividing q;

(4) The element τ(p) satisfies

τ^{∗}(p) = τ(p),

where ∗ is the antiautomorphism (1-18).

Proof. The ”second rows” of different matrices of the set are clearly not proportional modulo q in the sense of (3-13), and so, by Lemma 3.6, the matrices

belong to different left cosets modulo K. If ( \begin{matrix}∗ & ∗ \\ q_{1}C′ & D \end{matrix} ) ∈ K_{1}, then the matrix D

is invertible modulo q_{1}, and therefore, since q|q_{1}^{2}, it has also an inverse D′ modulo q. It follows that the pair (q_{1}C′, D) is proportional modulo q to the pairs

(q_{1}D′C′, D′D) ≡ (q_{1}C, E_{n}) (mod q),

which concludes the proof of the part (1).

Since all of the products

( \begin{matrix}E & B \\ 0 & pE \end{matrix} ) ( \begin{matrix}E & 0 \\ q_{1}C & E \end{matrix} ) ( \begin{matrix}E & B \\ 0 & pE \end{matrix} )^{−1} = ( \begin{matrix}E + q_{1}BC & −(q_{1}/p)BCB \\ qC & E − q_{1}CB \end{matrix} ) ,

where E = E_{n}, and B, C ∈ 𝕊_{n}, belong to the group K, it follows that

Π(p)τ(p) = p^{−⟨n⟩} ∑_{B,C∈𝕊/p𝕊} (K ( \begin{matrix}E & B \\ 0 & pE \end{matrix} ) ( \begin{matrix}E & 0 \\ q_{1}C & E \end{matrix} ))

= ∑_{B∈𝕊/p𝕊} (K ( \begin{matrix}E & B \\ 0 & pE \end{matrix} )) = Π(p),

where 𝕊 = 𝕊_{n}, which proves the part (2).

One can assume that all of the matrices C in (3-41) are divisible by p′. Then we have

p^{⟨n⟩}Π(p′)τ(p) = ∑_{\begin{matrix}B∈𝕊/p′𝕊, \\ C∈𝕊/p𝕊 \end{matrix}} (K ( \begin{matrix}E & B \\ 0 & p′E \end{matrix} ) ( \begin{matrix}E & 0 \\ q_{1}p′C & E \end{matrix} ))

= ∑_{B, C} (K ( \begin{matrix}E + q_{1}p′BC & −q_{1}BCB \\ q_{1}(p′)^{2}C & E − q_{1}p′CB \end{matrix} ) ( \begin{matrix}E & B \\ 0 & p′E \end{matrix} ))

= p^{⟨n⟩}τ(p)Π(p′),

since the matrix

( \begin{matrix}E & B \\ 0 & p′E \end{matrix} ) ( \begin{matrix}E & 0 \\ q_{1}p′C & E \end{matrix} ) ( \begin{matrix}E & B \\ 0 & p′E \end{matrix} )^{−1} = ( \begin{matrix}E + q_{1}p′BC & −q_{1}BCB \\ q_{1}(p′)^{2}C & E − q_{1}p′CB \end{matrix} )

belongs to K_{1} and runs a complete system of representatives for K\K_{1}, when C runs a set 𝕊/p𝕊.

The part (4) easily follows from definitions.

□

In next section we shall see how the neutral rings work in singular theory.

§3.3. Unramified embeddings of local rings

A q-singular prime number p is said to be unramified, if it does not divide the number q_{1} = q/p, otherwise it is called ramified. In this section we shall show that the Frobenius elements Π(p) for unramified primes numbers p are invertible in an extension of the singular HS–ring H_{q}(K) of the group K = Γ_{0}^{n}(q). This will allow us to define a natural homomorphic embeddings of the (regular) p-rings

(3-42)  H_{p}(q_{1}) = H_{p}^{n}(q_{1}) = HS(Γ_{0}^{n}(q_{1}), R_{p}(Γ_{0}^{n}(q_{1})))

of the groups Γ_{0}^{n}(q_{1}) into the extension. The embeddings replace parabolic embeddings of the regular theory and have a number of similar properties.

Lemma 3.13. Let p be a nonramified prime divisor of q . Then the matrix

(3-43)  H(p) = H^{n}(p, q) = ( \begin{matrix}apE_{n} & bE_{n} \\ qE_{n} & pE_{n} \end{matrix} ) ,

where a, b are rational integers satisfying ap − bq_{1} = 1, is a symplectic matrix with the multiplier μ(H(p)) = p, which satisfies the following rules:

(3-44)  H(p)^{2} = γ_{p} · (pE_{2n})  with γ_{p} ∈ K,

(3-45)  KH(p)K = H(p)K.

The proof is a routine exercise on multiplication of block matrices, which we leave to the reader.

Lemma 3.14. Let p be a q–singular unramified prime number. Then the double coset

(3-46)  Ω(p) = Ω^{n}(p, q) = (H^{n}(p, q))_{K} ∈ H_{q}(K)

has the following properties:

(3-47)  Ω(p) = (KH(p));

(3-48)  Ω(p)^{2} = [p],

where [p] has the form (2-8);

(3-49)  Ω(p)^{∗} = Ω(p),

where star stands for the antiautomorphism (1-18);

(3-50)  Π(p) = Ω(p)ξ_{n}(p),

where Π(p) and ξ_{n}(p) = ξ_{n}^{n}(p) were defined by (3-2) and (3-31), respectively;

(3-51)  Ω(p)Ω(p′) = Ω(p′)Ω(p),

if p′ is another unramified prime;

(3-52)  Ω(p)Π(p′) = Π(p′)Ω(p),

if p′ is a singular prime number different from p.

Proof. The decomposition (3-47) follows from (3-45). The relation (3-48) follows from (3-47) and (3-44). By definition of star map and (3-44), we have

Ω(p)^{∗} = (pH(p)^{−1}) = (γ_{p}H(p)) = Ω(p).

By (3-47), (3-32), and (3-4), we get

Ω(p)ξ_{n}(p) = ∑_{B∈𝕊_{n}/p𝕊_{n}}(KH(p)M_{n} ( \begin{matrix}E_{n} & B \\ 0 & E_{n} \end{matrix} ))

= ∑_{B}(KH(p)M_{n} ( \begin{matrix}E & 0 \\ 0 & pE \end{matrix} )^{−1} ( \begin{matrix}E & B \\ 0 & pE \end{matrix} )) = Π(p),

since the matrix

H(p)M_{n} ( \begin{matrix}E & 0 \\ 0 & pE \end{matrix} )^{−1} = ( \begin{matrix}(a^{2} + bq_{1})E & (ab + b)E \\ q(a + 1)E & (q_{1}b + p)E \end{matrix} ) ,

where E = E_{n} and q_{1} = q/p, belongs to K.

Further, one can easily verify that the matrix H(p)H(p′)H(p)^{−1}H(p′)^{−1} belongs to K, which implies (3-51).

Finally, one can easily check that the product

H(p) ( \begin{matrix}E & B \\ 0 & p′E \end{matrix} ) H(p)^{−1} ( \begin{matrix}E & B′ \\ 0 & p′ \end{matrix} )^{−1}

with B, B′ ∈ 𝕊_{n} belongs to the group K, if

B′ ≡ aB − p^{−1}bE (mod p′).

It follows that

Ω(p)Π(p′) = ∑_{B∈𝕊_{n}/p′𝕊_{n}}(KH(p) ( \begin{matrix}E & B \\ 0 & p′E \end{matrix} ))

= ∑_{B′∈𝕊_{n}/p′𝕊_{n}}(K ( \begin{matrix}E & B′ \\ 0 & p′E \end{matrix} ) H(p)) = Π(p′)Ω(p).

□

Now we shall take a small break in order to remind a simple construction which we shall need later. If 𝔸 is an associative ring with the unity element, and a_{1}, …, a_{r} are not zero divisors belonging to the center of 𝔸, then one can produce the smallest extension of 𝔸, where all of the elements a_{1}, …, a_{r} are invertible, using the standard procedure of localization with respect to the multiplicative system

S = {a_{1}^{n_{1}} ⋯ a_{r}^{n_{r}}; n_{1}, …, n_{r} = 0, 1, … }.

We shall denote the extension by

𝔸[S^{−1}] = 𝔸[a_{1}^{−1}, …, a_{r}^{−1}].

(The notation has been already used in Proposition 2.11.)

Proposition 3.15. Let p be a prime divisor of q not dividing the quotient q/p, then the Frobenius element Π(p) = Π_{K}^{n}(p) is invertible in the extension

H′_{p}(K) = H_{p}(K)[[p]^{−1}]

of the ring H_{p}(K), and

(3-53)  Π(p)^{−1} = [p]^{−1} · ξ_{n}(p)^{−1} · Ω(p).

Proof. The statement follows from (3-50), (3-48), and Lemma 3.11.

□

With the help of the inverse Frobenius element (3-53) we are now going to define a homomorphism of the local p–subring L_{p} = L_{p}^{n} of the triangular ring (2-28), where p is a q–singular unramified prime number, into an extension of the local p–ring of the group K = Γ_{0}^{n}(q).

Let X ∈ L_{p}. According to Proposition 2.11, ”projections” X · Π̂(p)^{d} belongs to the centralizer Ĉ_{p} of Π̂(p) in L_{p},

(3-54)  X · Π̂(p)^{d} ∈ Ĉ_{p},

if d is sufficiently big. By Theorem 3.3 (3), (4), the ring Ĉ_{p} is the isomorphic image under the parabolic embedding (2-30) of the centralizer C_{p} of Π(p) = Π_{K}(p) in the ring S_{p}(q),

j(C_{p}) = Ĉ_{p},

and so there is unique element

(3-55)  i(X · Π̂(p)^{d}) ∈ C_{p},

where i stands for the inverse of j on Ĉ_{p}. We put then

(3-56)  X̃ = Π(p)^{−d} · i(X · Π̂(p)^{d}) ∈ C′_{p} = C_{p}[Π(p)^{−1}] ∈ H′_{p}(K),

where Π(p)^{−d} = (Π(p)^{−1})^{d}.

Theorem 3.16. Let p be a q–singular unramified prime number, i.e. p|q, but p^{2} ∤ q. Then the following assertions hold:

(1) For every X ∈ L_{p} = L_{p}^{n} the element X̃ defined by (3-56) is independent of the choice of d ∈ ℕ satisfying (3-54);

(2) The map

(3-57)  X → X̃ ∈ C_{p}[Π(p)^{−1}]  (X ∈ L_{p})

is a ℚ–linear ring homomorphism;

(3) The restriction of the map (3-57) on the image Ĥ_{p}(q_{1}) ∈ L_{p} of the local HS–ring (3-42) of the group Γ_{0}^{n}(q_{1}) with q_{1} = q/p under the parabolic embedding (2-30) is an isomorphic embedding of the ring Ĥ_{p}(q_{1}) into C_{p}[Π(p)^{−1}].

Proof. (1) If d_{1} is another exponent satisfying (3-54), and, say, d_{1} > d, then

Π^{−d_{1}} · i(XΠ̂^{d_{1}}) = Π^{−d_{1}} · i(XΠ̂^{d} · Π̂^{d_{1}−d}) = Π^{−d_{1}} · i(Π̂^{d_{1}−d} · i(XΠ̂^{d}))
= Π^{−d_{1}}Π^{d_{1}−d} · i(XΠ̂^{d}) = Π^{−d} · i(XΠ̂^{d}),

where Π = Π(p)

(2) If X, X_{1} ∈ L_{p}, and d satisfies (3-54) both for X and X_{1}, then we have

{ \atop XX_{1}} = Π^{−2d} · i(XX_{1}Π̂^{2d}) = Π^{−2d} · i(X(X_{1}Π̂^{d})Π̂^{d})
= Π^{−2d} · i((XΠ̂^{d}) · (X_{1}Π̂^{d})) = Π^{−d} · i(XΠ̂^{d}) · Π^{−d} · i(X_{1}Π̂^{d}),

since i(XΠ̂^{d}) belongs to C_{p} and therefore commutes with Π and Π^{−1}.

(3) If X is a nonzero element of the ring Ĥ_{p}(q_{1}), then XΠ̂(p)^{d} ≠ 0, by Proposition 2.11 and (2-51), and so X̃ ≠ 0, since the element Π(p)^{−d} is not a zero divisor in H′_{p}(K).

□

It follows from the definition of parabolic embeddings and the last part of Theorem 3.16, that the map

(3-58)  T → T̂ → T̃ ∈ C_{p}[Π(p)^{−1}]  (T ∈ H_{p}(q_{1}))

is a composition of two isomorphic embeddings and therefore is an isomorphic embedding by itself. We shall say that the map (3-58) is unramified embedding of the ring H_{p}^{n}(q_{1}) and its image

(3-59)  H_{p}^{n}(q) = H_{p}^{n}(q) = H̃_{p}^{n}(q_{1})

will be called the unramified (singular) p–ring of the group K = Γ_{0}^{n}(q).

Theorem 3.17. Let p be a q–singular unramified prime number. Then the following assertions hold:

(1) Every element of the p–ring (3-59) commutes with every element of the regular reducible ring (2-3);

(2) Every element of the p–ring (3-59) commutes with every element of others unramified p′–rings H_{p′}(q) as well as with every of singular Frobenius elements Π(a) with a|q^{∞};

(3) The element Π(p) satisfies the equation

(3-60)  ∑_{j=0}^{m} q̃_{j}^{n}(p)Π(p)^{m−j} = 0,  (m = 2^{n})

whose coefficients q̃_{j}^{n}(p) ∈ H_{p}(q) are images under the map (3-58) of the corresponding coefficients q_{j}^{n}(p) ∈ H_{p}(q_{1}) of the polynomial (2-27) with q_{1} = q/p in place of q; the coefficients q̃_{j}^{n}(p) satisfy the relations

(3-61)  \begin{matrix}{ \atop q_{m−i}^{n}}(p) = (p^{⟨n⟩}[p]_{K})^{(m/2)−i}q̃_{i}^{n}(p),  (0 ≤ i ≤ m), \\ q̃_{0}^{n}(p) = [1]_{K}, q̃_{m}^{n}(p) = (p^{⟨n⟩}[p]_{K})^{m/2}. \end{matrix}

Proof. By Theorem 2.3 and Theorem 2.5, it is sufficient to prove the part (1) for a local regular reducible ring D instead of H_{rr}(q). In this case it follows from Proposition 2.12 that each element of D commutes with each element of C_{p}. In particular, it commutes with Π(p) and so with Π(p)^{−1}. It follows that each element of D commutes with every element of the ring C_{p}[Π(p)^{−1}], which contains the p–ring H_{p}(q). The part (1) is proved.

Since all of H_{p}(q), H_{p′}(q), and Π(a) for unramified p, p′, and a|q^{∞} are contained in a commutative extension of the ring C_{q}, the part (2) follows.

The relation (3-60) follows from (2-52) by applying to both parts the homomorphism (3-57). Similarly, the relations (3-61) follow from the corresponding relations for the coefficients of the polynomial (2-27), and obvious relations

(3-62)  ˜[p^{δ}]_{K_{1}} = ˜[p^{δ}]_{Γ_{0}} = [p^{δ}]_{K},

where K_{1} = Γ_{0}^{n}(q_{1}), and δ = 0, 1, ….

□

The unramified embeddings (3-58) and the homomorphisms (3-57) make it possible to transfer all of the factorizations of polynomials over local HS–rings of the group Γ = Γ^{n} with factors over HS–rings L_{p} of the triangular subgroups, obtained, for example, in [An(87),§3.5], into corresponding factorizations of polynomials over p–rings H_{p}(q) with factors over rings C′_{p} = C_{p}[Π(p)^{−1}], if p is unramified. It is interesting to note that, although the order of factors over L_{p} is strictly fixed, the order of their images is not significant, since their coefficients belong to the commutative ring C′_{p}.

Let p be an unramified prime, and let

(3-63)  Q̃_{p}(v) = Q̃_{p}^{n}(v) = ∑_{i=0}^{m}(−1)^{i}q̃_{i}^{n}(p)v^{i}  (m = 2^{n})

be the polynomial over the p–ring H_{p}(q), whose coefficients are images of corresponding coefficients of the polynomial (2-27) over H_{p}(q_{1}) under the unramified embedding. Then Proposition 3.5.11 of [An(87)] implies the factorization

Q̃_{p}(v) = Q′(v)([1] − Π(p)v),

where Q′(v) is a polynomial of degree 2^{n}−1 over C′_{p}, which follows also from (3-60). For small n more explicit results can be obtained:

Proposition 3.18. Let p be a q–singular unramified prime number, and let n = 1. Then the polynomial Q̃_{p}^{1} has the following factorization over the ring (C_{p}^{1})′:

(3-64)  \begin{aligned}Q̃_{p}^{1}(v) & = [1] − T̃^{1}(p)v + p[p]v^{2} \\ & = ([1] − p[p]Π(p)^{−1}v)([1] − Π(p)v), \end{aligned}

where [1] = [1]_{K}, [p] = [p]_{K} with K = Γ_{0}^{1}(q), and where T^{1}(p) ∈ H_{p}^{1}(q_{1}) has the form (2-20). In particular, the middle coefficient T̃^{1}(p) can be written in the form

(3-65)  \begin{aligned}T̃^{1}(p) & = p[p]Π(p)^{−1} + Π(p) \\ & = (1 − p)Ω(p) + Π(p)^{∗} + Π(p), \end{aligned}

where Ω(p) = Ω^{1}(p, q) is the element (3-46), and where star stands for the antiautomorphism (1-18).

Proof. Applying the map (3-57) coefficient-wise to the factorization of Q̂_{p}^{1}(v) given in Proposition 3.5.13 of [An(87)], we get

Q̃_{p} = ([1] − Π̃_{0}^{1}(p)v)([1] − Π(p)v),

which proves (3-64), since

Π̃_{0}^{1}(p) = Π(p)^{−1} · i(Π_{0}^{1}(p)Π̂(p)) = Π(p)^{−1} · p[p],

by (3-62) and relation (3.4.13) of [An(87)]. The formulas (3-65) follow from (3-64), (3-53), and (3-39), since

ξ_{1}(p)Ω(p) = ξ_{1}(p)^{∗}Ω(p)^{∗} = (Ω(p)ξ_{1}(p))^{∗} = Π(p)^{∗},

by (3-34) and (3-49).

□

For n = 2 formulas are more complicated:

Proposition 3.19. Let p be a q–singular unramified prime number and n = 2. Then the polynomial Q̃_{p}^{2}(v) has the following factorization over C_{p}[Π(p)^{−1}]:

(3-66)

\begin{aligned}Q̃_{p}^{2}(v) & = [1] − T̃^{2}(p)v + q̃_{2}(p)v^{2} − p^{3}[p]T̃^{2}(p)v^{3} + p^{6}[p^{2}]v^{4} \\ & = ([1] − p^{3}[p]Π(p)^{−1}v)([1] − pΠ(p)^{−1}Π_{1}(p^{2})v + p^{3}[p]v^{2})([1] − Π(p)v), \end{aligned}

where [p^{δ}] = [p^{δ}]_{K} with K = Γ_{0}^{2}(q), T^{2}(p) and q_{2}(p) were defined in (2-20) and (2-26), and where

(3-67)  Π_{1}(p^{2}) = (diag(p, 1, p, p^{2}))_{K} ∈ C_{p}.

Proof. It follows from the factorization of the polynomial Q̂_{p}^{2}(v) given by Proposition 3.5.14 of [An(87)], where Π_{2,0}^{(2)} (a misprint) must be replaced by Π_{2,0}^{(0)}, that

Q̃_{p}^{2}(v) = ([1] − Π̃_{−}v)([1] − Π̃v + p(Π̃_{2,0}^{(1)} + Π̃_{2,0}^{(0)})v^{2})([1] − Π̃_{+}v),

where we have used the notation of the proposition. Using (3-56) and the relations (3.4.14), (3.5.23), and (3.5.25) of [An(87)], we obtain

Π̃_{−} = Π(p)^{−1} · i(Π_{−}Π_{+}) = p^{3}[p]Π(p)^{−1},
Π̃ = Π(p)^{−1} · i(ΠΠ_{+}) = Π(p)^{−1} · i(pΠ_{1,1}^{(0)}) = pΠ(p)^{−1} · Π_{1}(p^{2}),
Π̃_{2,0}^{(1)} + Π̃_{2,0}^{(0)} = Π(p)^{−2} · i(Π_{2,0}^{(1)}Π_{+}^{2} + Π_{2,0}^{(0)}Π_{+}^{2}) = Π(p)^{−2} · i(p^{2}[p]Π_{+}^{2}) = p^{2}[p],

and

Π̃_{+} = Π(p)^{−1} · i(Π_{+}Π_{+}) = Π(p).
□

We shall say that an element T of a HS–ring of the group K = Γ_{0}^{n}(q) is symmetric if it is invariant under the star map (1-18):

(3-68)  T^{∗} = T.

It looks likely that local p–rings H_{p}(q) in unramified cases consist of symmetric elements, as it was in regular cases according to Theorem 2.3. Since the rings H_{p}(q) together with H_{p}(q/p) are commutative polynomial rings, it would sufficient to check (3-68) for a set of generators, say, for

(3-69) T̃(p) = Π(p)^{−1} · i(T̂^{n}(p)Π̂(p)),  T̃_{j}^{n}(p^{2}) = Π(p)^{−2} · i(T̂_{j}^{n}(p^{2})Π(p)^{2}),

where j = 1, …, n (see Theorem 2.6 for H_{p}^{n}(q/p)). However, the check seems to be quite informal, requires detailed information on products of double cosets in the corresponding HS– rings, and extent of necessary computations grows tremendously with increase of n. Therefore we shall restrict ourselves only to the cases n = 1 and n = 2.

Proposition 3.20. Let p be a q–singular unramified prime number. Then the p–rings H_{p}^{1}(q) and H_{p}^{2}(q) consist of symmetric elements.

Proof. By (2-20) and (3-62), we have

T̃_{1}^{1}(p^{2}) = [p]_{Γ_{0}^{1}(q)} and T̃_{2}^{2}(p^{2}) = [p]_{Γ_{0}^{2}(q)},

which elements are clearly symmetric. The element T̃^{1}(p) is symmetric, by formulas (3-65) and (3-49), which concludes the story for n = 1. If n = 2, as we have seen, it is sufficient to show that elements

(3-70)  T̃(p) = T̃^{2}(p) and q̃_{2}(p) = pT̃_{1}^{2}(p^{2}) + p(p^{2} + 1)[p]_{Γ_{0}^{2}(q)}

(see (2-26)) are symmetric, which is far from being easy and requires more information on multiplication of certain double cosets in the ring

(3-71)  H_{p}^{2}(q_{1}, q) = HS(K, Σ_{p}^{2}(q_{1})),

where K = Γ_{0}^{2}(q), q_{1} = q/p, and

(3-72)  Σ_{p}^{2}(q_{1}) = \{M = ( \begin{matrix}A & B \\ C & D \end{matrix} ) ∈ Σ_{p}^{2}; C ≡ 0 (mod q_{1})\} .

More concretely, we shall need formulas for products of double cosets of multiplier 1 by double cosets of multiplier 1 and p.

As to products of double cosets of multiplier 1, the cosets belong to the neutral subring N^{2}(q_{1}, q) of the ring (3-71) and, by Lemma 3.10, we have the formulas

(3-73)  \begin{matrix}ξ · ξ_{1} = ξ_{1} · ξ = p^{2}ξ_{1} + (p^{2} − 1)ξ, \\ ξ^{2} = p^{3}[1] + p^{2}(p − 1)ξ_{1} + p^{2}(p − 1)ξ, \end{matrix}

where ξ_{1} = ξ_{1}^{2}(p), ξ = ξ_{2}^{2}(p) are the elements (3-31), which imply the relations

(3-74)  ξ^{−1}ξ_{1} = ((1 − p^{2})[1] + ξ_{1})/p^{2},

since

ξ((1 − p^{2})[1] + ξ_{1})/p^{2} = ((1 − p^{2})ξ + p^{2}ξ_{1} + (p^{2} − 1)ξ)/p^{2} = ξ_{1}.

In order to consider double cosets of multiplier p we have first to present in a convenient form representatives of all left cosets KM, where M ∈ Σ_{p}^{2}(q_{1}) and μ(M) = p. For that one can multiply a system of representatives for cosets K\Γ_{0}^{2}(q_{1}) by a system of representatives for cosets

(3-75)  Γ_{0}^{2}(q_{1})\ {M ∈ Σ_{p}^{2}(q_{1}); μ(M) = p} .

As to the first system, we shall take the system of representatives described in Lemma 3.8, that is the union of the three sets:W^{0} = {E_{4}},

W^{1} = M_{1} \{( \begin{matrix}ε & δ ·^{t}ε^{−1} \\ 0 & ^{t}ε^{−1} \end{matrix} ); ε = ( \begin{matrix}1 & c \\ 0 & 1 \end{matrix} ) , ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) , δ = ( \begin{matrix}d & 0 \\ 0 & 1 \end{matrix} ) , c, d mod p\}
= M_{1} · R_{1},

and

W^{2} = M_{2} \{( \begin{matrix}E & B \\ 0 & E \end{matrix} ); B ∈ 𝕊_{2}/p𝕊_{2}\} = M_{2} · R_{2},

where E = E_{2},

(3-76)  M_{1} = ( \begin{matrix}a & 0 & b & 0 \\ 0 & 1 & 0 & 0 \\ q_{1} & 0 & p & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ) , and M_{2} = ( \begin{matrix}aE & bE \\ q_{1}E & pE \end{matrix} )

with a and b satisfying ap − bq_{1} = 1.

Since p does not divide q_{1}, we could take, by Lemma 2.1, as representatives for the cosets (3-75) the set of triangular representatives for cosets

Γ_{0}^{2}\ {M ∈ Σ_{0,p}^{2}; μ(M) = p} ,

which can be described using Lemma 3.3.32 and Lemma 3.2.7 of [An(87)] as the union of three sets:

{D_{0}}, D_{1} · R_{1}, and D_{2} · R_{2},

where

D_{0} = diag(p, p, 1, 1), D_{1} = diag(1, p, p, 1), D_{2} = diag(1, 1, p, p).

But it will be more convenient for further calculations to modify the representatives on replacing the matrices D_{0} and D_{1} by matrices D′_{0} = M_{2}D_{0} and

D′_{1} = ( \begin{matrix}I & 0 \\ 0 & I \end{matrix} ) M_{1} ( \begin{matrix}I & 0 \\ 0 & I \end{matrix} ) D_{1} = M′_{1}D_{1} with I = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) ,

respectively, where M_{1} and M_{2} are the matrices (3-76). Note that

(3-77)  D′_{0} = H(p) and D′_{1} = ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & pa & 0 & b \\ 0 & 0 & p & 0 \\ 0 & q & 0 & p \end{matrix} ) ,

where H(p) = H^{2}(p, q) is the matrix (3-43) for n = 2.

If we multiply each of three sets of the chosen system of representatives for cosets K\Γ_{0}^{2}(q_{1}) by each of three sets of the modified system of representatives for cosets (3-75) and take into account formulas (3-77), we obtain a complete system of representatives of different left cosets of the set

{M ∈ Σ_{p}^{2}(q_{1}); μ(M) = p}

modulo K in the form of union of the following nine sets:

S_{1} = {H} ,  S_{2} = {D′_{1}R_{1}} ,  S_{3} = {D_{2}R_{2}} ,
S_{4} = {M_{1}R_{1}H} ,  S_{5} = {M_{1}R_{1}D′_{1}R_{1}} ,  S_{6} = {M_{1}R_{1}D_{2}R_{2}} ,
S_{7} = {M_{2}R_{2}H} ,  S_{8} = {M_{2}R_{2}D′_{1}R_{1}} ,  S_{9} = {M_{2}R_{2}D_{2}R_{2}} .

Let ( \begin{matrix}A & B \\ 0 & D \end{matrix} ) be a matrix of Γ_{0}^{n}, then the matrices

H^{−1} ( \begin{matrix}A & B \\ 0 & D \end{matrix} ) H = ( \begin{matrix}apA + qB − bq_{1}D & bA + pB − bD \\ −aqA − q_{1}qB + aqD & −q_{1}bA − qB + apD \end{matrix} )

and

D_{2}^{−1} ( \begin{matrix}A & B \\ 0 & D \end{matrix} ) D_{2} = ( \begin{matrix}A & pB \\ 0 & D \end{matrix} )

belong to K. As to the matrix (D′_{1})^{−1} ( \begin{matrix}A & B \\ 0 & D \end{matrix} ) D′_{1}, one can easily check after

routine multiplication of the matrices that it belongs to K if ( \begin{matrix}A & B \\ 0 & D \end{matrix} ) ∈ R_{2} or

( \begin{matrix}A & B \\ 0 & D \end{matrix} ) ∈ R′_{1} = \{( \begin{matrix}ε & δ ·^{t}ε^{−1} \\ 0 & ^{t}ε^{−1} \end{matrix} ); ε = ( \begin{matrix}1 & c \\ 0 & 1 \end{matrix} ) , δ = ( \begin{matrix}d & 0 \\ 0 & 1 \end{matrix} ) , c, d mod p\} .

Therefore we have the inclusions:

(3-78)  \begin{aligned}S_{1} & ⊂ KHK, S_{2} ⊂ KD′_{1}K, S_{3} ⊂ KD_{2}K, \\ S_{4} & = \{M_{1}HH^{−1}R_{1}H\} ⊂ KM_{1}HK, \\ S_{6} & = \{M_{1}D_{2}D_{2}^{−1}R_{1}D_{2}R_{2}\} ⊂ KM_{1}D_{2}K, \\ S_{7} & = \{M_{2}HH^{−1}R_{2}H\} ⊂ KM_{2}HK, \\ S_{8} & = \{M_{2}D′_{1}(D′_{1})^{−1}R_{2}D′_{1}R_{1}\} ⊂ KM_{2}D′_{1}K, \\ S_{9} & = \{M_{2}D_{2}D_{2}^{−1}R_{2}D_{2}R_{2}\} ⊂ KM_{2}D_{2}K. \end{aligned}

As to the set S_{5}, we divide the set R_{1} into the subset R′_{1} defined above and the complementary subset

R′′_{1} = \{( \begin{matrix}I & δI \\ 0 & I \end{matrix} ) = ( \begin{matrix}I & 0 \\ 0 & I \end{matrix} ) ( \begin{matrix}E & δ′ \\ 0 & E \end{matrix} ); δ′ = IδI = ( \begin{matrix}1 & 0 \\ 0 & d \end{matrix} ) , d mod p\} .

Then we have

(3-79)

S_{5} = {M_{1}R′_{1}D′_{1}R_{1}} ⋃ {M_{1}R′′_{1}D′_{1}R_{1}}

= \{M_{1}D′_{1}(D′_{1})^{−1}R′_{1}D′_{1}R_{1}\} ⋃ \{M_{1} ( \begin{matrix}I & 0 \\ 0 & I \end{matrix} ) D′_{1}(D′_{1})^{−1} ( \begin{matrix}E & δ′ \\ 0 & E \end{matrix} ) D′_{1}R_{1}\}

= S′_{5} ⋃ S′′_{5}

⊂ KM_{1}D′_{1}K ⋃ KM_{1} ( \begin{matrix}I & 0 \\ 0 & I \end{matrix} ) D′_{1}K.

On the other hand, if we multiply corresponding double cosets using decompositions (3-47), (3-32), (3-4), (3-50), and the relation

D′_{1} = HQM_{1} with Q = H^{−1}D′_{1}M_{1}^{−1} = ( \begin{matrix}∗ & ∗ \\ Q_{2,1} & ∗ \end{matrix} ) ∈ K,

since Q_{2,1} = ( \begin{matrix}−q(a + 1) & 0 \\ 0 & 0 \end{matrix} ), we obtain the following decompositions:

(3-80)

Ω = (KH),  Ωξ_{1} = ∑_{M∈S_{2}}(KM),  Ωξ = ∑_{M∈S_{3}}(KM),
ξ_{1}Ω = ∑_{M∈S_{4}}(KM),  ξ_{1}Ωξ_{1} = ∑_{M∈S_{5}}(KM),  ξ_{1}Ωξ = ∑_{M∈S_{6}}(KM)
ξΩ = ∑_{M∈S_{7}}(KM),  ξΩξ_{1} = ∑_{M∈S_{8}}(KM),  ξΩξ = ∑_{M∈S_{9}}(KM),

where Ω = Ω^{2}(p, q) is the double coset (3-46), ξ_{1} = ξ_{1}^{2}(p) and ξ = ξ_{2}^{2}(p) are the cosets (3-31). By comparing the decompositions (3-80) with inclusions (3-78) and (3-79), we get the relations

Ωξ_{1} = (D′_{1})_{K},  Ωξ = (D_{2})_{K},  ξ_{1}Ω = (M_{1}H)_{K},  ξ_{1}Ωξ = (M_{1}D_{2})_{K},
ξΩ = (M_{2}H)_{K},  ξΩξ_{1} = (M_{2}D′_{1})_{K},  ξΩξ = (M_{2}D_{2})_{K},

and

(3-81)  ξ_{1}Ωξ_{1} = Φ + Ψ,

where

(3-82)  Φ = ∑_{M∈S′_{5}}(KM) = (P)_{K}, and Ψ = ∑_{M∈S′′_{5}}(KM) = (P′)_{K}

with

P = M_{1}D′_{1} = ( \begin{matrix}∗ & ∗ \\ P_{2,1} & ∗ \end{matrix} ) , and P′ = M_{1} ( \begin{matrix}I & 0 \\ 0 & I \end{matrix} ) D′_{1} = ( \begin{matrix}∗ & ∗ \\ P′_{2,1} & ∗ \end{matrix} ) ,

where

P_{2,1} = ( \begin{matrix}q_{1} & 0 \\ 0 & q \end{matrix} ) , P′_{2,1} = ( \begin{matrix}0 & q(a + p) \\ 0 & 0 \end{matrix} ) .

Note that the double cosets Φ and Ψ are different, since the blocks P_{2,1} and P′_{2,1} have different ranks modulo p.

With the knowledge of all ten double cosets of multiplier p of the ring (3-71) together with their decompositions into left cosets we can deduce all necessary properties of the double cosets. First of all, by (3-34) and (3-49), we have

Φ^{∗} + Ψ^{∗} = ξ_{1}^{∗}Ω^{∗}ξ_{1}^{∗} = ξ_{1}Ωξ_{1} = Φ + Ψ,

whence

(3-83)  Φ^{∗} = Φ and Ψ^{∗} = Ψ,

since Φ^{∗} ≠ Ψ, by the same reason that we have used to show that Φ ≠ Ψ. Verification of the following three relations is nothing else but a standard but tiresome exercise on multiplication of double cosets in HS–rings, which we leave to the reader:

ξΨ = pΦ + (p − 1)ξΩξ_{1}; (3-84) ΩΨ = ΨΩ; (3-85)

Π_{1}(p^{2}) = ΩΦ, (3-86)

where Π_{1}(p^{2}) is the element (3-67). From (3-84) follows the relation

(3-87)  ξ^{−1}Φ = (Ψ + (1 − p)Ωξ_{1})/p,

since

ξ(Ψ + (1 − p)Ωξ_{1})/p = (pΦ + (p − 1)ξΩξ_{1} + (1 − p)ξΩξ_{1})/p = Φ.

Finally, by (3-74), (3-87), and (3-81), we obtain

(3-88)

ξ^{−1}Ψ = ξ^{−1}(ξ_{1}Ωξ_{1} − Φ) = ((1 − p^{2})[1] + ξ_{1})Ωξ_{1}/p^{2} − (Ψ + (1 − p)Ωξ_{1})/p
= (Φ + (1 − p)Ψ + (1 − p)Ωξ_{1})/p^{2}.

Returning to elements (3-70), we can now present them in symmetric form. By (3-66), we have

T̃^{2}(p) = p^{3}[p]Π^{−1} + pΠ^{−1}Π_{1}(p^{2}) + Π
= p^{3}ξ^{−1}Ω + pξ^{−1}Ω^{−1}ΩΦ + Ωξ,

where we have used the relations (3-53), (3-86), and (3-50). By (3-40) and (3-87), last expression can be rewritten in the form

(3-89)

T̃^{2}(p) = (ξ + (1 − p)ξ_{1} + (1 − p)[1])Ω + (Ψ + (1 − p)Ωξ_{1}) + Ωξ
= ξΩ + Ωξ + (1 − p)(ξ_{1}Ω + Ωξ_{1}) + Ψ + (1 − p)Ω,

which is symmetric, by (3-34), (3-49), and (3-83). Again by (3-66), we have

q̃_{2}(p) = p^{4}[p]Π^{−1}Π^{−1}Π_{1}(p^{2}) + pΠ^{−1}Π_{1}(p^{2})Π + 2p^{3}[p]
= p^{4}ξ^{−1}Ωξ^{−1}Ω^{−1}ΩΦ + pΩΦ + 2p^{3}[p],

where we have used the relations (3-53), (3-48), (3-86), and Theorem 3.3(4). By (3-87), (3-85), and (3-88), the last expression can be rewritten in the form

(3-90)

q̃_{2}(p) = p^{3}ξ^{−1}Ω(Ψ + (1 − p)Ωξ_{1}) + pΩΦ + 2p^{3}[p]
= p^{3}ξ^{−1}ΨΩ + p^{3}(1 − p)[p]ξ^{−1}ξ_{1} + pΩΦ + 2p^{3}[p]
= p(Φ + (1 − p)Ψ + (1 − p)Ωξ_{1})Ω + p(1 − p)[p]((1 − p^{2})[1] + ξ_{1})
+ pΩΦ + 2p^{3}[p]

= p(ΦΩ + ΩΦ) + p(1 − p)(ΨΩ + Ωξ_{1}Ω + [p]ξ_{1}) + p(p^{3} + p^{2} − p + 1)[p],

which is symmetric, by (3-49), (3-83), (3-85), and (3-34).

□

Chapter 4

Old forms, new forms, eigenforms

In this chapter we shall define old and new cusp forms for the group Γ_{0}^{n}(q) and prove, in particular, that the spaces of new forms of genus n ≤ 2 are spanned by common eigenfunctions of all Hecke operators corresponding to elements of the regular reducible ring H_{rr}^{n}(q), of the singular unramified ring

(4-1)  H_{su}^{n}(q) = ∏_{p|q,p^{2}∤q} H_{p}^{n}(q)

generated by all singular unramified p–rings, and to all of the singular Frobenius elements Π(a) (see Theorem 4.15). If n = 1, the result for regular operators belongs to Petersson (see [Pet(39)]); action of singular Frobenius elements was considered by Atkin and Lehner in [A-L(70)].

We restrict ourselves here only to the case of cusp forms. One can also consider old and new Eisenstein–Klingen series, but it is another story.

§4.1. Trace–idempotents

We shall intoduce here a simple but useful abstract tool.

Let G be a multiplicative group and Γ ⊂ G a subgroup of finite index ν = [G : Γ]. Let us consider the element of HS–ring H = HS_{ℚ}(Γ, G) given by

(4-2)  τ = τ(Γ\G) = ν^{−1} ∑_{g_{i}∈Γ\G}(Γg_{i}).

Lemma 4.1. The element (4-2) satisfies the following rules:

τg = ν^{−1} ∑_{g_{i}∈Γ\G}(Γg_{i}g) = τ, (g ∈ G) (4-3)

(4-4) τ · (g)_{Γ} = τ · ∑_{g_{j}∈Γ\ΓgΓ}(Γg_{j}) = #(Γ\ΓgΓ) · τ,

and τ^{2} = τ.

Proof. Since {g_{i}g} is just another system of representatives for Γ\G, the relation (4-3) follows. The relation (4-4) is a consequence of (4-3). The last relation follows from (4-3) and definition of products in HS–rings.

□

We shall call the element τ = τ(Γ\G) the trace–idempotent of the pair (Γ, G)

Suppose now that the group G acts on a ℚ–vector space V by linear operators

|g : v → v|g, (v ∈ V, g ∈ G)

satisfying

v|g|g′ = v|gg′ (v ∈ V, g, g′ ∈ G).

Denote by V (Γ) and V (G) the subspaces of Γ–invariant and G–invariant elements of V, respectively. Then V (G) ⊂ V (Γ), and HS–ring H acts on the space V (Γ) by Hecke operators. Since the element (4-2) is an idempotent, it follows that the space V (Γ) is the direct sum of subspaces

V_{1}(Γ) = V (Γ)|τ, V_{2}(Γ) = {v ∈ V (Γ); v|τ = 0} :
V (Γ) = V_{1}(Γ) + V_{2}(Γ), V_{1}(Γ) ⋂ V_{2}(Γ) = {0}.

Lemma 4.2. V_{1}(Γ) = V (G).

Proof. The inclusion ⊂ follows from (4-3). The inverse inclusion is obvious.

□

Finally, let us assume that the space V is equipped with a nondegenerate Hermitian scalar product (v, v′) satisfying

(v|g, v′) = (v, v′|g^{−1}) (v, v′ ∈ V, g ∈ G).

Then the subspace V_{2}(Γ) can be characterised as the orthogonal complement to the subspace V (G) in V (Γ):

Lemma 4.3. V_{2}(Γ) = V (G)^{⊥} = {v ∈ V (Γ); (v, V (G)) = 0}.

Proof. Let v ∈ V_{2}(Γ) and v′ = v′|τ ∈ V (G) = V_{1}(Γ). Then we have

(v, v′) = (v, v′|τ) = ν^{−1} ∑_{g_{i}∈Γ\G}(v, v′|τg_{i}^{−1})

= ν^{−1} ∑_{g_{i}∈Γ\G}(v|g_{i}, v′|τ) = (v|τ, v′) = 0,

and so v ∈ V (G)^{⊥}.Conversely, if v ∈ V (G)^{⊥}, then we have

(v|τ, v|τ) = ν^{−1} ∑_{g_{i}∈Γ\G}(v|τ, v|g_{i})

= ν^{−1} ∑_{g_{i}∈Γ\G}(v|τg_{i}^{−1}, v) = (v|τ, v) = 0,

and so v|τ = 0.

□

Summing up, we obtain the following theorem:

Theorem 4.4. With the above notation and assumptions, the subspace V (G) ⊂ V (Γ) and its ortogonal complement V (G)^{⊥} can be characterized by

(4-5)  V (G) = V (Γ)|τ(Γ\G),  V (G)^{⊥} = {v ∈ V (Γ); , v|τ(Γ\G) = 0} .

§4.2. Definition of old and new forms

We are considering the representation of the HS–ring H(K) = HS_{ℚ}(K, Σ^{n}) of the group K = Γ_{0}^{n}(q) on the space

𝔑 = 𝔑(q) = 𝔑_{k}^{n}(q) = 𝔑_{k}(K)

of cusp forms of a fixed integral weight k for the group K given by the Hecke operators (1-17) or normalized Hecke operators (1-20). For definiteness, we shall speak on the operators (1-17),although all the definitions, results and proofs (with relations (1-21) in place of Theorem 1.9(2)) remains valid also for normalized operators. The space 𝔑 is equipped with the Petersson scalar product (1-13), which is Hermitian, nondegenerate, and satisfies relations (1-14).

Let p be a prime divisor of q, and let Π(p) and Π^{∗}(p) be the corresponding Frobenius element (3-2) and its dual with respect to the antiautomorphism (1-18). We shall define the subspace of p–old forms of 𝔑 by

(4-6)  O_{p} = O_{k}^{n}(q)_{p} = ∑_{i=0}^{m−1} 𝔑(q/p)|Π^{∗}(p)^{i},  (m = 2^{n}),

if p is unramified (i.e. p^{2} ∤ q), and by

(4-7)  O_{p} = O_{k}^{n}(q)_{p} = 𝔑(q/p) + 𝔑(q/p)|Π^{∗}(p),

if p is ramified (i.e. p^{2}|q), where

(4-8)  𝔑(q/p) = 𝔑_{k}(Γ_{0}^{n}(q/p)),

and | = |_{k} stand for the Hecke operators (1-17).

Let

(4-9)  τ(p) = τ^{n}(p, q) = [K′ : K]^{−1} ∑_{M_{i}∈K\K′}(KM_{i}),

where K′ = Γ_{0}^{n}(q/p), be the trace–idempotent (4-2) of the pair (K, K′),then, by Theorem 4.4, we can write

(4-10)  O_{p} = ∑_{i=0}^{m−1} 𝔑|τ(p) · Π^{∗}(p)^{i},  (m = 2^{n})

if p^{2} does not divide q, and

(4-11)  O_{p} = 𝔑|τ(p) + 𝔑|τ(p)Π^{∗}(p),

if p^{2} divides q.

The space

(4-12)  O = O_{k}^{n}(q) = ∑_{p prime,p|q} O_{p}

will be called the (global) subspace of old forms of 𝔑.

We shall define the subspace N_{p} of p–new forms and the subspace N of (global) new forms as orthogonal complements with respect to Petersson scalar product of the subspace of p–old forms and the subspace of old forms, respectively:

(4-13)  N_{p} = N_{k}^{n}(q)_{p} = {F ∈ 𝔑; (F, O_{p}) = 0} ,

(4-14)  N = N_{k}^{n}(q) = {F ∈ 𝔑; (F, O) = 0} .

Then we have

(4-15)  N = ⋂_{p prime,p|q} N_{p},

and, by Theorem 4.4 and Theorem 1.9(2) each of the subspaces N_{p} can be presented in the form

(4-16)  N_{p} = {F ∈ 𝔑; F |Π(p)^{i}τ(p) = 0, i = 0, 1, …, 2^{n} − 1} ,

if p^{2} does not divide q, or in the form

(4-17)  N_{p} = {F ∈ 𝔑; F |τ(p) = 0, F |Π(p)τ(p) = 0} ,

if p^{2} divides q.

§4.3. Invariant subspaces and eigenforms

Here we shall consider the questions of invariance of subspaces of old and new forms under Hecke operators and show that under certain conditions the subspaces of new forms are spanned by common eigenfunctions of the operators. We preserve the notation introduced in §4.2.

Lemma 4.5. Let p and p′ be two prime numbers, such that p divides q, but p′ does not divide q. Then the following statements hold:

(1) The subspace 𝔑(q/p) ⊂ 𝔑 is invariant under each of the Hecke operators |T with T ∈ H_{p′}(q);

(2) Every element of the ring H_{p′}(q) commutes with the Frobenius element Π(p) and with its dual Π^{∗}(p).

Proof. By Theorem 2.6, the rings H_{p′}(q) and H_{p′}(q/p) are isomorphic, and, if T ∈ H_{p′}(q), T′ ∈ H_{p′}(q/p) is a pair of elements related by the isomorphism, then their images T̂ and T̂′ under parabolic embedding (2-30) coincide. It follows that

𝔑(q/p)|T = 𝔑(q/p)|T̂ = 𝔑(q/p)T̂′
= 𝔑(q/p)|T′ ⊂ 𝔑(q/p),

which proves the first part.

As to the second part, by Proposition 2.12(1), we have

Π̂T = Π̂T̂ = T̂Π̂ = T̂ Π (Π = Π(p))

for every T ∈ H_{p′}(q), whence ΠT = T Π. It follows that

T^{∗}Π^{∗} = (ΠT)^{∗} = (T Π)^{∗} = Π^{∗}T^{∗},

which together with Theorem 2.3 concludes the proof of the lemma.

□

Theorem 4.6. (1) Each of the local subspaces O_{p} and N_{p} with p|q, as well as the global subspaces O and N are invariant under all of the Hecke operators corresponding to elements of the regular reducible ring H_{rr}^{n}(q) defined by (2-3);

(2) Each of the listed subspaces is spanned by common eigenfunctions of all operators of the part (1).

Proof. By Theorem 2.5, it is sufficient to prove the theorem for a local ring H_{p′}(q) with p′ ∤ q instead of H_{rr}^{n}(q). It follows from Lemma 4.5 and the definitions (4-6), (4-7) of subspaces of old forms, that every of O_{p} is invariant under |T with T ∈ H_{p′}(q), and so does their sum (4-12). The invariance of N_{p} and N follows by (4-13), (4-14), (2-6), and Theorem 1.9(2). The part (1) is proved.

By Theorem 2.3 and Theorem 1.9(2), the operators |T with T ∈ H_{rr}^{n}(q) commute with each other and are selfadjoint. Since the space 𝔑 is finite–dimensional, the second part follows from the first one, by a well-known theorem of linear algebra.

□

Let us turn now to singular Hecke operators.

Proposition 4.7. Let p be an unramified prime divisor of q. Then the relation

(4-18)  F |T̃ = F |T

holds for every F ∈ 𝔑(q/p) and every T ∈ H_{p}(q/p), where T → T̃ is the mapping (3-58). In particular, the space 𝔑(q/p) is invariant under all of the Hecke operators correponding to elements of the unramified p–ring H_{p}(q).

Proof. It is sufficient to check the relations (4-18) for generators (3-69) of H_{p}(q). By the definition of elements T^{n}(p) and T_{j}^{n}(p^{2}) ( see (2-20)), and Proposition 2.11, we have inclusions

T̂^{n}(p)Π̂(p), T̂_{j}^{n}(p^{2})Π̂(p)^{2} ∈ Ĉ_{p},

and so, by (3-56),

T̃^{n}(p) = Π(p)^{−1} · i (T̂^{n}(p)Π̂(p)) , T̃_{j}^{n}(p^{2}) = Π(p)^{−2} · (T̂_{j}^{n}(p^{2})Π̂(p)^{2}) .

Since Hecke operators on 𝔑 corresponding to elements T̂ ∈ Ĉ_{p} and i(T̂) ∈ C_{p} clearly coincide, the relations (4-18) for the generators will follow from (2-49) and (2-50), if we can prove the formulas

(4-19)  F |Π(p)^{−1} = F |Π_{+}^{−1}

and

(4-20)  F |Π(p)^{−2} = F |Π_{+}^{−2}

for every F ∈ 𝔑(q/p), where

Π_{+}^{−1}, Π_{+}^{−2} ∈ L′_{p} = L_{p}[[p]^{−1}]

are first ”negative powers” of the Frobenius element Π_{+} = Π̂(p).

According to Proposition 3.9, we obtain the relations

F |ξ_{n}(p) = p^{⟨n⟩}F, (F ∈ 𝔑(q/p))

which imply the relations

(4-21)  F |ξ_{n}(p)^{−1} = p^{−⟨n⟩}F.  (F ∈ 𝔑(q/p))

Then, by (3-53) and (3-43), we get

F |Π(p)^{−1} = F |[p]^{−1}ξ_{n}(p)^{−1}Ω(p)

= F |(p^{⟨n⟩}[p])^{−1} ( \begin{matrix}aE_{n} & bE_{n} \\ (q/p)E_{n} & pE_{n} \end{matrix} ) ( \begin{matrix}pE_{n} & 0 \\ 0 & E_{n} \end{matrix} )

= F |(p^{⟨n⟩}[p])^{−1} ( \begin{matrix}pE_{n} & 0 \\ 0 & E_{n} \end{matrix} ) ,

which is equal to F |Π_{+}^{−1} ( see, for example, [An(87), Exercise 3.5.7(3)]. We have proved the relations (4-19), which, in particular imply the inclusion

𝔑(q/p)|Π_{+}^{−1} ⊂ 𝔑.

Let us show also that

(4-22)  𝔑(q/p)|Π_{+}^{−2} ⊂ 𝔑.

By the definition of ”negative powers” ( see [An(87), §3.5 ]), we have

Π_{+}^{−2} = −q̂_{m}^{−1} (∑_{i=1}^{m}(−1)^{i}{ \atop q_{m−i}^{n}}(p)Π_{+}^{i−2}) ,

where m = 2^{n}, and q̂_{j}^{n}(p) are images under parabolic embedding of corresponding coefficients of the polynomial (2-27) over the ring H_{p}(q/p). It follows that

𝔑(q/p)|Π_{+}^{−2} ⊂ ∑_{i=1}^{m} 𝔑(q/p)|q̂_{m}^{n}(p)^{−1}{ \atop q_{m−i}^{n}}(p)Π_{+}^{i−2}

⊂ 𝔑(q/p)|Π_{+}^{−1} + 𝔑(q/p) + ∑_{i=3}^{m} 𝔑(q/p)|Π_{+}^{i−2}

⊂ 𝔑,

since, by definitions and Theorem 3.3,

𝔑(q/p)|Π_{+}^{d} = 𝔑(q/p)|Π(p)^{d} ⊂ 𝔑,

if d ≥ 1. We can prove now the relation (4-20). Let F ∈ 𝔑(q/p). Set F_{1} = F |Π(p)^{−2}, F_{2} = F |Π_{+}^{−2}.

Since the operator |Π(p)^{−2} maps the space 𝔑 into itself, it follows that F_{1} ∈ 𝔑. By (4-22), we have inclusions F_{2} ∈ 𝔑. Also

F_{1}|Π(p)^{2} = F |Π(p)^{−2}Π(p)^{2} = F,

and

F_{2}|Π(p)^{2} = F |Π_{+}^{−2}|Π_{+}^{2} = F |Π_{+}^{−2}Π_{+}^{2} = F,

by (2-38), (3-4), the definition of Hecke operators, and relation (2-50) with d = 2 and X = [1]. Since the operator |Π(p)^{2} is invertible on 𝔑, we conclude that F_{1} = F_{2}.

□

We can now prove the following theorem:

Theorem 4.8. Suppose that n = 1 or n = 2. Then the following asserions holds:

(1) Each of the local subspaces O_{p} and N_{p} with prime p dividing q, as well as the global subspaces O and N are invariant under all of the Hecke operators corresponding to elements of the singular unramified ring H_{su}^{n}(q) defined by (4-1);

(2) Each of the listed subspaces is spanned by common eigenfunctions of all of the operators.

Proof. According to Theorem 3.17(2) and (4-1), it is sufficient to prove the theorem for a local singular unramified ring H_{p′}(q) instead of H_{su}(q).

Let p be a singular prime number, and let T ∈ H_{p′}(q). Then T^{∗} = T, by Proposition 3.20, and so, by Theorem 3.17(2), we have Π(p)T = T Π(p), and

(4-23)  T · Π^{∗}(p) = T^{∗} · Π^{∗}(p) = (Π(p) · T)^{∗} = (T · Π(p))^{∗} = Π^{∗}(p) · T.

It follows from (4-23) and (4-6) or (4-7), that subspace O_{p} is invariant under operator |T, if the inclusions

(4-24)  𝔑(q/p)|T ⊂ 𝔑(q/p)  (T ∈ H_{p′}(q))

are valid. By Proposition 4.7, the inclusions are true,if p′ = p. So we can assume that p′ ≠ p. By Theorem 4.4, for every singular prime number p we can write

(4-25)  𝔑(q/p) = 𝔑|τ(p),

where τ(p) is the trace–idempotent (4-9). On the other hand, every element T ∈ H_{p′}(q) has the form T = Π(p′)^{−d}T_{1} with d ∈ ℕ and T_{1} ∈ C_{p′}. By Proposition 3.9(4), if p^{2} does not divide q, and Proposition 3.12(3), if p^{2} divides q, we have

(4-26)  τ(p)Π(p′) = Π(p′)τ(p),

whence

(4-27)  τ(p)Π(p′)^{−1} = Π(p′)^{−1}τ(p).

It follows from (4-27) and (4-25) that

𝔑(q/p)|Π(p′)^{−d} = 𝔑|τ(p)Π(p′)^{−d} = (𝔑|Π(p′)^{−d})|τ(p) ⊂ 𝔑(q/p).

Since p′ divides q/p, it follows from Theorem 3.3 that every operator |T_{1} with T_{1} ∈ C_{p′} maps the space 𝔑(q/p) into itself. The inclusion (4-24) and the invariance of the subspace O_{p} follows. This implies the invariance of the subspace O. The invariance of N_{p} and N follows by (4-13), (4-15), Theorem 1.9(2), and Proposition 3.20. The part (1) is proved.

By Theorem 3.17(2), the ring H_{su}^{n} is commutative. It follows then, by Proposition 3-20, that it consist of symmetric elements. The part (2) follows from Theorem 1.9(2), by the same tautology, as end of the proof of Theorem 4.6.

□

Remark 4.9. The same proof goes through without any changes for a given n, if one knows that all all of unramified local rings H_{p}^{n}(q) consist of symmetric elements.

We turn now to Hecke operators corresponding to singular Frobenius elements (3-2).

Lemma 4.10. Let p be an unramified prime divisor of q and n = 1 or n = 2. Then the subspaces O_{p} and N_{p} are both invariant under Hecke operators corresponding to elements Π(p)^{±1}, Π^{∗}(p)^{±1}, ξ_{n}(p), and Ω(p).

Proof. By Theorem 1.9(2), it is sufficient to consider only the case of subspace O_{p}. By (4-6), we have

(4-28)  O_{p}|Π^{∗}(p) = ∑_{i=1}^{m} 𝔑(q/p)|Π^{∗}(p)^{i} ⊂ O_{p} + 𝔑(q/p)|Π^{∗}(p)^{m}.

By Theorem 3.17(2),(3) and Proposition 3.20, we get the relation

(4-29)  \begin{aligned}(∑_{j=0}^{m}(−1)^{j}q̃_{j}^{n}(p)Π(p)^{m−j})^{∗} & = (∑_{j=0}^{m}(−1)^{j}Π(p)^{m−j}q̃_{j}^{n}(p)^{∗})^{∗} \\ & = ∑_{j=0}^{m} q̃_{j}^{n}(p)Π^{∗}(p)^{m−j} = 0, \end{aligned}

where the coefficients q̃_{j}^{m}(p) belong to H_{p}(q). It follows then, by (3-61), that

Π^{∗}(p)^{m} = − ∑_{j=1}^{m}(−1)^{j}q̃_{j}^{n}(p)Π^{∗}(p)^{m−j},

whence, by Proposition 4.7, we get

𝔑(q/p)|Π^{∗}(p)^{m} ⊂ ∑_{j=1}^{m} 𝔑(q/p)|q̃_{j}^{n}(p)Π^{∗}(p)^{m−j}

⊂ ∑_{j=1}^{m} 𝔑(q/p)|Π^{∗}(p)^{m−j} = O_{p},

which together with (4-28) proves the invariance of O_{p} under Π^{∗}(p).

Since, by (3-50), (3-49), and (3-34), we have

(4-30)  Π^{∗}(p) = (Ω(p)ξ_{n}(p))^{∗} = ξ_{n}(p)Ω(p),

it follows from Lemma 3.11 and (3-48) that the element Π^{∗}(p) is invertible and

(4-31)  Π^{∗}(p)^{−1} = [p]^{−1}Ω(p)ξ_{n}(p)^{−1}.

In particular, the inclusion O_{p}|Π^{∗}(p) ⊂ O_{p} implies the equalities

(4-32)  O_{p}|Π^{∗}(p) = O_{p},  O_{p}|Π^{∗}(p)^{−1} = O_{p},

since we are only dealing with finite-dimensional spaces.

By (3-53), (3-39), and (3-40), we obtain the relations

(4-33)  p[p]Π(p)^{−1} = ((1 − p)(ξ_{0}(p) + ξ_{1}(p)) + pξ_{1}(p)) Ω(p),

if n = 1, and the relation

(4-34)  p^{3}[p]Π(p)^{−1} = ((1 − p)(ξ_{0}(p) + ξ_{1}(p) + ξ_{2}(p)) + pξ_{2}(p))Ω(p),

if n = 2. By Proposition 3.9, we have the relation

(4-35)  ξ_{0}(p) + ξ_{1}(p) + ⋯ + ξ_{n}(p) = ν(p)τ(p),

where ν(p) = [Γ_{0}^{n}(q/p) : Γ_{0}^{n}(q)] and τ(p) is the trace–idempotent (4-9). Then, by (4-33), (4-34), (4-35), (4-30), and (4-4), we get the relation

(4-36)  \begin{aligned}p^{⟨n⟩}[p]Π(p)^{−1} & = (1 − p)ν(p)τ(p)Ω(p) + pΠ^{∗}(p) \\ & = c(p)τ(p)ξ_{n}(p)Ω(p) + pΠ^{∗}(p) \\ & = c(p)τ(p)Π^{∗}(p) + pΠ^{∗}(p) \end{aligned}

with a nonzero constant c(p), valid for n = 1 and n = 2. (Unfortunately, it is not valid, if n ≥ 3.) It follows from (4-36), (4-32), and (4-25), that

O_{p}|Π(p)^{−1} ⊂ O|τ(p)Π^{∗}(p) + O_{p}|Π^{∗}(p)
⊂ 𝔑(q/p)|Π^{∗}(p) + O_{p} ⊂ O_{p},

whence

(4-37)  O_{p}|Π(p)^{−1} = O_{p},  O_{p}|Π(p) = O_{p}.

By (4-6) and (4-5), we get

O_{p} = ∑_{i=0}^{m−1} 𝔑(q/p)|(Π^{∗})^{i} = ∑_{i=0}^{m−1} 𝔑|τ(Π^{∗})^{i},

where τ = τ(p) and Π^{∗} = Π^{∗}(p), whence, by (4-32), we obtain

O_{p} = 𝔑|τ(Π^{∗})^{−1} + ∑_{i=1}^{m−1} 𝔑|τ(Π^{∗})^{i−1}
= 𝔑|τΩξ^{−1} + 𝔑|τ = 𝔑|τξΩξ^{−1} + 𝔑|τ,

if n = 1, and

O_{p} = 𝔑|τΩξ^{−1} + 𝔑|τ + 𝔑|τξΩ + 𝔑|τξΩξΩ
= 𝔑|τξΩξ^{−1} + 𝔑|τ + 𝔑|τΩ + 𝔑|τΩξΩ,

if n = 2, where ξ = ξ_{n}(p), Ω = Ω(p) and where we have used several times the relation (4-4). It follows, by (4-25), (4-31), and (3-50), that

O_{p}|ξ = 𝔑|τξΩ + 𝔑|τξ = 𝔑(q/p)|Π^{∗} + 𝔑(q/p),

if n = 1, and

O_{p}|ξ = 𝔑|τξΩ + 𝔑|τξ + 𝔑|τΩξ + 𝔑|τΩξΩξ
= 𝔑(q/p)|Π^{∗} + 𝔑(q/p) + 𝔑(q/p)|Π + 𝔑(q/p)|Π^{2},

if n = 2, where Π = Π(p). In both cases these relations together with the relations (4-32) and (4-37) imply the inclusion O_{p}|ξ ⊂ O_{p}, and so the relations

(4-38)  O_{p}|ξ = O_{p}|ξ_{n}(p) = O_{p}|ξ_{n}(p)^{−1} = O_{p},

since element ξ = ξ_{n}(p) is invertible. Finally, by (4-32) and (4-38), we have

(4-39)  O_{p}|Ω(p) = O_{p}|ξ_{n}(p)^{−1}ξ_{n}(p)Ω = O_{p}|Π^{∗}(p) = O_{p}.

The corresponding equalities for N_{p} follows by Theorem 1.9(2).

□

Lemma 4.11. Let p be an uramified prime number and n = 1 or n = 2. Then the Hecke operators |Π(p) and |Π^{∗}(p) on the space N_{p} of p–new forms satisfy the following relations:

(4-40)  F |Π(p) = F |Π^{∗}(p) = −F |Ω(p),  (F ∈ N_{p})

if n = 1, and

(4-41)  F |Π(p)Π^{∗}(p) = F |Π^{∗}(p)Π(p) = p^{2}F |[p],  (F ∈ N_{p})

if n = 2.

Proof. If n = 1 and F ∈ N_{p}, then, by (4-16), (4-35), and (3-50), we get the relations

F |τ(p) = ν(p)^{−1}F |(ξ_{0}(p) + ξ_{1}(p)) = 0,

and

F |Π(p)τ(p) = F |Ω(p)ξ_{1}τ(p) = F |Ω(p)p(ξ_{0}(p) + ξ_{1}(p)) = 0,

whence

F |ξ_{1}(p) = −F, and F |Π(p) = −F |Ω(p).

On the other hand, by (4-30), we have

F |Π^{∗}(p) = F |ξ_{1}(p)Ω(p) = −F |Ω(p).

But if n = 2, then, by (3-36) and (4-35), we can write

ξ_{2}(p)^{2} = p^{2}(p − 1)(ξ_{0}(p) + ξ_{1}(p) + ξ_{2}(p)) + p^{2}ξ_{0}(p) = p^{2}(p − 1)ν(p)^{−1}τ(p) + p^{2}ξ_{0}(p),

whence, by (3-50), (4-30), and (3-38), we get

F |Π(p)Π^{∗}(p) = F |Ω(p)ξ_{2}(p)^{2}Ω(p) = F_{1}|ξ_{2}(p)^{2}Ω(p)
= p^{2}F_{1}|Ω(p) = p^{2}F |Ω(p)^{2} = p^{2}F |[p],

since F_{1} = F |Ω(p) ∈ N_{p}, by previous lemma. Similarly,

F |Π^{∗}(p)Π(p) = F |ξ_{2}(p)Ω(p)^{2}ξ_{2}(p) = F |ξ_{2}(p)^{2}[p] = p^{2}F |[p].
□

Lemma 4.12. Let n ∈ ℕ and let p, p′ be two different prime divisors of q, where p^{2} does not divide q. Then the following assertions hold:

(1) The Frobenius element Π(p) and its dual Π^{∗}(p) commute both with Π(p′) and Π^{∗}(p);

(2) The subspaces O_{p′} and N_{p′} are invariant under operators |Π(p) and |Π^{∗}(p).

Proof. The elements Π(p) and Π(p′) both belong to the commutative ring C_{q} and so commute. Hence, by duality, Π^{∗}(p) commutes with Π^{∗}(p′). By (3-50) and (3-52), we obtain

Ω(p)ξ_{n}(p)Π(p′) = Π(p′)Ω(p)ξ_{n}(p) = Ω(p)Π(p′)ξ_{n}(p).

Since the element Ω(p) is invertible, it implies the relation

ξ_{n}(p)Π(p′) = Π(p′)ξ_{n}(p),

whence, by (4-30),

Π^{∗}(p)Π(p′) = ξ_{n}(p)Ω(p)Π(p′) = ξ_{n}(p)Π(p′)Ω(p) = Π(p′)Π^{∗}(p).

Element Π(p) commutes with Π^{∗}(p′) by duality. The first part is proved.

In order to prove the second part, it is sufficient, by (4-16), (4-17), and the above, to check that Π(p) and Π^{∗}(p) both commute with the trace–idempotent τ(p′), which is true, by Proposition 3.9(4), if p^{2} does not divide q, and by Proposition 3.12(3), if p^{2} divides q.

□

Lemma 4.13. Let p, p′ be two prime divisors of q and p^{2} divides q. Then for every genus n the subspaces BN_{p′} and N are invariant under the operator |Π(p), and

(4-42)  N_{p}|Π(p) = N|Π(p) = {0}.

Proof. Let us suppose first that p′ ≠ p. Then, since Π(p) and Π(p′) commute, the invariance of N_{p′} follows from (4-16) and Proposition 3.9(4), if p′ is unramified, and from (4-17) and Proposition 3.12(3), if p′ is ramified.

It remains to prove (4-42). If F ∈ N_{p}, then F |Π(p)τ(p) = 0, by (4-17). On the other hand, Π(p)τ(p) = Π(p), by Proposition 3.12(2). The relation (4-42) follows.

□

Theorem 4.14. Let p be a prime divisor of q. Suppose that n = 1 or n = 2. Then the following assertions hold:

(1) Each of the local subspaces N_{p}, as well as the global subspace N are invariant under all of the Hecke operators |Π(p′) for prime divisors p′ of q and operators |Π^{∗}(p′) for prime divisors p′ of q not dividing q/p′. The operators on N commute with each other.

(2) The operators |Π(p) on N are selfadjoint, if n = 1, are normal, if p^{2} ∤ q and n = 2, and are equal to the zero operator, if p^{2}|q.

(3) The subspace N has a basis of common eigenfunctions of all Frobenius operators |Π(p) with primes p dividing q and all dual operators |Π^{∗}(p) for prime divisors p of q not dividing q/p.

Proof. The invariance of N_{p} under |Π(p′) and |Π^{∗}(p′) for unramified p′ was proved in Lemma 4.10 and Lemma 4.12. The invariance of N_{p} under |Π(p′) for ramified p′ was proved in Lemma 4.13. Since all of the Frobenius elements Π(p′) commute with each other, the rest of part (1) follows from Lemma 4.11 and Lemma 4.12(1).

The part (2) follows from Lemma 4.11 and Lemma 4.13.

The part (3) is a consequence of parts (1) and (2) and well-known theorems of linear algebra.

□

Now we can join together all groups of operators considered in Theorems 4.6, 4.8, and 4.14 in order to obtain the following final theorem on eigenfunctions of Hecke operators on the subspace N of new forms of 𝔑.

Theorem 4.15. The subspace

N = N_{k}^{n}(q) ⊂ 𝔑 = 𝔑_{k}(K)

of all new cusp forms of an integral weight k for the group K = Γ_{0}^{n}(q), where n = 1 or n = 2, is invariant under all of the Hecke operators corresponding to elements of the regular reducible ring H_{rr}^{n}(q), of the singular unramified ring H_{su}^{n}(q), to all of the singular Frobenius elements Π(p) with primes p dividing q, and all duals Π^{∗}(p) for prime divisors p of q not dividing q/p. The subspace is spanned by common eigenfunctions of the operators.

Proof. All of the operators are generated by local operators which can be divided into the following four groups of operators corresponding, respectively, to the elements

(i) T ∈ H_{p}(q) for primes p not dividing q;

(ii) T ∈ H_{p}(q) for primes p dividing q, but not dividing q/p;

(iii) Π(p) for primes p dividing q;

(iv) Π^{∗}(p) for primes p dividing q, but not dividing q/p.

The invariance of the space N and the existence of basises of common eigenfunctions for all of the operators of each of the four groups is true by Theorems 4.6, 4.8, 4.15. So according to linear algebra, the only thing we have to check in order to conclude the proof is that the operators of different groups commute with each other. We have: elements of the groups (i) and (ii) commute, by Theorem 3.17(1); elements of the groups (i) and (iii) commute, by Proposition 2.12, Theorem 2.8, and Theorem 3.3; it implies, by duality and Theorem 2.3, that elements of the groups (i) and (iv) commute; elements of the group (ii) and (iii) commute by Theorem 3.17(2) ; it implies, by duality and Proposition 3.20, that elements of the groups (ii) and (iv) commute; finally, by Theorem 4.14(1), the operators on N correspondig to elements of the groups (iii) and (iv) commute.

□

Conclusion

Speaking on possible extension of the results on eigenfunctions, I should like to note that in order to extend Theorem 4.8 to a genus n > 2, it would be sufficient to prove that rings H_{p}^{n}(q) consist of symmetric elements for all q–singular unramified prime numbers p. On the other hand, it is doubtful that the part of Theorem 4.14 related to unramified primes can be generalized to a genus n > 2. The reason is that the relations (4-36) and (4-41) are not valid, if n ≥ 3.

Perhaps, on considering not only Frobenius elements, one could tell more on ramified primes, than I did. It would be interesting for consideration of standard zeta functions for the groups Γ_{0}^{n}(q) in ramified cases.

The first and foremost application of the results on eigenfunctions of Frobenius operators will be to extend my results obtained nearly 25 years ago in [An(74)] on spinor zeta functions of the group Γ^{2} = Sp_{2}(ℤ) to the case of the groups Γ_{0}^{2}(q), which is our next task.

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