twisting of siegel modular forms with characters, … · twisting of siegel modular forms with...

Algebra i analiz St. Petersburg Math. J.Tom 20 (2008), 6 Vol. 20 (2009), No. 6, Pages 851–871

S 1061-0022(09)01076-0Article electronically published on October 1, 2009

TWISTING OF SIEGEL MODULAR FORMS WITH CHARACTERS,

AND L-FUNCTIONS

A. ANDRIANOV

Abstract. Linear twistings of Siegel modular forms with Dirichlet characters areconsidered. It is shown that the twisting operators transform modular forms tomodular forms. Commutation of twisting operators and Hecke operators is exam-ined. It is proved that under certain conditions the spinor zeta-function of a twistedmodular form can be interpreted as the L-function of the initial modular form with

twisting character. As an illustration of the twist techniques, analytic properties ofL-functions of cusp forms of genus n = 1 are considered.

Introduction

Consider a function, say, a Siegel modular form of genus n ≥ 1, given on the upperhalf-plane

Hn =Z = X + iY ∈ Cn

n

∣∣∣ tZ = Z, Y > 0 (i =√−1)

by an absolutely convergent Fourier series of the form

(1) F (Z) =∑

A∈An,A≥0

f(A) e2πiTr(AZ)

with constant Fourier coefficients f(A), where the series is assumed to converge uniformlyon compact sets, and A ranges over all positive semidefinite matrices belonging to theset

An =

A = (aαβ) ∈

1

2Znn

∣∣∣ tA = A, a11, a22, . . . , ann ∈ Z

of all symmetric half-integer matrices of order n with integral entries on the principaldiagonal. Investigation of Euler products corresponding to modular forms leads to thenatural question, which was once proposed by A. A. Panchishkin, on twisting the se-ries (1) with multiplicative functions such as characters. This paper is an attempt toaddress this question.

If n = 1, then it is natural to define the twisting of a Fourier series∑a∈Z, a≥0

f(a) e2πiaz

with a character χ by the Fourier series∑a∈Z, a≥0

χ(a)f(a) e2πiaz.

But if n > 1, then, while considering linear twistings, i.e., such that a character of alinear form in coefficients of the running matrix A ∈ An is involved, there is no a priori

2000 Mathematics Subject Classification. Primary 11F46, 11F60, 11F66.Key words and phrases. Hecke operators, Siegel modular forms, zeta-functions of modular forms.Supported in part by RFBR (grant no. 08-01-00233).

c©2009 American Mathematical Society

851

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

852 A. ANDRIANOV

reason to prefer one linear form to another. Therefore, it is quite natural to considertwistings with characters of arbitrary integral linear forms in coefficients of the matrix A.Obviously, every such form can be written as L(A) = Tr(LA) with an integral symmetricmatrix

(2) L = tL ∈ Znn

of order n. Thus, we get the twisting of Fourier series F with a character χ and ap-matrix (parameter-matrix) L; this twisting is defined by the series

(3) (F |T (χ, L))(Z) =∑

A∈An, A≥0

χ(Tr(LA))f(A) e2πiTr(AZ) (Z ∈ Hn).

The operator

(4) |T (χ, L) : F → F |T (χ, L)

will be called the twisting operator with character χ and p-matrix L.Zeta-functions and L-functions of modular forms of genus n = 1 (elliptic modular

forms) were investigated by Erich Hecke and his followers. In particular, it was provedthat these functions have meromorphic analytic continuation to the entire s-plane andsatisfy a functional equation (see, e.g., [6, Chapters 1 and 5]). Analytic continuation tothe entire s-plane and a functional equation for the zeta-function of Siegel modular formsof genus n = 2 for the full modular group Γ2 = Sp2(Z) were proved in [2] and [3]. Areasonably full investigation of zeta-functions and L-functions of modular forms of genusn > 2 is a problem for the distant future. Here we make an initial step in this direction.

Contents. In §§1–3 we consider the action of twisting operators on Siegel modular formsof an arbitrary genus n ≥ 1. In particular, in §1 we show that, under certain assumptions,the twisting operators take modular forms to modular forms. In §2 we treat commutationrelations between twisting operators and Hecke operators. In §3 we prove that, undercertain conditions, the zeta-function of a twisted modular form can be interpreted as theL-function of the initial modular form with the character of twisting. As an illustration,in §4 we show how the twisting techniques can be used to investigate analytic propertiesof L-functions of cusp forms of genus n = 1, viewed as zeta-functions of twisted forms. Inforthcoming papers we intend to apply this approach to the investigation of L-functionsof modular forms of genus n = 2.

Notation. The letters N, Z, Q, R, and C are reserved for the set of positive rationalintegers, the ring of rational integers, the field of rational numbers, the field of realnumbers, and the field of complex numbers, respectively.

Xmn is the set of all (m × n)-matrices with entries in a set X. If M is a matrix, tM

always denotes the transpose of M , and for a square matrix M , Tr(M) is the sum ofdiagonal elements of M . If Y is a real symmetric matrix, then Y > 0 (respectively,Y ≥ 0) means that Y is positive definite (respectively, positive semidefinite). 1n = 1and 0n = 0 denote the unit matrix and the zero matrix of order n, respectively. The barover a complex number or character means complex conjugation.

We use the notation

(5) A[B] = tBAB

for two matrices of suitable dimension, and the abbreviation

(6) 〈n〉 = n(n+ 1)/2.


TWISTING OF SIEGEL MODULAR FORMS WITH CHARACTERS 853

§1. Twisting of modular forms with characters

In this section we show, in particular, that under certain conditions the twistingoperators take modular forms to modular forms, and we examine the correspondinggroups.

Let

Gn =M ∈ R2n

2n

∣∣∣ tMJnM = µ(M)Jn, µ(M) > 0

with Jn =

(0n 1n−1n 0n

)be the connected component of the unit in the general real symplectic group of genus n.The group Gn acts as a group of analytic automorphisms of the upper half-plane Hn bythe rule

Gn M =

(A BC D

): Z → M〈Z〉 = (AZ +B)(CZ +D)−1 (Z ∈ Hn).

Acting on the upper half-plane, the group Gn operates on functions F : Hn → C by(normalized) Petersson operators of integral weights k,

(1.1)Gn M =

(A BC D

): F → F |kM

= µ(M)nk−〈n〉 det(CZ +D)−kF (M〈Z〉)

(see (6)). The normalizing factor µ(M)nk−〈n〉, going back for n = 1 to Hecke, is notsignificant by itself, but it simplifies a number of formulas related to Hecke operators.The Petersson operators transform isomorphically the space of functions on Hn into itself,map holomorphic functions to holomorphic functions, and satisfy the rule

(1.2) F |kMM ′ = F |kM |kM ′ for all M, M ′ ∈ Gn

(see, e.g., [4, Lemmas 1.4.1, 1.4.2]).In the above notation, the following statement is true.

Lemma 1.1. If F is a function on Hn with Fourier expansion (1), χ is a Dirichletcharacter modulo m ∈ N, and L is a matrix of the form (2), then the twisting of F withcharacter χ and p-matrix L can be written with the help of the Petersson operators (1.1)of an arbitrary integral weight k in the form

(1.3) F |T (χ, L) =1

m

∑r,lmodm

χ(r)e−2πirl

m F |kU(m−1lL),

where, for a real symmetric matrix B of order n, we use the notation

(1.4) U(B) =

(1 B0 1

)∈ Gn.

Proof. The definitions and the Fourier expansion (1) imply that

(F |kU(m−1lL))(Z) = F (Z +m−1lL) =∑

A∈A,A≥0

f(A) e2πi lm Tr(LA)e2πiTr(AZ).


854 A. ANDRIANOV

Consequently,

1

m

∑r,lmodm

χ(r) e−2πirl

m F |kU(m−1lL)

=1

m

∑r, lmodm

χ(r)e−2π irl

m

∑A∈A,A≥0

f(A) e2πilm Tr(LA)e2πiTr(AZ)

=∑

rmodm

χ(r)∑

A∈A,A≥0

1

m

∑lmodm

e2πilm (−r+Tr(LA))f(A) e2πiTr(AZ)

=∑

A∈A, A≥0

χ(Tr(LA))f(A) e2πiTr(AZ) = (F |T (χ, L))(Z).

For a Dirichlet character χ modulo m ∈ N and an integer l ∈ Z, we introduce theGaussian sums

g(χ, l) =∑

rmodm

χ(r) e2πi lr

m and g(χ) = g(χ, 1).

The character χ is called a primitive character modulo m if it is a character modulo m,but not modulo any proper divisor of m. If χ is a primitive character modulo m, thenthe Gaussian sums satisfy the relations

(1.5) g(χ, r) = χ(r)g(χ), |g(χ)| =√m

(see, e.g., [6, Proposition 21]).

Lemma 1.2. In the notation of the preceding lemma, if, moreover, the character χ isprimitive modulo m, then formula (1.3) can be written in the form

(1.6) F |T (χ, L) =g(χ)

m

∑lmodm

χ(−l)F |kU(m−1lL),

where g(χ) = g(χ, 1) is the Gaussian sum and χ is the complex conjugate of the charac-ter χ.

Proof. If χ is primitive, then, by Lemma 1.1 and (1.5),

F |T (χ, L) =1

m

∑lmodm

( ∑rmodm

χ(r)e−2πirl

m

)F |kU(m−1lL)

=1

m

∑lmodm

χ(−l)g(χ)F |kU(m−1lL).

We recall that a (holomorphic) modular form of genus n ≥ 1 and integral weight kfor a subgroup Λ of finite index in the modular group Spn(Z) of genus n is defined as aholomorphic function F on Hn that satisfies

(1.7) F |kM = F for each M ∈ Λ,

and, if n = 1, is regular at all cusps of Λ. All such functions form a finite-dimensionallinear space Mk(Λ) over C. The forms equal to zero at all cusps are called cusp forms;they constitute the subspace of cusp forms Nk(Λ).

In this paper we consider modular forms for the full modular group

(1.8) Γ = Γn = Spn(Z) =M ∈ Gn ∩ Z2n

2n

∣∣∣ µ(M) = 1,

for congruence subgroups of type

Γ0(m) = Γn0 (m) =

M =

(A BC D

)∈ Γ

∣∣∣ C ≡ 0 (mod m)



and

(1.9) Γ[0](m) = Γn[0](m) =

M =

(A BC D

)∈ Γ0(m)

∣∣∣ B ≡ 0 (mod m)

,

for the principal congruence subgroup of level m,

Γ(m) = Γn(m) =M ∈ Γn

∣∣∣ M ≡ 12n (mod m),

and for subgroups conjugate to Γ[0](m) and Γ(m),

(1.10) Γ[0](m) = Γn[0](m) = V (m)Γ[0](m)V (m)−1 = Γ0(m

2)

and

Γ(m) = Γn(m) = V (m)Γ(m)V (m)−1

=

(A BC D

)∈ Γ

∣∣∣ A ≡ D ≡ 1 (mod m), C ≡ 0 (mod m2)

,(1.11)

where, for fixed n, m ≥ 1, we set

(1.12) V (m) = V n(m) =

(1 00 m1

).

We shall consider the spaces Mk(Λ) of all modular forms and the spaces Nk(Λ) ofall cusp forms of a fixed integral weight k for the groups Λ introduced above. In thefollowing lemma, we list useful properties of the matrices (1.4) and the correspondingPetersson operators.

Lemma 1.3. 1) Every matrix M = (A BC D ) of order 2n with blocks of order n and

arbitrary matrices of the form (1.4) satisfy the relation

(1.13) U(B′)

(A BC D

)U(B′′)−1 =

(A+B′C B′D −AB′′ +B −B′CB′′

C −CB′′ +D

).

2) For every matrix of the form (1.4) with B = m−1L, where L is as in (2), the group

Γ(m) satisfies the relation

(1.14) U(m−1L)Γ(m)U(m−1L)−1 = Γ(m).

3) The operators |kU(m−1L) with L as in (2) map the spaces Mk(Γ(m)) and Nk(Γ(m))onto themselves.

Proof. Formula (1.13) is a result of multiplication of block matrices. By this formula,

for M = (A BC D ) ∈ Γ(m), we obtain the matrix

U(m−1L)

(A BC D

)U(m−1L)−1

=

(A+m−1LC m−1(LD −AL) +B −m−2LCL

C −m−1CL+D

),

which, obviously, belongs to Γ(m). This proves that the left-hand side of (1.14) is con-tained in the right-hand side. The reverse inclusion follows similarly, because, clearly,U(B)−1 = U(−B).

Finally, to prove statement 3), actually it suffices to check, e.g., that for the function

F ′ = F |kU(m−1L) with F ∈ Mk(Γ(m)) we have F ′|kM = F ′ for all M ∈ Γ(m). By(1.2),

F ′|kM = F |kU(m−1L)M = F |kU(m−1L)MU(m−1L)−1|kU(m−1L) = F ′,

because U(m−1L)MU(m−1L)−1 ∈ Γ(m) by (1.14).


856 A. ANDRIANOV

Proposition 1.4. For every Dirichlet character χ modulo m and every p-matrix L of

the form (2), the twisting operator |T (χ, L) maps the spaces Mk(Γ(m)) and Nk(Γ(m))

of modular forms and cusp forms of weight k for the group Γ(m) into themselves.

Proof. Note that every function in these spaces has a Fourier expansion of the form (1).The assertion directly follows from formula (1.3) and statement 3) of Lemma 1.3 withlL in place of L.

Proposition 1.5. If F ∈ Mk(Γ0(m2)), χ is a character modulo m, and L is a matrix

of the form (2), then the twist of the form F with character χ and p-matrix L satisfiesthe rules

(1.15) (F |T (χ, L))|k(A BC D

)= F |T (χ, L[D]),

(A BC D

)∈ Γ0(m

2),

where L[D] = tDLD. If, moreover, the block D satisfies the congruence

(1.16) L[D] ≡ ν(D)L (mod m) with a scalar ν(D) = νL(D),

then for the function F |T (χ, L) we have

(1.17) (F |T (χ, L))|k(A BC D

)= χ(ν(D))F |T (χ, L).

Proof. If M = (A BC D ) ∈ Γ0(m

2) and l ∈ Z, then, by (1.3) and (1.2), we can write

(F |T (χ, L))|kM =1

m

∑r,lmodm

χ(r)e−2πirl

m F |kU(m−1lL)|kM

=1

m

∑r,lmodm

χ(r)e−2πirl

m F |kU(m−1lL)MU(m−1lL[D])−1U(m−1lL[D])

=1

m

∑r,lmodm

χ(r)e−2πirl

m F |kM ′l |kU(m−1lL[D]),

where M ′l = U(m−1lL)MU(m−1l tDLD)−1. By statement 1) of Lemma 1.3 with B′ =

m−1lL and B′′ = m−1lL[D], we obtain

M ′l =

(A+m−1lLC m−1l(LD −AL[D]) +B −m−2l2LCL[D]

C −m−1lCL[D] +D

)=

(A+m−1lLC −m−1l(−A tD + 1n)LD +B −m−2l2LC tDLD

C −m−1lC tDLD +D

).

Clearly, this matrix is integral. Moreover, it belongs to the group Gn, and satisfiesµ(M ′

l ) = µ(M) = 1. Since C ≡ 0 (mod m2), it follows that M ′l ∈ Γ0(m

2). Hence,F |kM ′

l = F , and applying formula (1.3) once again, we get

(F |T (χ, L))|kM =1

m

∑r,lmodm

χ(r)e−2πirl

m F |kU(m−1lL[D]) = F |T (χ, L[D]).

Turning to (1.17), we use (1.15) to obtain

(F |T (χ, L))|k(A BC D

)=

∑A∈A, A≥0

χ(Tr(L[D]A))f(A)e2πi(Tr(AZ))

=∑

A∈A, A≥0

χ(ν(D)(Tr(LA))f(A)e2πi(Tr(AZ))

= χ(ν(D))F |T (χ, L).



Lemma 3.3.2(1) in [4] implies that the natural mapping of the group Γ0(m2) to the

group GLn(Z/mZ) of nonsingular matrices of order n over the residue ring Z/mZ givenby

Γ0(m2) M =

(A BC D

)→ D/modm ∈ GLn(Z/mZ)

is an epimorphic homomorphism with the kernel Γ(m). For a matrix D in the groupGLn(Z/mZ), we denote by

(1.18) (D) ∈ Γ0(m2)

some inverse image ofD under this mapping. Since Γ(m) is a normal subgroup of Γ0(m2),

the cosets

(1.19) Γ(m)(D) = (D)Γ(m) = Γ(m)(D)Γ(m)

are independent of the choice of D ∈ GLn(Z/mZ). Hence, each operator

(1.20) |k(D) : F → F |k(D)(D ∈ GLn(Z/mZ), F ∈ Mk(Γ(m))

)depends only on the matrix D modulo m and maps the space Mk(Γ(m)) onto itself.Moreover, these operators satisfy

(1.21) |k(D)|k(D′) = |k(DD′) (D, D′ ∈ GLn(Z/mZ)).

Now, Proposition 1.5 can be reformulated in terms of the operators |k(D).

Proposition 1.6. If F ∈ Mk(Γ0(m2)), then

(1.22) (F |T (χ, L))|k(D) = F |T (χ, L[D]) for all D ∈ GLn(Z/mZ).

If, moreover, the matrix D satisfies (1.16), then this formula turns into

(1.23) (F |T (χ, L))|k(D) = χ(ν(D))F |T (χ, L).

In particular,

(F |T (χ, L))|kτ (d) = χ(d2)F |T (χ, L) for all d ∈ N prime to m,

where

(1.24) τ (d) = τn(d) = (d · 1n).

Along with operators (1.20), we shall also use the Petersson operators correspondingto matrices of order 2n of the form

(1.25) Ω = Ωn(m) =

(0 m−11

−m1 0

).

It is easily seen that Ω ∈ Gn, µ(Ω) = 1,

(1.26) Ω−1 = −Ω, and Ω2 = −12n.

The obvious relations

(1.27) Ω

(A BC D

)Ω−1 = Ω−1

(A BC D

)Ω =

(D − C

m2

−m2B A

),

valid for every 2n-matrix (A BC D ) with n-blocks A, B, C, D, and the definitions of the

groups Λ = Γ0(m2) and Γ(m) show that for these groups we have

(1.28) ΩΛΩ−1 = Ω−1ΛΩ = Λ.

Coming back to modular forms, we have the following assertions.


858 A. ANDRIANOV

Proposition 1.7. 1) In the notation (1.1) and (1.25), the linear operator

(1.29) |kΩ : F → F ∗ = F |kΩ

maps the spaces Mk(Γ0(m2)), Mk(Γ(m))), Nk(Γ0(m

2)), and Nk(Γ(m)) onto themselves,and

(1.30) (F ∗)∗ = (−1)kF.

2) The operators (1.29) and (1.20) on the spaces Mk(Γ(m)) and Nk(Γ(m)) (respec-tively) satisfy the relation

(1.31) |kΩ|k(D) = |k( tD−1)|kΩ,where D−1 is the inverse of D modulo m.

3) In the notation of Proposition 1.4, we have

(1.32) (F |T (χ, L))∗|k(D) = (F |T (χ, L[ tD−1]))∗;

if, moreover, the matrix D satisfies (1.16), then formula (1.32) turns into

(1.33) (F |T (χ, L))∗|k(D) = χ(ν(D))(F |T (χ, L))∗;

in particular,

(1.34) (F |T (χ, L))∗|kτ (d) = χ(d2)(F |T (χ, L))∗

for all d ∈ N prime to m, where τ (d) denotes a matrix of the form (1.24).

Proof. Using (1.29), (1.28), and (1.2), for M ∈ Λ = Γ0(m2) or Γ(m) and F ∈ Mk(Λ) or

Nk(Λ), we get

F ∗|kM = F |kΩM = F |kΩMΩ−1Ω = F |kΩMΩ−1|kΩ = F |kΩ = F ∗.

Formula (1.30) follows from (1.26). The remaining part of statement 1) is clear.If (D) = (A B

C D ), then, by (1.27) and the definitions, we have

Ω(D) = Ω(D)Ω−1Ω = (A)Ω = ( tD−1)Ω,

because tAD ≡ 1n (mod m). The remaining part of 2) follows from (1.2)Identities (1.32)–(1.34) follow from (1.23), (1.31), and the definitions.

§2. Twisting operators and Hecke operators

Here we shall consider the interaction between the Hecke operators and the twistingoperators |T (χ, L), where χ is a Dirichlet character modulo m and L is a parametermatrix (2). First, we briefly recall the definitions of the Hecke–Shimura rings and theHecke operators. For the details, see [4, Chapters 3 and 4].

Let ∆ be a multiplicative semigroup and Λ a subgroup of ∆ such that every doublecoset ΛMΛ of ∆ modulo Λ is a finite union of left cosets ΛM ′. Consider the vectorspace over a field (say, the field C of complex numbers) consisting of all formal finitelinear combinations with coefficients in C of the symbols (ΛM), M ∈ ∆, that are inone-to-one correspondence with the left cosets ΛM of the set ∆ modulo Λ. The groupΛ acts naturally on this space by the right multiplication defined on the symbols (ΛM)by (ΛM)λ = (ΛMλ) with M ∈ ∆ and λ ∈ Λ. We denote by

H(Λ, ∆) = HC(Λ, ∆)

the subspace of all Λ-invariant elements. The multiplication of elements of H(Λ, ∆),given by the formula(∑

α

aα(ΛMα))(∑

β

bβ(ΛM′β))=

∑α,β

aαbβ(ΛMαM′β),



is independent of the choice of representatives Mα and Nβ in the corresponding leftcosets and turns the linear space H(Λ, ∆) into an associative algebra over C with theunity element (Λ1Λ), called the Hecke–Shimura ring of ∆ relative to Λ (over C). Theelements

(2.1) T (M) = T (M)Λ = (ΛMΛ) =∑

M ′∈Λ\ΛMΛ

(ΛM ′) (M ∈ ∆),

which are in one-to-one correspondence with the double cosets of ∆ modulo Λ, belong toH(Λ, ∆) and form a free basis of that ring over C. For brevity, the symbols (ΛM) andT (M) will be referred to as the left and double classes (of ∆ modulo Λ), respectively.

Now, suppose that the semigroup

(2.2) ∆ = ∆n = Z2n2n ∩Gn

consists of all integral matrices contained in the group Gn, and the group Λ is a subgroupof finite index in the modular group Γn. Then the necessary conditions of the definitionare satisfied, and we can define the Hecke–Shimura ring

(2.3) H(Λ) = H(Λ, ∆n).

Next, we define a linear representation of this ring on the space Mk(Λ) of modular formsof weight k for the group Λ by Hecke operators:

(2.4) H(Λ) T =∑α

aα(ΛMα) : F → F |T = F |kT =∑α

aαF |kMα,

where the |kMα are the Petersson operators (1.1). The Hecke operators are independentof the choice of representatives in the corresponding left cosets and map the spacesMk(Λ)and Nk(Λ) into themselves.

Largely, we shall be interested not in the entire Hecke–Shimura ring (2.3), but ratherin certain subrings, called the m-regular subrings or simply the regular subrings, defined

for the group Λ of the form Γ = Γn, Γ0(m2) = Γn

0 (m2), and Γ(m) = Γn(m) as the

Hecke–Shimura rings

(2.5) H(m)(Λ) = H(Λ, ∆(m)(Λ)),

of the group Λ and the m-regular semigroups ∆(m)(Λ) given, respectively, by the condi-tions

∆(m) = ∆(m)(Γ) =M ∈ ∆n

∣∣∣ gcd(m,µ(M)) = 1,(2.6)

∆(m)(Γ0(m2)) =

M =

(A BC D

)∈ ∆(m)(Γ)

∣∣∣ C ≡ 0 (mod m2)

,(2.7)

and

∆(m)(Γ(m)) =

M =

(A BC D

)∈ ∆(m)(Γ0(m

2))∣∣∣ A ≡ 1 (mod m)

.(2.8)

The corresponding Hecke operators are called the regular Hecke operators. It turns outthat the rings (2.5) for these three groups are naturally isomorphic. More precisely, aneasy modification of the proof of [4, Theorem 3.3.3] yields the following statement.

Proposition 2.1. 1) The linear maps of the m-regular subrings (2.6)–(2.8) defined on the

double classes (2.1) for each of the pairs (Λ, Λ′) of the form (Γ(m), Γ0(m2)), (Γ(m), Γ),

and (Γ0(m2), Γ) by

(2.9) H(m)(Λ) TΛ(M) → TΛ′(M) ∈ H(m)(Λ′) (M ∈ ∆(m)(Λ))


860 A. ANDRIANOV

are ring isomorphisms of the corresponding Hecke–Shimura rings. Moreover, the de-compositions of the related double classes into left classes correspond naturally to eachother:

(2.10) TΛ(M) =∑α

(ΛMα) =⇒ TΛ′(M) =∑α

(Λ′Mα) (M ∈ ∆(m)(Λ)).

2) The regular Hecke operators are compatible with the mappings (2.9) and the na-tural embeddings of the corresponding spaces of modular forms of weight k; namely, ifF ∈ Mk(Λ

′) ⊂ Mk(Λ) and TΛ → TΛ′ , then

(2.11) F |TΛ = F |TΛ′ .

Proposition 2.2. Each of the regular rings (2.5) for the groups Λ of the form Γ =

Γn, Γ0(m2) = Γn

0 (m2), or Γ(m) = Γn(m) is a commutative integral domain generated

over C by the algebraically independent elements⎧⎪⎪⎪⎨⎪⎪⎪⎩TΛ(p) = TΛ

(diag(1, . . . , 1︸︷︷︸

n

, p, . . . , p︸︷︷︸n

)),

T jΛ(p

2) = TΛ

(j(p) diag(1, . . . , 1︸︷︷︸

n−j

, p, . . . , p︸︷︷︸j

, p2, . . . , p2︸︷︷︸n−j

, p, . . . , p︸︷︷︸j

))

(1 ≤ j ≤ n),

where p runs over all prime numbers not dividing m, and the elements

j(p) = nj (p) = (diag(1, . . . , 1︸︷︷︸

n−j

, p, . . . , p︸︷︷︸j

))

(1 ≤ j ≤ n)

have the form (1.18).

Proof. For Λ = Γn this claim was proved in [4, Theorem 3.3.23(1)]. The other casesfollow from this by Proposition 2.1.

Theorem 2.3. For the groups Λ = Γ(m) and Λ′ = Γ0(m2), let M ∈ ∆(m)(Λ), and

let F ∈ Mk(Λ′) be a modular form of weight k for Λ′. Let χ be a primitive Dirichlet

character modulo m and L a matrix of the form (2). Then the following commutationrelation holds true for the action of the Hecke operator TΛ(M) and the twist operatorT (χ, L) on the form F :

(2.12) (F |T (χ, L))|TΛ(M) = χ(µ(M))(F |TΛ′(M))|T (χ, L).

Proof. By formulas (1.6), (2.1), (2.4), and (1.2), we obtain

(2.13)

(T (χ,L)F )|TΛ(M) =g(χ)

m

∑lmodm

∑M ′∈Λ\ΛMΛ

χ(−l)F |kU(m−1lL)M ′

=g(χ)

m

m−1∑l=0

∑M ′∈Λ\ΛMΛ

χ(µ(M))χ(−lµ(M))

× (F |kU(m−1lL)M ′U(−m−1lµ(M)L))|kU(m−1lµ(M)L)

= χ(µ(M))g(χ)

m

m−1∑l=0

χ(−l′)

×( ∑

M ′∈Λ\ΛMΛ

F |kU(m−1lL)M ′U(−m−1l′L))

)|kU(m−1l′L),

where l′ = lµ(M). By (1.13), for every matrix

M ′ =

(A′ B′

C ′ D′

)∈ ΛMΛ ⊂ ∆(m)(Λ)



and every l = 0, 1, . . . ,m− 1, we have(2.14)

M ′l = U(m−1lL)M ′U(−m−1l′L)

=

(A′ +m−1lLC ′ −m−1(lLD′ − l′A′L) +B′ −m−2ll′LC ′L

C ′ −m−1l′C ′L+D′

)=

(A′

l B′l

C ′l D′

l

).

Every such matrix belongs to ∆(m)(Λ), because, clearly, the matrices A′l, C

′l , and D′

l areintegral matrices satisfying the congruences A′

l ≡ A′ ≡ 1n (mod m) and C ′l = C ′ ≡ 0n

(mod m2), and the matrix B′l is integral, because lLD′ − l′A′L ≡ lµ(M)L − l′L =

l′(L− L) ≡ 0 (mod m). By statement 2) of Lemma 1.3, we have

U(m−1lL)ΛU(−m−1lL) = Λ and U(m−1l′L)ΛU(−m−1l′L) = Λ.

Thus, the matrix M ′l ranges over the set

U(m−1lL)(Λ \ ΛMΛ

)U(−m−1l′L) = Λ \ Λ(U(m−1lL)MU(−m−1l′L))Λ.

It follows that for each l we can write∑M ′∈Λ\ΛMΛ

F |kU(m−1lL)M ′U(−m−1l′L) = F |TΛ(U(m−1lL)MU(−m−1l′L)).

By (2.11), the last-written image is equal to

F |TΛ′(U(m−1lL)MU(−m−1l′L)),

and in order to complete the proof it suffices to show that, for every l, we have

(2.15) Λ′(U(m−1lL)MU(−m−1l′L))Λ′ = Λ′MΛ′

with l′ = lµ(M), because then each inner sum on the right in (2.21) is equal to F |TΛ′(M).As we have seen, the two matrices Ml = U(m−1lL)MU(−m−1l′L) and M belong tothe set ∆(m)(Λ) ⊂ ∆(m)(Λ

′). Moreover, µ(Ml) = µ(M). By [4, Lemma 3.3.6], (2.15)will be proved if we check that Ml and M have equal matrices of symplectic divisors,sd(Ml) = sd(M). Put

mU(m−1lL) =

(m · 1 lL0 m · 1

)= N, mU(−m−1l′L) =

(m · 1 −l′L0 m · 1

)= N ′.

Then

U(m−1lL) =1

mγ sd(N)γ1, U(−m−1l′L) =

1

mγ2 sd(N

′)γ3,

where γ, γ1, γ2, γ3 ∈ Γ. Hence, Ml =1m2 γ sd(N)γ1Mγ2 sd(N

′)γ3. It follows that

(2.16) m2 sd(N)−1γ−1Mlγ−13 = γ1Mγ2 sd(N

′).

Since µ(N) = µ(N ′) = m2, the definition of symplectic divisors yields sd(m2 sd(N)−1) =sd(N), sd(γ−1Mlγ

−13 ) = sd(Ml), sd(γ1Mγ2) = sd(M), and sd(sd(N ′)) = sd(N ′). Since

the number µ(Ml) = µ(M) is coprime to µ(N) = µ(N ′) = m2, relation (2.16) and theknown properties of the matrices of symplectic divisors imply that

sd(m2 sd(N)−1) sd(γ−1Mlγ−13 ) = sd(γ1Mγ2) sd(sd(N

′)),

i.e.,

sd(N) sd(Ml) = sd(M) sd(N ′).

Obviously, this coincidence of diagonal matrices implies sd(N) = sd(N ′) and sd(Ml) =sd(M), which proves (2.15) and the theorem.


862 A. ANDRIANOV

Corollary 2.4. In the notation of Theorem 2.3, if a modular form F ∈ Mk(Λ′) is an

eigenfunction for the Hecke operator |TΛ′(M), M ∈ ∆(m)(Λ), with the eigenvalue λ(M),then the function F |T (χ, L) ∈ Mk(Λ) is an eigenfunction for the operator |TΛ(M) withthe eigenvalue χ(µ(M))λ(M).

Now we turn to relationships between regular Hecke operators and the star operator(1.29) defined with the help of a matrix Ω = Ωn(m) of the form (1.25) on the space

Mk(Γ(m)). For this, it will be convenient to split this space into invariant subspacesof the operators |kτ (d) = |kτn(d) = |k(d1) with all d prime to m (see (1.24)). Themapping d → |kτ (d) determines a representation of the multiplicative Abelian group

GL1(Z/mZ) on the space Mk(Γ(m)). Thus, this space is a direct sum of one-dimensionalinvariant subspaces. If F |kτ (d) = ψ(d)F with all d prime to m, then ψ is a character ofGL1(Z/mZ), which can be viewed as a Dirichlet character modulo m. Then we have thefollowing direct sum decomposition:

(2.17) Mk(Γ(m)) =⊕

ψ∈Char(GL1(Z/mZ))

Mk(Γ(m), ψ),

where

(2.18) Mk(Γ(m), ψ) =F ∈ Mk(Γ(m))

∣∣∣ F |kτ (d) = ψ(d)F, gcd(d, m) = 1.

Note that, by Proposition 1.6, the image of the subspace M(Γ0(m2)) under the twist

operator T (χ, L) with a character χ modulo m and p-matrix L is contained in thesubspace (2.18) with ψ = χ2:

(2.19) Mk(Γ0(m2))|T (χ, L) ⊂ Mk(Γ(m), χ2).

For regular Hecke operators on spaces of the form Mk(Γ(m), ψ), we now prove thefollowing proposition.

Proposition 2.5. The following assertions are valid for the group Λ = Γ(m).1) Suppose F ∈ Mk(Λ, ψ) and M ∈ ∆(m)(Λ). Then

(2.20) (F |TΛ(M))∗ = ψ(µ(M))F ∗|TΛ(τ (µ(M))M∗),

where τ (µ) = τn(µ), G∗ = G|kΩ, and

(2.21) M∗ = Ω−1MΩ (Ω = Ωn(m)).

2) The matrix τ (µ(M))M∗ belongs to ∆(m)(Λ) together with M , and the mapping

(2.22) M → M = τ (µ(M))M∗

determines a bijection of the set ∆(m)(Λ) that is identical on the sets of double cosetsmodulo Λ contained in ∆(m)(Λ), so that

(2.23) TΛ(M) = TΛ(M) for all M ∈ ∆(m)(Λ).

In particular, relation (2.20) can be rewritten in the form

(2.24) (F |TΛ(M))∗ = ψ(µ(M))F ∗|TΛ(M)(M ∈ ∆(m)(Λ)

).



Proof. Using (1.29), (2.1), (2.4), (1.31), and (1.2), we obtain

(F |TΛ(M))∗ = F |TΛ(M)|Ω =∑

M ′∈Λ\ΛMΛ

F |kM ′Ω

=∑

M ′∈Λ\ΛMΛ

F |kΩτ (µ(M))−1Ω−1Ωτ (µ(M))Ω−1M ′Ω

=∑

M ′∈Λ\ΛMΛ

F |kτ (µ(M))Ωτ (µ(M))(M ′)∗

= ψ(µ(M))∑

M ′∈Λ\ΛMΛ

F ∗|kτ (µ(M))(M ′)∗

= ψ(µ(M))∑

M ′′∈Λ\Λτ(µ(M))M∗Λ

F ∗|kM ′′ = ψ(µ(M))F ∗|TΛ(τ (M)M∗),

because Ω−1ΛΩ = Λ, which proves part 1).The fact that M ∈ ∆(m)(Λ) follows from the definitions. By (1.26) and (1.31), we

have (M∗)∗ = M and

(2.25) ˇM = M for all M ∈ ∆(m)(Λ).

Thus, the mapping (2.22) is a bijection. By Proposition 2.1 for the pair (Γ(m), Γ), forthe proof of (2.23) it suffices to check that M ∈ ΓMΓ, or that the matrices M and Mhave equal matrices of symplectic divisors, sd(M) = sd(M) (see [4, Lemma 3.3.6]). Thedefinition shows that MmΩ−1 = τ (µ(M))mΩ−1M . Since the multiplier µ(M) = µ(M)is coprime to the multiplier µ(mΩ−1) = m2, we can use the well-known properties ofsymplectic divisors to obtain

sd(MmΩ−1) = sd(M) sd(mΩ−1) = sd(τ (µ(M))mΩ−1M) = sd(mΩ−1) sd(M).

Hence, sd(M) = sd(M). Note that relation (1.31) implies

(2.26) F ∗ ∈ Mk(Γ(m), ψ) if F ∈ Mk(Γ(m), ψ),

where ψ denotes the character conjugate to ψ.

Corollary 2.6. A modular form F ∈ Mk(Γ(m), ψ) is an eigenfunction for the Hecke

operator |TΛ(M), where Λ = Γ(m) and M ∈ ∆(m)(Λ), with the eigenvalue λF (M) if and

only if the function F ∗ ∈ M(Γ(m), ψ) is an eigenfunction for the operator |TΛ(M) withthe eigenvalue

(2.27) λF∗(M) = ψ(µ(M))λF (M).

Proof. Since λF (M)F = F |TΛ(M), (2.24) yields

λF (M)F ∗ = (λF (M)F )∗ = (F |TΛ(M))∗ = ψ(µ(M))F ∗|TΛ(M),

which proves (2.27).

§3. Regular zeta-functions of twisted forms and L-functions

We consider sums of all different double cosets of fixed multipliers prime to m ofm-regular Hecke–Shimura rings H(m)(Λ) of the form (2.5) for the groups Λ defined byconditions (1.8), (1.10), or (1.11):

(3.1) TΛ(a) =∑

M∈Λ\∆(m)(Λ)/Λ, µ(M)=a

TΛ(M) with a prime to m.


864 A. ANDRIANOV

Theorem 3.1. The elements (3.1) for the groups Λ of the form Γ, Γ0(m2), or Γ(m)

satisfy the following rules:

(3.2) TΛ(a)TΛ(a′) = TΛ(aa

′) if gcd(a, a′) = 1;

for each prime number p not dividing m, the formal power series over the ring H(m)(Λ)

with the coefficients TΛ(1), TΛ(p), TΛ(p2), . . . is formally equal to a rational fraction with

coefficients in H(m)(Λ) and with denominator and numerator of degree 2n and at most2n − 2, respectively,

(3.3)

∞∑δ=0

TΛ(pδ)tδ = Qp,Λ(t)

−1Rp,Λ(t),

where

(3.4) Qp,Λ(t) =2n∑i=0

(−1)iqiΛ(p)t

i, Rp,Λ(t) =2n−2∑i=0

(−1)iriΛ(p)ti,

and the coefficients qiΛ(p) and riΛ(p) belong to H(m)(Λ). Moreover, the coefficients of the

denominator Qp,Λ(t) satisfy the relations

(3.5) q0Λ(p) = [1]Λ, q1

Λ(p) = TΛ(p), q2n

Λ (p) =(pn(n+1)/2[p]Λ

)2n−1

and the symmetry relations

(3.6) q2n−iΛ (p) =

(pn(n+1)/2[p]Λ

)2n−1−i

qiΛ(p) (0 ≤ i ≤ 2n),

where

(3.7) [a]Λ = TΛ (aτ (a))

and τ (a) = τn(a) has the form (1.24).

Proof. In accordance with the isomorphism of the rings H(m)(Λ) and H(m)(Γ) describedin Proposition 2.1, it suffices to prove the theorem for the group Λ = Γ = Γn only. Butfor the group Γn all the claims are well known: an analog of (3.2) was proved in [7], ananalog of the summation formula (3.3) was established in [1], and relations similar to(3.5)–(3.6) were checked in [4, §3.3.3].

From (3.2) and (3.3) it follows that the formal Dirichlet series with the coefficientsη(a)TΛ(a) for a prime to m, where η(a) is a completely multiplicative complex-valuedfunction of a, can be expanded in a formal Euler product:

(3.8)

DΛ(s, η) =∑

a∈N, gcd(a,m)=1

η(a)TΛ(a)

as

=∏

primes p m

∞∑δ=0

η(p)δTΛ(pδ)p−δs =

∏primes p m

Qp,Λ(η(p)p−s)−1Rp,Λ(η(p)p

−s)

=

( ∏primes p m

Qp,Λ(η(p)p−s)

)−1( ∏primes p m

Rp,Λ(η(p)p−s)

),

where as is regarded as a formal quasicharacter of the multiplicative semigroup N ofpositive integers. (We recall that, by Proposition 2.2, the ring H(m)(Λ) is commutative.)



Now we turn to regular Hecke operators on the spaces Mk(Λ) of modular forms ofweight k for these groups Λ. Suppose we are given an eigenfunction F ∈ Mk(Λ) for allHecke operators |T with T ∈ H(m)(Λ),

(3.9) F |T = λF (T )F (T ∈ H(m)(Λ)).

We set λF (TΛ(a)) = λF (a), so that

(3.10) F |TΛ(a) = λF (a)F (gcd(a,m) = 1).

Replacing all coefficients T ∈ H(m)(Λ) in the formal identity (3.8) with the eigenvaluesλF (T ) of the corresponding Hecke operators acting on the eigenfunction F , we obtain aformal Euler product expansion over C of the form

(3.11)

DF (s, η) =∑

a∈N, gcd(a,m)=1

η(a)λF (a)

as

=

( ∏primes p m

Qp, F (η(p)p−s)

)−1( ∏primes p m

Rp, F (η(p)p−s)

),

where

(3.12) Qp, F (t) =2n∑i=0

(−1)iλF

(qiΛ(p)

)ti, Rp, F (t) =

2n−2∑i=0

(−1)iλF

(riΛ(p)

)ti.

Assuming now that the function η(a) grows not faster than a constant power of a, andusing the known estimates of Fourier coefficients of modular forms, it is not hard to showthat the infinite series and products occurring in the formal identity (3.11) convergeabsolutely and uniformly in a right half-plane Re s > c = c(F, η) of the complex variables, thus yielding holomorphic functions in s there. (For the case of the group Λ = Γn, see[3, §1.3].) We shall call the function

(3.13) LF (s, η) =

( ∏primes p m

Qp, F (η(p)p−s)

)−1

the (regular) L-function of the eigenfunction F with “character” η and call the function

(3.14) ZF (s) = LF (s, 1) =

( ∏primes p m

Qp, F (p−s)

)−1

the (regular) zeta-function of the eigenfunction F . Proposition 2.1 shows that the Eulerproducts (3.13) and (3.14) do not depend on the choice of a group Λ such that the spaceMk(Λ) contains the eigenfunction F .

Theorem 3.2. For the group Λ′ = Γ0(m2) = Γn

0 (m2), let F ∈ Mk(Λ

′) be an eigen-function for all Hecke operators |TΛ′(M) with M ∈ ∆(m)(Λ

′). Then, for every primitiveDirichlet character χ modulo m and every p-matrix L, the twisted form F |T (χ, L) ∈Mk(Λ), where Λ = Γ(m), is an eigenfunction for all Hecke operators |TΛ(M) withM ∈ ∆(m)(Λ), and the zeta-function of the twisted form in every domain of absoluteconvergence is equal to the L-function of the form F with character χ:

(3.15) ZF |T (χ,L)(s) = LF (s, χ).

Moreover, the form(F |T (χ, L)

)∗, where the star denotes the mapping (1.29), is also an

eigenfunction for all Hecke operators |TΛ(M) with M ∈ ∆(m)(Λ), and the zeta-function


866 A. ANDRIANOV

of this form in every domain of absolute convergence is equal to the L-function of theform F with the conjugate character χ:

(3.16) Z(F |T (χ, L))∗(s) = LF (s, χ).

Proof. By Corollary 2.4, the relation F |TΛ′(a) = λF (a)F implies (F |T (χ, L))TΛ(a) =χ(a)λF (a)F for each a prime to m. Hence, by (3.11) and (3.14), for each prime p notdividing m we obtain

(3.17) Qp, F |T (χ,L)(t) = Qp, F (χ(p)t)

and

ZF |T (χ, L)(s) =

( ∏primes p m

Qp, F |T (χ, L)(p−s)

)−1

=

( ∏primes p m

Qp, F (χ(p)p−s)

)−1

= LF (s, χ).

In order to prove (3.16), we note that, by (2.19) and Corollary 2.6, for each a prime tom we get

λ(F |T (χ, L))∗(a) = χ2(a)λF |T (χ, L)(a).

Therefore, by (3.17), for each prime p not dividing m we have

Qp, (F |T (χ,L))∗(t) = Qp, F |T (χ, L)(χ2(p)t) = Qp, F (χ(p)χ2(p))t) = Qp, F (χ(p)t),

and (3.16) is true.

Note that the definition (2.18) of the subspaces Mk(Γ(m), ψ) implies easily that ev-

ery modular form F ∈ Mk(Γ(m), ψ) is an eigenfunction of each Hecke operator |[a]Λcorresponding to the element (3.7) for Λ = Γ(m) with the eigenvalue

ψ(a)a2(nk−〈n〉) det(a · 1)−k = ψ(a)ank−n(n+1),

F |[a]Λ = ψ(a)ank−n(n+1)F (F ∈ Mk(Γ(m), ψ) and a prime to m);(3.18)

see (6). By (2.19), the twist with character χ of every modular form of this space or the

space Mk(Γ0(m2)) belongs to Mk(Γ(m), χ2). Hence, for the twisted forms we obtain

(3.19) (F |T (χ, L))|[a]Λ = χ(a2)ank−n(n+1)(F |T (χ, L)) for all a prime to m.

These relations together with (3.5), (3.6) allow us to compute the constant and theleading coefficients in the denominators of p-factors of zeta-functions for the twistedform G = F |T (χ, L) of an eigenfunction F ∈ Mk(Γ0(m

2)):

(3.20) λG

(q0Λ(p)

)= 1, λG

(q2n

Λ (p))=

(pnk−〈n〉χ(p2)

)2n−1

and to write the general symmetry relations for the coefficients of the denominators:

(3.21) λG

(q2n−iΛ (p)

)=

(pnk−〈n〉χ(p2)

)2n−1−i

λG

(qiΛ(p)

)(0 ≤ i ≤ 2n).



§4. L-functions of cusp forms of genus one

Here we apply Theorem 3.2 on reduction to the simplest case of modular forms in onevariable and reprove some consequences of the Atkin–Lehner theory of new forms [5]. Inthis section we assume that the genus n is equal to 1.

Let F ∈ Nk(Γ0(m2)) be a cusp form of an integral weight k for the congruence

subgroup of the modular group Γ1 of the form Γ0(m2) = Γ1

0(m2) with Fourier expansion

(1) for n = 1,

(4.1) F (z) =

∞∑a=1

f(a)e2πiaz (z = x+ iy ∈ H = H1).

We assume that F is an eigenfunction for all Hecke operators |T = |TΛ′ with T ∈H(m)(Λ

′), where Λ′ = Γ0(m2):

(4.2) F |T = λF (T )F (T = TΛ′ ∈ H(m)(Λ′)).

Next, we denote by G the image of F under the twist operator (4) with a fixed primitiveDirichlet character χ modulo m and the p-matrix L = l = 1; by definition, G has theFourier expansion

(4.3) G(z) =

∞∑a=1

g(a)e2πiaz, where g(a) = χ(a)f(a),

and, by Proposition 1.4, it belongs to the space Nk(Γ(m)) of cusp forms of weight k for

the group Γ(m) = Γ1(m) of the form (1.11),

(4.4) G = F |T (χ) = F |T (χ, 1) ∈ Nk(Γ(m)).

Then, by Corollary 2.4, the form G is an eigenfunction for all Hecke operators |TΛ

with Λ = Γ(m) and TΛ ∈ H(m)(Λ). Moreover, the eigenvalues of the Hecke operators

corresponding to the elements (3.1) for Λ = Γ(m) and Λ′ = Γ0(m2) acting on G and F ,

respectively, satisfy

(4.5) λG(a) = χ(a)λF (a) (a ∈ N, gcd(a,m) = 1).

Thus, we have two sequences of complex numbers associated with each of the eigen-functions: the sequence of Fourier coefficients, and the sequence of eigenvalues of Heckeoperators. The natural question is whether these sequences are related to each other. Thequestion is interesting in two respects: first, because of multiplicative properties of Heckeoperators and their eigenvalues, such relations could reveal multiplicative properties ofFourier coefficients, which often presents an arithmetical interest, and second, analyticproperties of modular forms as the generating series for their Fourier coefficients maypossibly be transferred to analytic properties of generating functions for the eigenvaluesand corresponding Euler products.

The relationship between Fourier coefficients of an eigenfunction and the correspond-ing eigenvalues, for modular forms in one variable, were discovered by Hecke and arequite simple.

Lemma 4.1 (Hecke). Suppose that a modular form F ∈ Nk(Γ0(m2)) with Fourier ex-

pansion (4.1) is an eigenfunction of the Hecke operator T (a) for the group Γ0(m2) with a

prime to m, and let λF (a) be the corresponding eigenvalue. Then, for every b = 1, 2, . . . ,we have

(4.6) λF (a)f(b) =∑d|a, b

dk−1f

(ab

d2

),


868 A. ANDRIANOV

where d ranges over all positive common divisors of a and b; in particular,

(4.7) f(a) = f(1)λF (a).

Proof. The proof for an arbitrary m is actually the same as that in the case where m = 1;see, e.g., [6, Chapter II, Proposition 7].

From this lemma and the well-known estimates of Fourier coefficients of cusp formswe obtain the following Euler product expansions for the generating Dirichlet series ofFourier coefficients of eigenforms.

Proposition 4.2. Suppose that a cusp form F ∈ Nk(Γ0(m2)) with Fourier expansion

(4.1) is an eigenfunction of all regular Hecke operators of the form T (a) for the groupΓ0(m

2) with the eigenvalues λF (a), and χ is a primitive Dirichlet character modulom ≥ 1. Then the following identity is valid in the right half-plane Re s > k/2 + 1:

(4.8)

∞∑a=1

χ(a)f(a)

as= f(1)

∏primes pm

(1− χ(p)λF (p)p

−s + χ2(p)pk−1−2s)−1

= f(1)LF (s, χ),

where LF (s, χ) is the L-function (3.13) of the form F .

Proof. Using (4.7), we get

∞∑a=1

χ(a)f(a)

as= f(1)

∞∑a=1

χ(a)λF (a)

as= f(1)LF (s, χ)

(note that χ(a) = 0 if a is not prime to m).

Identity (4.8) makes it possible to investigate the analytic properties of the L-functionsLF (s, χ). By Theorem 3.2, it suffices to consider the zeta-function of the twisted formG.

Proposition 4.3. Suppose that a cusp form F of weight k for the group Γ0(m2) is an

eigenfunction of all regular Hecke operators for that group. Let G = F |T (χ) ∈ Mk(Γ(m))be the twist (4.3) of F with a primitive Dirichlet character χ modulo m ≥ 1. Then thefollowing is true.

1) The zeta-function (3.14) of the eigenfunction G in the half-plane Re s > k/2 + 1satisfies the identity(4.9)

ΨG(s) = ms−k/2(2π)−sΓ(s)ZG(s)

=1

g(1)

(ms−k/2

∫ ∞

1/m

ys−1G(iy) dy + (−i)kmk/2−s

∫ ∞

1/m

yk−1−sG∗(iy) dy

),

where g(1) is the first coefficient of the Fourier expansion (4.4) of G, Γ(s) is the gamma-function, and G → G∗ is the mapping (1.29).

2) The right-hand side of (4.9) is holomorphic for all s. Thus, the function ΨG(s) hasanalytic continuation to the entire s-plane as a holomorphic function.

3) The function ΨG(s) satisfies the functional equation

(4.10) ΨG(k − s) = (−i)kg∗(1)

g(1)ΨG∗(s),

where, for the eigenfunction G∗ = (F |T (χ))∗ ∈ Nk(Γ(m)) of all regular Hecke operators,



we set

(4.11) ΨG∗(s) = ms−k/2(2π)−sΓ(s)ZG∗(s),

ZG∗(s) is the zeta-function of G∗, and g∗(1) is the first Fourier coefficient of G∗.

Proof. By Theorem 3.2 and formulas (4.3), we can write identity (4.8) in the form

(4.12) RG(s) =∞∑a=1

g(a)

as=

∞∑a=1

χ(a)f(a)

as= f(1)LF (s, χ) = f(1)ZG(s).

Using the Euler integral∫ ∞

0

ys−1e−αy dy = Γ(s)α−s (α > 0, Re s > 0),

where Γ(s) is the gamma-function, we obtain∫ ∞

0

ys−1G(iy) dy =

∞∑a=1

g(a)

∫ ∞

0

ys−1e−2πay dy

= (2π)−sΓ(s)

∞∑a=1

g(a)

as= (2π)−sΓ(s)RG(s) (Re s > k/2 + 1).

Hence, by (4.12),

(4.13)

ΨG(s) =1

g(1)ms−k/2(2π)−sΓ(s)RG(s) =

1

g(1)ms−k/2

∫ ∞

0

ys−1G(iy) dy

=1

g(1)

(ms−k/2

∫ ∞

1/m

ys−1G(iy) dy +ms−k/2

∫ 1/m

0

ys−1G(iy) dy

).

Replacing y by 1/m2y, we can write∫ 1/m

0

ys−1G(iy) dy =

∫ 1/m

∞(m2y)1−sG(i/m2y)

−dy

m2y2

= (−i)kmk−2s

∫ ∞

1/m

yk−1−s(−imy)−kG(i/m2y) dy

= (−i)kmk−2s

∫ ∞

1/m

yk−1−sG∗(iy) dy,

because, by definition,

G∗(z) = G|k(0 1/mm 0

)= (−mz)−kG(−1/m2z).

Substituting this in the right-hand side of (4.13), we arrive at (4.9).Since G is a cusp form, the absolute values of the two integrands on the right in

(4.9) decay exponentially as y → +∞. Hence, both integrals converge absolutely anduniformly for all s ∈ C and determine functions holomorphic everywhere.


870 A. ANDRIANOV

Finally, by formula (4.9) with G∗ in place of G and k − s in place of s, we get

ΨG∗(k − s) =1

g∗(1)

(mk−s−k/2

∫ ∞

1/m

yk−s−1G∗(iy) dy

+(−i)kmk/2−(k−s)

∫ ∞

1/m

yk−1−(k−s)G∗∗(iy) dy

)= ik

g(1)

g∗(1)

1

g(1)

((−i)k

∫ ∞

1/m

yk−s−1G∗(iy) dy +ms−k/2

∫ ∞

1/m

ys−1G(iy) dy

)= ik

g(1)

g∗(1)ΨG(s),

because, by (1.26),

G∗∗ = G|kΩ|kΩ = G|kΩ2G|k(−12) = (−1)kG.

The functional equation (4.10) follows.

Returning to L-functions, now we can prove the following theorem.

Theorem 4.4. In the notation and under the assumptions of Proposition 4.2, the func-tion

(4.14) ΨF (s, χ) = ms−k/2(2π)−sΓ(s)LF (s, χ)

has analytic continuation to the entire s-plane as a holomorphic function and satisfiesthe functional equation

(4.15) ΨF (k − s, χ) = (−i)kg∗(1)

f(1)ΨF (s, χ),

where f(1) and g∗(1) are the first Fourier coefficients of F and of (F |T (χ))∗, respectively.

Proof. By Theorem 3.2, in the notation of Proposition 4.3 we have ΨF (s, χ) = ΨG(s),and the functional equation (4.15) follows from (4.10), because

g(1) = χ(1)f(1) = f(1)

by (4.3), and

ZG∗(s) = L(s, χ)

by (3.16).

References

[1] A. N. Andrianov, Spherical functions for GLn over local fields, and the summation of Hecke series,Mat. Sb. (N. S.) 83(125) (1970), no. 3, 429–451; English transl., Math. USSR-Sb. 12 (1970), 429–452. MR0282982 (44:216)

[2] , Dirichlet series with Euler product in the theory of Siegel modular forms of genus two,Trudy Mat. Inst. Steklov. 112 (1971), 73–94; English transl. in Proc. Steklov Inst. Math. 1971.MR0340178 (49:4934)

[3] , Euler products that correspond to Siegel’s modular forms of genus 2, Uspekhi Mat. Nauk29 (1974), no. 3, 43–110; English transl. in Russian Math. Surveys 29 (1974), no. 3. MR0432552(55:5540)

[4] , Quadratic forms and Hecke operators, Grundlehren Math. Wiss., Bd. 286, Springer-Verlag,Berlin, 1987. MR0884891 (88g:11028)

[5] A. O. L. Atkin and J. Lehner, Hecke operators on Γ0(m), Math. Ann. 185 (1970), 134–160.MR0268123 (42:3022)


http://www.ams.org/mathscinet-getitem?mr=0282982











[6] A. Ogg, Modular forms and Dirichlet series, W. A. Benjamin, Inc., New York–Amsterdam, 1969.MR0256993 (41:1648)

[7] G. Shimura, On modular correspondences for Sp(n, Z) and their congruence relations, Proc. Nat.Acad. Sci. U.S.A. 49 (1963), no. 6, 824–828. MR0157009 (28:250)

St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences,

Fontanka 27, 191023 St. Petersburg, Russia

E-mail address: [email protected] address: [email protected]

Received 2/JUN/2008

Translated by THE AUTHOR






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