Special Values of the Standard Zeta Functions for Elliptic Modular Forms

Special Values of the Standard Zeta Functions for Elliptic Modular Forms

Hidenori Katsurada

CONTENTS 1. Introduction

2. Fourier Coefficients of Siegel-Eisenstein Series 3. Pullback Formula

4. Computation ofL(f, l, χ)

5. Numerical Examples and Comments Acknowledgments

References

2000 AMS Subject Classification:Primary 11F67 Keywords: Standard zeta functions, special values

We give an algorithm for computing the special values of twisted standard zeta functions of elliptic modular forms by using the pullback formula for the Siegel-Eisenstein series of degree 2.

1. INTRODUCTION

Let M and k be positive integers and φ a Dirichlet character modulo M. For a normalized cuspidal Hecke eigenformf of weightkand Nebentypus φwith respect to Γ₀(M), and a Dirichlet character χ modulo N, let L(f, s, χ) be the standard zeta function off twisted byχ.

(For the precise definition of the standard zeta function, see the paragraph immediately preceding Theorem 3.3.) The twisted standard zeta function of an elliptic modular form is sometimes called a twisted symmetric-square L function, an important subject in number theory, and is related to many other areas, especially to Galois repre- sentations. For examples, see [Doi et al. 98] and [Dummi- gan 01]. The special values of the standard zeta function are particularly important. To be more precise, assume thatk is even, and set

L^∗(f, m, χ) = L(f, m, χ) π^k+2mf, f

for a positive integer m ≤ k−1 such that (−1)^m−1 = χ(−1),where–,–is the normalized Petersson product.

As is well known, these values are algebraic numbers and their qualitative natures have been fully investigated by many people (see [Sturm 80, Shimura 00, B¨ocherer and Schmidt 00]). To investigate various problems related to these values, it is important to compute these values ex- actly. Several people have considered algorithms for com- puting these values and have carried out the computa- tions. Sturm [Sturm 80] gave an algorithm for computing these values for a generalχ.However, it seems difficult to give exact values by direct use of his method. Zagier [Za- gier 77] gave an explicit formula expressing L^∗(f, m, χ)

c A K Peters, Ltd.

1058-6458/2005$0.50 per page Experimental Mathematics14:1, page 27

in the case whereM is a square-free positive integer con- gruent to 1 modulo 4, φ is the Kronecker character (^M_∗) corresponding to the extensionQ(√

M)/Q,andχis triv- ial. Stopple [Stopple 96] gave an explicit formula express- ing L^∗(f, m, χ) in the caseM = 1 and χ is a quadratic character of prime conductor q≡1 mod 4.

In [Katsurada 03], we announced some formulas which seem useful for the computation ofL^∗(f, m, χ) in the case where M = 1 or a prime number congruent to 1 modulo 4, φ = (^M_∗ ), and χ is not necessarily a quadratic char- acter of prime conductor psuch thatχ(−1) = 1.In this paper, we give a complete proof of these formulas un- der more general settings. The main tool is the pullback formula of the Siegel-Eisenstein series of degree 2 due to B¨ocherer and Schmidt [B¨ocherer and Schmidt 00] and Shimura [Shimura 00]. Such a formula has been used to study the qualitative nature of the special values of the standard zeta function. However, as far as the au- thor knows, no one has used the formula to give its exact values. In this paper, we carry out such a computation.

To explain our method briefly, for simplicity M = p. Let k and l be even positive integers such that l ≤ k. Then we define a certain Siegel-Eisenstein se- ries E_2,l^∗ (Z, M p², φ¯χ, s) in Section 2. We write e(u) = exp(2π√

−1u) for a complex numberu.Then, as is well known, ifl ≥4, E_2,l^∗ (Z, M p², φχ,¯ 0) becomes a holomor- phic modular form of weight l and of Nebentypus φ¯χ;

and has a Fourier expansion of the following form:

E_2,l^∗ (Z;M p², φχ,¯ 0) =

A

c_n,l(A, M p², φ¯χ,0)e(tr(AZ)), where A runs over all positive definite half-integral ma- trices of degree 2, and tr(–) denotes the trace of a matrix.

Set

˜

c_2,l(A,0) = ˜c_2,l(A, M p², φ¯χ,0)

=A(l,0)⁻¹c_2,l(A, M p², φ¯χ,0)

with a suitable normalizing factorA(l,0)⁻¹(see Theorem 2.1). For two positive integers m₁, m₂ set

(m₁, m₂;l,0) =

r²≤4m1m₂

˜ c_2,l

m₁ r/2 r/2 m₂

,0

G^k−l_l (m₁m₂, r)χ(r)τ( ¯χ), whereG^k−l_l (u, v) is the polynomial introduced by Zagier [Zagier 77], andτ( ¯χ) is the Gauss sum (see Section 3).

Furthermore, set

t(m;l,0) =(p, p²m;l,0)−φ(p)p^k−2(p, m;l,0), and

Fp,p(z) = ∞ m=1

t(m;l,0)e(mz).

Then by the holomorphy of the Eisenstein series and the theory of differential operators on modular forms, due to Ibukiyama [Ibukiyama 99], Fp,p(z) belongs to Sk(Γ₀(M p), φ) (see Sections 3 and 4). Now, take a ba- sis{fi}^d_i=1¹ ofSk(Γ₀(M), φ) consisting of primitive forms, and write

fi(z) = ∞ m=1

ai(m)e(mz)

with ai(1) = 1. Then, by the pullback formula due to B¨ocherer and Schmidt [B¨ocherer and Schmidt 00], we have

Fp,p(z) =γk,l,p,M d₁

i=1

L^∗(fi, l−1, χ) ¯ci2f˜i(z), whereγk,l,p,M is a rational number explicitly determined by k, l, p, M; and c_i is a certain algebraic number with absolute norm 1; and

f˜i(z) = ∞ m=1

ai(pm)e(mz)

(see (2) of Theorem 4.2). We restate an explicit form of

˜

c_2,l(A,0) (see Theorem 2.1).

Thus, by the above formula combined with the trace formula of Hecke operators, we can compute the norm NKf,χ(L^∗(f, m, χ)) for a primitive form f ∈ S_k(Γ₀(M), φ) and for an odd integer m such that 3 ≤ m ≤ k − 1. Here Kf,χ is the field over Q gener- ated by all the eigenvalues of Hecke operators relative to f and all the values of χ (see Theorem 4.6). If χ² is not trivial, E_2,2^∗ (Z;M p², φ¯χ,0) becomes holomor- phic, and by the same procedure, we obtain an ex- act value for NKf,χ(L^∗(f,1, χ)). On the other hand, if χ² is trivial, E_2,2^∗ (Z;M p², φχ,¯ 0) is not holomorphic.

However, E_2,2^∗ (Z;M p², φχ,¯ −1/2) is holomorphic, and by the same procedure, we obtain an exact value of NKf,χ(L^∗(f,0, χ)); and by the functional equation due to Li [Li 79], we can also computeN_Kf,χ(L^∗(f,1, χ)) (see (2) of Proposition 4.7). In the case M = p we obtain similar results (see (1) of Theorem 4.2 and (1) of Theo- rem 4.6). In Section 5, we give some numerical examples, and discuss some related topics.

As an application of Theorem 4.2, we show that a prime factor of the denominator of L^∗(f, m, χ) gives a congruence between f and another primitive form (see Theorem 4.10).

By using the method in this paper, we expect more fruitful results about the special values of standard zeta functions of other modular forms, for example, of Siegel modular forms and of Hilbert modular forms. We will discuss these topics in subsequent papers.

2. FOURIER COEFFICIENTS OF SIEGEL-EISENSTEIN SERIES

LetGSp⁺_n(R) be the group of proper symplectic simili- tudes of degreen,andHnSiegel’s upper half space of de- green. As is usual, we writeγ(Z) = (AZ+B)(CZ+D)⁻¹ andj(γ, Z) = det(CZ+D) for

γ=

A B

C D

∈GSp⁺_n(R).

We write f|kγ(z) = (detγ)^k/2j(γ, z)^−kf(γ(z)) for γ ∈ GSp⁺_n(R) and aC^∞-functionf onHn.We simply write f|γforf|kγ, if there is no confusion. LetSpn(Z) be the Siegel modular group of degreen.For a positive integer M, we denote by Γ⁽ⁿ⁾₀ (M) (respectively Γ₀⁽ⁿ⁾(M)) the subgroup of Sp_n(Z) consisting of matrices whose lower leftn×nblock (respectively upper rightn×nblock) is congruent toO moduloM.

For a Dirichlet character φ modulo M, we denote by ˜φ(respectively ˜φ) the character of Γ⁽ⁿ⁾₀ (M) (respec- tively Γ₀⁽ⁿ⁾(M)) defined by ˜φ(γ) = φ(detD) (respec- tively ˜φ(γ) =φ(detA)) for

γ=

A B

C D

.

We denote by 1M the trivial character modulo M and, in particular, set 1 = 11. For a Dirichlet character φ moduloM, we denote by Mk(Γ⁽ⁿ⁾₀ (M), φ) (respectively M_k^∞(Γ⁽ⁿ⁾₀ (M), φ)) the space of holomorphic (respectively C^∞-) modular forms of weightkand Nebentypusφwith respect to Γ⁽ⁿ⁾₀ (M),and bySk(Γ⁽ⁿ⁾₀ (M), φ) the subspace ofMk(Γ⁽ⁿ⁾₀ (M), φ) consisting of cusp forms. In particu- lar, ifφ=1M,we writeSk(Γ₀⁽ⁿ⁾(M)) forSk(Γ⁽ⁿ⁾₀ (M), φ).

Furthermore, for a subgroup Γ ofSp_n(Z) we denote by Γ_∞the subgroup of Γ consisting of matrices whose lower leftn×nblock isO.

For a function f on Hn we write f^c(Z) = f(−Z).¯ Letdvdenote the invariant volume element onHn given bydv= det(Im(Z))⁻ⁿ⁻¹∧1≤j≤l≤n(dx_jl∧dy_jl).Here, for Z∈Hnwe writeZ = (xjl)+√

−1(yjl) with real matrices (x_jl) and (y_jl). For two C^∞-modular forms f and g of weightkand Nebentypusφwith respect to Γ⁽ⁿ⁾₀ (M),we define the Petersson scalar productf, g_Γ(n)

0 (M)off and gby

f, g_Γ(n)

0 (M)=

Γ⁽ⁿ⁾₀ (M)\Hn

f(Z)g(Z) det(Im(Z))^kdv, provided the integral converges.

Letφbe a Dirichlet character moduloL.For a positive integerM such thatL|M,we denote byφM the Dirichlet

character moduloM induced by φ. Letf and g be ele- ments of M_k^∞(Γ⁽ⁿ⁾₀ (M₁), φM₁) and M_k^∞(Γ⁽ⁿ⁾₀ (M₂), φM₂), respectively. Let N be any common multiple of M₁ andM₂.Then,f andgbelong toM_k^∞(Γ⁽ⁿ⁾₀ (N), φN),and the valuem(Φ_Γ(n)

0 (N))⁻¹f, g_Γ(n)

0 (N)does not depend on the choice of N, where Φ_Γ(n)

0 (N) is the fundamental do- main for

Hn modulo Γ⁽ⁿ⁾₀ (N), andm(Φ_Γ(n)

0 (N)) =

Γ⁽ⁿ⁾₀ (N)\Hndv.We denote this value byf, gand call it the normalized Petersson product of f andg.

For a Dirichlet characterψ, we denote byL(s, ψ) the DirichletL-function associated withψ. Letn, l, and M be positive integers. For a Dirichlet characterφmodulo M such that φ(−1) = (−1)^l, we define the Eisenstein seriesE_n,l (Z;M, φ, s) by

E_n,l (Z;M, φ, s) = det Im(Z)^sL(l+ 2s, φ)

[n/2]

i=1

L(2l+ 4s−2i, φ²)

×

γ∈Γ₀⁽ⁿ⁾(M)∞\Γ₀⁽ⁿ⁾(M)

φ˜(γ)j(γ, Z)^−l|j(γ, Z)|^−2s.

We then defineE_n,l^∗ (Z;M, φ, s) by

E_n,l^∗ (Z;M, φ, s) =j(ι, Z)^−lE_n,l (ι(Z);M, φ, s), where

ι=

0n −1n

1n 0n

.

Let Hn(Z) denote the set of half-integral matrices of degree n over Z, and denote by Hn(Z)>0 (respec- tively Hn(Z)_≥0) the subset of Hn(Z) consisting of pos- itive definite (respectively semipositive definite) matri- ces. Then, it is well known thatE_n,l^∗ (Z;M, φ, s) belongs to M_l^∞(Γ⁽ⁿ⁾₀ (M), φ) and has a Fourier expansion of the following form:

E_n,l^∗ (X+√

−1Y;M, φ, s) =

A∈Hn(Z)

cn,l(A, Y, M, φ, s)e(tr(AX)).

In particular, if E_n,l^∗ (Z;M, φ, s) belongs to Ml(Γ⁽ⁿ⁾₀ (M), φ), it has the following Fourier expan- sion:

E_n,l^∗ (Z;M, φ, s) =

A∈Hn(Z)≥0

cn,l(A, M, φ, s)e(tr(AZ)).

Throughout the rest of this paper, we exclusively con- sider the case n= 2.

Let l be an even positive integer. Let M > 1 be an integer, and let φ be a Dirichlet character modulo M such that φ(−1) = 1. Then, E_2,l^∗ (Z;M, φ,0) be- longs to M_l(Γ⁽²⁾₀ (M), φ) in the case l ≥4.Furthermore, E_2,2^∗ (Z;M, φ,0) belongs to M₂(Γ⁽²⁾₀ (M), φ) if φ² =1M. We remark that E_2,2^∗ (Z;M, φ,0) is neither holomorphic nor nearly holomorphic in the sense of [Shimura 00]

if φ² = 1M. However, E^∗_2,l(Z;M, φ,−1/2) belongs to M₂(Γ⁽²⁾₀ (M), φ) in this case.

Now, to see the Fourier coefficient of the Eisenstein series, for an element

A=

a₁₁ a₁₂/2 a₁₂/2 a₂₂

∈ H2(Z),

set e = eA = GCD(a₁₁, a₁₂, a₂₂). For an element A ∈ H2(Z) such that rankA= 1 and for each prime number p, define a polynomialFp(A, X) as

F_p(A, X) =

ordp(eA) i=0

(pX)ⁱ,

where ordpdenotes the normalized additive valuation on the field ofp-adic numbers. For an element

A=

a₁₁ a₁₂/2 a₁₂/2 a₂₂

∈ H2(Z)>0,

write−4 detA=_A²_A with_Athe fundamental discrimi- nant ofQ(√

−detA) and_A a positive integer. Further- more, let χ_A = (_∗^A) be the Kronecker character cor- responding to Q(√

−detA)/Q. For a prime numberp, define a polynomialF_p(A, X) as

Fp(A, X) =

ordp(eA) i=0

(p²X)ⁱ

ordp(_A)−i j=0

(p³X²)^j−χA(p)pX

×

ordp(eA) i=0

(p²X)ⁱ

ordp(_A)−i−1 j=0

(p³X²)^j. For a Dirichlet character ψ moduloL,letm_ψ denote its conductor, and ψ⁽⁰⁾ the associated primitive char- acter. Furthermore, let B_m,ψ be the mth generalized Bernoulli number associated withψ,and letτ(ψ) be the Gauss sum defined by

τ(ψ) =

XmodL

ψ(X)e(X/L).

Letlbe an even positive integer, ands= 0 or−1/2.Let φbe a Dirichlet character such thatφ(−1) = 1.Now as- sume that the triple (l, s, φ) satisfies one of the following conditions:

(h–1) l≥4 ands= 0;

(h–2) l= 2, s= 0 andφ² is not trivial;

(h–3) l= 2 ands=−1/2.

Theorem 2.1.[Katsurada 99, Shimura 00]. LetM >1 be an integer, and φa Dirichlet character modulo M such thatφ(−1) = 1.Letlbe an even positive integer, ands= 0or−1/2.Assume that the triple(l, s, φ)satisfies one of the Conditions (h–1), (h–2), or (h–3). First, assume that (l, s, φ) satisfies either Condition (h–1) or (h–2). Then forA∈ H2(Z)_≥0 set

˜

c_2,l(A,0) = ˜c_2,l(A;M, φ,0) =



















(4 detA)^l−3/2

p|_AF_p(A, φ(p)p^−l)B_l−1,(φχ

A)⁽⁰⁾

×τ((φχA)⁽⁰⁾)(−√

−1)m¹⁻_(φχ^l_A)(0)

×

p|M(1−(φχ_A)⁽⁰⁾(p)p^1−l)

A >0

0 otherwise.

Next assume that(l, s, φ)satisfies Condition (h–3). Then forA∈ H2(Z)_≥0 set

˜

c_2,2(A,−1/2) = ˜c_2,2(A;M, φ,−1/2) =



























−

p|_AF_p(A, φ(p)p⁻¹)B_1,(φχA)(0)

×

p|M(1−(φχA)⁽⁰⁾(p)) A >0

−1/2

p|eAF_p(A, φ(p)p⁻¹)

×

p|M(1−(φ²)⁽⁰⁾(p)p)B_2,(φ2)⁽⁰⁾ rankA= 1 1/8

p|M{(1−(φ²)⁽⁰⁾(p)p)(1−(φ)⁽⁰⁾(p)p)}

×B_2,(φ2)⁽⁰⁾B_2,(φ)(0) A=O Let

A(l, s) =(−1)^l/22^lπ^3l−3/2 Γ(l)²Γ(l−1/2) forl≥2 ands= 0, and let

A(l, s) = 8π^5/2 Γ(3/2)

for l = 2 and s = −1/2, where Γ (–) is the Gamma function. Then we have

c_2,l(A;M, φ, s) =A(l, s)˜c_2,l(A;M, φ, s).

Remark 2.2. Assume that (l, s, φ) satisfies Condition (h–1) or (h–2). Letm be the conductor ofφ, and write

A = _A˜

A with _A =

p|mp^ordp(A) and (˜_A, m) = 1.

Then, by the functional equation ofF_p(A, X) (see [Kat- surada 99]), we can rewrite ˜c_2,l(A,0) as

˜

c_2,l(A,0) = (|A|_A²)^l−3/2φ(˜_A)²

p|˜

A

Fp(A,φ(p)p¯ ^l−3)

×B_l−1,(φχ

A)⁽⁰⁾(−√

−1)τ((φχ_A)⁽⁰⁾)m¹⁻_(φχ^l

A)⁽⁰⁾

×

p|M

(1−(φχA)⁽⁰⁾(p)p^1−l). (2–1) Let φ be a Dirichlet character modulo M with con- ductormsuch thatφ(−1) = 1.Setm =M/m.If|A|is prime tom,we have

τ((φχA)⁽⁰⁾) =√

−1φ(A)χA(m)τ(φ)|A|^1/2, m_(φχA)(0)=m|A|,

and

p|M

(1−(φχ_A)⁽⁰⁾(p)p^1−l) =

p|m

(1−φ⁽⁰⁾χ_A(p)p^1−l).

Thus if 4 detAis prime toM, we have

˜

c_2,l(A,0) =φ(−4 detA)

p|_A

Fp(A,φ(p)p¯ ^l−3)

×B_l−1,(φχ

A)⁽⁰⁾χA(m)τ(φ)m^1−l

×

p|m

(1−(φχ_A)⁽⁰⁾(p)p^1−l). (2–2) In particular, ifM is a square-free odd positive integer dividingm₁m₂ and r is an integer prime to m₁m₂, we have

˜

c_2,l(A,0) =φ(r)²

p|A

F_p(A,φ(p)p¯ ^l−3)

×B_l−1,(φχ

A)⁽⁰⁾

×τ(φ)m^1−l

p|m

(1−(φχ_A)⁽⁰⁾(p)p^1−l) (2–3) for

A=

m₁ r/2 r/2 m₂

. On the other hand, we have

˜

c_2,l(A,0) =

p|A

F_p(A, φ(p)p^l−3)(|A|_A²)^l−3/2

×B_l−1,(φχ

A)⁽⁰⁾m^3/2−_(φχ ^l

A)⁽⁰⁾

×

p|M

(1−(φχA)⁽⁰⁾(p)p^1−l), (2–4)

ifφ²=1M. Thus ifφ²=1M and|A|is prime tom, we have

˜

c_2,l(A,0) =

p|A

F_p(A, φ(p)p^l−3)B_l−1,(φχ

A)⁽⁰⁾m^3/2−l

×

p|m

(1−(φχ_A)⁽⁰⁾(p)p^1−l). (2–5) Assume that M is a square-free odd positive integer and that|A| is prime tom.Ifφis primitive,

p|M

(1−(φχA)⁽⁰⁾(p)p^1−l) = 1.

On the other hand, letφ=1M.Let A=

m₁ r/2 r/2 m₂

∈ H2(Z)>0

withm₁m₂ divided byM andrprime tom₁m₂.Then,

˜

c_2,2(A;M,1M,−1/2) = 0. (2–6) Furthermore, for any positive integerl≥2,we have

˜

c_2,l(A;M,1M,0) =−

p|A

Fp(A, p^l−3)B_1,χ(0) A

p|M

(1−p^1−l), (2–7) provided (l, s, φ) satisfies Condition (h–1) or (h–2). On the other hand, ifφ²=1M but φ=1M, we have

˜

c_2,2(A;M, φ,−1/2) =













−₁₂¹

p|eAFp(A, φ(p)p⁻¹)

p|M(1−p) rankA= 1

1 48

p|M{(1−p)(1−(φ)⁽⁰⁾(p)p)}B_2,(φ)(0)

A=O.

(2–8)

3. PULLBACK FORMULA

Now we define B¨ocherer’s differential operator. For de- tails, see [B¨ocherer and Schmidt 00]. First we define the differential operatorDαon the module C^∞(H2) ofC^∞- functions onH2by

Dα(f) =−(α−1/2)∂f /∂z₁₂ +z₁₂

∂²f /∂z₁₁∂z₂₂−1

4∂²f /∂z₁₂²

forf ∈C^∞(H2) and Z =

z₁₁ z₁₂ z₁₂ z₂₂

∈H2.

For a nonnegative integerν define the differential opera- torsD^ν_αand ˜D^ν_αby

D^να=Dα+ν−1....Dα, and

D˜α^ν =Dα^ν|z₁₂=0.

Furthermore, fors∈Candf ∈C^∞(H2),we define ˜D_α,s^ν by

D˜_α,s^ν (f)(z₁₁, z₂₂) = (y₁₁y₂₂)^sD˜^ν_α+s(detY^−sf(Z)), where Z=X+√

−1Y ∈H2 and Y =

y₁₁ y₁₂ y₁₂ y₂₂

.

Let φ be a Dirichlet character modulo M. Then, it is well known that ˜D_l^ν and ˜D^ν_l,s map M_l^∞(Γ⁽²⁾₀ (M), φ) into M_l+ν^∞ (Γ⁽¹⁾₀ (M), φ) ⊗ M_l+ν^∞ (Γ⁽¹⁾₀ (M), φ). Further- more, ˜D_l^ν mapsMl(Γ⁽²⁾₀ (M), φ) intoM_l+ν(Γ⁽¹⁾₀ (M), φ)⊗ M_l+ν(Γ⁽¹⁾₀ (M), φ), and, in particular, if ν > 0, its im- age is contained inS_l+ν(Γ⁽¹⁾₀ (M), φ)⊗S_l+ν(Γ⁽¹⁾₀ (M), φ).

Clearly these two operators, D˜^ν_l and ˜D_l,s^ν , coincide with each other if s = 0. Furthermore, for F(Z) ∈ M_l^∞(Γ⁽²⁾₀ (M), φ) andg(z₁)∈ S_l+ν(Γ⁽¹⁾₀ (M), φ) we have the following identity as a function of z₂:

D˜^ν_l(F)(–, z₂), g=dl,ν,sD˜^ν_l,s(F)(–, z₂), g (3–1) if both sides are cusp forms (see [B¨ocherer and Schmidt 00, (1.30)]).

Here, we take the inner product as a function of z₁ and

dl,ν,s= ν µ=1

l−1 +ν−µ/2 l+s−1 +ν−µ/2.

In addition to the above notation, let N ≥1 be a pos- itive integer, and χ a Dirichlet character modulo N.

Assume that N² divides M. For positive even integers l, k such that l ≤ k we define a function (z₁, z₂) =

2,k(z₁, z₂;l, M, φ, χ, s) onH1×H1:

2,k(z₁, z₂;l, M, φ, χ, s) = D˜_l,s^k−l





x∈Z/NZ

¯ χ(x)

×E_2,l^∗ (–;M, φχ, s)¯ |kR(x/N) z₁ 0 0 z₂

,

where

R(x) =







1 0 0 x

0 1 x 0

0 0 1 0

0 0 0 1





.

Then (see [B¨ocherer and Schmidt 00]),(z₁, z₂) belongs toM_k^∞(Γ⁽¹⁾₀ (M), φ)⊗M_k^∞(Γ⁽¹⁾₀ (M), φ) .

Now to see an explicit form of ˜D^ν_l,for an even positive integer l and nonnegative integerν we define a polyno- mialG^2ν_l (u, v) inu, v by

G^2ν_l (u, v) = ν µ=0

(−1)^µ(l+ 2ν−µ−2)!

(2ν−2µ)!µ! u^µv^2ν−2µ. This polynomial was introduced by Zagier [Zagier 77].

We define Ibukiyama’s differential operator G_l^2ν on C^∞(H2) by

Gl^2ν=G^2ν_l (∂²/∂z₁₁∂z₂₂, ∂/∂z₁₂)|z12=0. We note that

Gl^2ν(e(tr(AZ)) = G^2ν_l (a₁₁a₂₂, a₁₂)(2π√

−1)^2νe(a₁₁z₁₁+a₂₂z₂₂) (3–2) for

A=

a₁₁ a₁₂/2 a₁₂/2 a₂₂

and

Z =

z₁₁ z₁₂ z₁₂ z₂₂

.

It is well known thatG_l^2ν is a constant multiple of ˜D^2ν_l (see [Ibukiyama 99]). More precisely, by calculating G_l^2ν(z^2ν₁₂) and ˜D_l^2ν(z^2ν₁₂) for

Z =

z₁ z₁₂ z₁₂ z₂

∈H2,

we have

Gl^2ν= (l+ 2ν−2)!

_2ν

µ=1(µ/2)(l−1 + 2ν−µ/2)

D˜l^2ν. (3–3)

By (3–3), for F(Z) ∈ M_l^∞(Γ⁽²⁾₀ (M), φ) and g(z₁) ∈ S_l+ν(Γ⁽¹⁾₀ (M), φ), we have

Gl^2ν(F)(–, z₂), g=e_l,2ν,sD˜^νl,s(F)(–, z₂), g (3–4) if both sides are cusp forms, where

e_l,2ν,s= (l+ 2ν−2)!

_2ν

µ=1(µ/2)(l−1 + 2ν−s−µ/2). Now for even positive integersl, k such thatl≤k,set

E2,k(z₁, z₂;l, M, φ, χ, s) = (2π√

−1)^l−kG^k−l_l





x∈Z/NZ

¯ χ(x)

×E_2,l^∗ (–;M, φ¯χ, s)|kR(x/N) z₁ 0 0 z₂

.

Then, as is easily seen,E2,k(z₁, z₂;l, M, φ, χ, s) is a con- stant multiple of_2,k(z₁, z₂;l, M, φ, χ, s) as a function of z₁ andz₂,and therefore, it belongs toM_k^∞(Γ₀(M), φ)⊗ M_k^∞(Γ₀(M), φ).Furthermore, regarding the holomorphy and the cuspidality ofE2,k(z₁, z₂;l, M, φ, χ, s),by a care- ful examination of the behavior at cusps, we have Propo- sition 3.1.

Proposition 3.1. Let k and l be positive even inte- gers such that l ≤ k, and s = 0 or −1/2. Let φ and χ be Dirichlet characters modulo M and N, re- spectively, that satisfy the above conditions. Assume that the triple (l, s, φχ)¯ satisfies one of the Conditions (h–1), (h–2), or (h–3). Then E2,k(z₁, z₂;l, M, φ, χ, s) belongs to M_k(Γ₀(M), φ) ⊗ M_k(Γ₀(M), φ). Further- more, assume l < k, or k ≥ 4 and N > 1.

ThenE_2,k(z₁, z₂;l, M, φ, χ, s) belongs to S_k(Γ₀(M), φ)⊗ Sk(Γ₀(M), φ).

Remark 3.2.We remark that in the casek≥4 andN >

1, E2,k(z₁, z₂;k, M, φ, χ, s) belongs to Sk(Γ₀(M), φ)⊗ S_k(Γ₀(M), φ) even if χ = 1N. On the contrary, in the case k = 2 and χ = 1N, we easily see that it does not belong to S_k(Γ₀(M), φ)⊗S_k(Γ₀(M), φ) by observ- ing Fourier coefficients of E_2,2^∗ (Z;M, φ,−1/2) in Theo- rem 2.1. At present, we don’t know about the cuspidal- ity ofE2,2(z₁, z₂;l, M, φ, χ, s) for a generalχthat satisfies (h–2) and (h–3).

Now by (3–4), for any f ∈Sk(Γ₀(M), φ), we have f,E_2,k(z₁,−¯z₂;l, M, φ, χ, s)=

(2π√

−1)^l−kel,k−l,sf,2,k(z₁,−¯z₂;l, M, φ, χ, s) (3–5) if both sides are cusp forms. Furthermore, by (3–2) and [B¨ocherer and Schmidt 00, (6.11)], we have

E2,k(z₁, z₂;l, M, φ, χ, s) = ∞

m₁=−∞

∞ m₂=−∞

c_2,l

m₁ r/2 r/2 m₂

,

y₁ 0 0 y₂

, M, φχ, s¯

×G^k−l_l (m₁m₂, r)

×T(r,χ)e(m¯ ₁x₁)e(m₂x₂) where we writez₁=x₁+√

−1y1, z₂=x₁+√

−1y2,and T(r, φ) =

xmodN

φ(x)e(rx/N) for a Dirichlet characterφmoduloN.

From now on, let Γ₀(N) = Γ⁽¹⁾₀ (N). LetM and k be positive integers and φ a Dirichlet character moduloM

such thatφ(−1) = (−1)^k.Let f(z) =

∞ m=1

a(m)e(mz)

be a normalized cuspidal Hecke eigenform of weight k and Nebentypus φ with respect to Γ₀(M). Then, for a Dirichlet character χmoduloN,we define the standard zeta functionL(f, s, χ) twisted byχ as

L(f, s, χ) =

p

(1−χ(p)α_pβ_pp^−s−k+1)

×(1−χ(p)α²_pp^−s−k+1)

×(1−χ(p)β²_pp^−s−k+1)₋₁ , whereαp, βp are complex numbers such that

αp+βp=a(p), αpβp=φ(p)p^k−1 (3–6) for each prime number p. Then by [B¨ocherer and Schmidt 00, Theorem 3.1] and (3–5), we have the fol- lowing theorem:

Theorem 3.3. In addition to the notation and the as- sumptions as above, assume that M > 1, N² divides M, φ² = 1M, and χ(−1) = 1. Let f ∈ Sk(Γ₀(M), φ) be a common eigenfunction of all Hecke operators. Fur- thermore, assume one of the following conditions: (a) k = l, (b) s = 0, or (c) E_2,l^∗ (Z;M, φ¯χ, s) belongs to M_l(Γ⁽²⁾₀ (M), φ¯χ).Then we have

f,E2,k(–,−¯z;l, M, φ, χ,s))¯ Γ0(M)=κl,k(s)N^k+l+2s−2

×M^1−k/2L(f|WM, l+ 2s−1, χ)f|WM|T(M/N²)(z), where

κ_l,k(s) = (−1)^l/2 2^{−3+2k−l+2s}π^k−l−1

×Γ(k+s−1/2)Γ(k+s−1) Γ(l+s)Γ(l+s−1/2)

× Γ(k−1) _k−l

µ=1(µ/2)(k−1−s−µ/2), T(M/N²)is the Hecke operator, and

WM =

0 −1

M 0

.

Remark 3.4.We slightly change the notation in [B¨ocherer and Schmidt 00]. It is not certain that the assertion of [Katsurada 03, Theorem 3.1] holds in general. Thus we

impose some conditions here. This does not affect our main results. There is a minor misprint in [B¨ocherer and Schmidt 00, Theorem 3.1]. On page 1,339, line 9, “21+n(n+1)/2−2ns” should read “2¹⁻nl+n(n+3)/2−2ns,”

and this correction has been done in [Katsurada 03, The- orem 3.1].

Now, let φ be as in Theorem 3.3, and assume that the triple (l, s, φχ) satisfies one of the Condi-¯ tions (h–1), (h–2), or (h–3). Then we define a function E˜2,k(z₁, z₂;l, M, φ, χ, s) onH1×H1so that

E˜2,k(z₁, z₂;l, M, φ, χ, s) = ∞

m₁=0

∞ m₂=0

r²≤4m1m2

˜ c_2,l

m₁ r/2 r/2 m₂

, M, φχ, s¯

×G^k_l^−l(m₁m₂, r)T(r,χ)¯ e(m₁z₁)e(m₂z₂), where

˜ c_2,l

m₁ r/2 r/2 m₂

, M, φχ, s¯

is as defined in Theorem 2.1. Then, by Theorem 2.1 we have

E2,k(z₁, z₂;l, M, φ, χ, s) =A(l, s) ˜E2,k(z₁, z₂;l, M, φ, χ, s).

From now on, for a Dirichlet character ψmoduloM₀ we use the same symbol ψ to denote the character modulo M induced fromψifM₀dividesM.For a positive integer rlet

δ_r=

r 0 0 1

,

and let Sk(Γ₀(M), φ)^(r) = {f|δr;f ∈ Sk(Γ₀(M), φ)}, and let S_k(Γ₀(M), φ)^new be the space of new forms in Sk(Γ₀(M), φ). We note that Sk(Γ₀(M), φ)^new = Sk(Γ₀(M), φ) if φ is a primitive character of con- ductor M. Furthermore for a primitive form f in Sk(Γ₀(M), φ)^newletcf be the complex number such that f|W_M =c_ff^c.Letλ_f(m) be the eigenvalue of the Hecke operator T(m) for a positive integerm. For an odd pos- itive integerm≤k−1,let

Λ(f, m, χ) = Γ(k−1)Γ(k+m−1)Γ(m+ 1) Γ(k−m)

× L(f, m, χ) 2^2k+2m−4π^k+2mf, f, and

Λ(f,0, χ) = Γ(k−1) L(f,0, χ) 2^2k−3π^kf, f.

We note that m(Φ_Γ0(N))) = ^π₃[Γ : Γ₀(N)]. Thus by The- orem 3.3 we obtain the following two theorems:

Theorem 3.5. Let the notation and the assumptions be as before. Let p be a prime number such that p ≡ 1 mod 4, φ = (^p_∗), and χ a Dirichlet character modulo psuch thatχ(−1) = 1.

(1) Letf be a primitive form inSk(Γ₀(p²), φ)^new.Then, f,E˜2,k(–,−z;¯ l, p², φ, χ, s)=

3[Γ : Γ₀(p²)]⁻¹p^l+2sΛ(f^c, l+2s−1, χ)f, fcff^c(z).

(2) Letf be a primitive form inSk(Γ₀(p), φ).Then, we have

f,E˜2,k(–,−¯z;l, p², φ, χ, s)=

3[Γ : Γ₀(p²)]⁻¹p^l+2sΛ(f^c|δp, l+ 2s−1, χ)

× f|δp, f|δpcff^c|δp(z), and

f|δ_p,E˜2,k(–,−z;¯ l, p², φ, χ, s)=

3[Γ : Γ₀(p²)]⁻¹p^l+2sΛ(f^c, l+2s−1, χ)f, fc_ff^c(z).

Theorem 3.6. Let the notation and the assumptions be as before. Let p₀ = 1or a prime number such thatp₀≡ 1 mod 4, and φ = (^p_∗⁰). Furthermore, let p be a prime number different from p₀, and χ a Dirichlet character modulopsuch that χ(−1) = 1.

(1) Let f be a primitive form in S_k(Γ₀(p₀p²), φ)^new. Then, we have

f,E˜2,k(–,−z;¯ l, p₀p², φ, χ, s)=

3[Γ : Γ₀(p₀p²)]⁻¹p^1−k/2₀ p^l+2sΛ(f^c, l+ 2s−1, χ)

× f, fcfλf(p₀)f^c(z).

(2) Let f be a primitive form in Sk(Γ₀(p₀p), φ)^new. Then, we have

f,E˜2,k(–,−¯z;l, p₀p², φ, χ, s)=

3[Γ : Γ₀(p₀p²)]⁻¹p¹⁻₀ ^k/2p^l+2sΛ(f^c|δp, l+ 2s−1, χ)

× f|δ_p, f|δ_pc_fλ_f(p₀)f^c|δ_p(z), and

f|δp,E˜2,k(–,−¯z;l, p₀p², φ, χ, s)=

3[Γ : Γ₀(p₀p²)]⁻¹p¹⁻₀ ^k/2p^l+2sΛ(f^c, l+ 2s−1, χ)

× f, fc_fλ_f(p₀)f^c(z).