Exact values of the standard zeta functions

Exact values of the standard zeta functions

Hidenori Katsurada

1 Introduction

The standard zeta function of a modular form is one of the most important subjects in number theory. In particular, in the elliptic modular case, its special values are related to many other areas. As for this, we refer to the paper by Doi, Hida, and Ishii [D-H-I]. Several people have considered algorithms to compute the special values and have computed the exact values (see Section 5.) In this note, we give some formulas which seem useful for a computation of the special values of the standard zeta function of a primitive form belonging to Γ0(N) twisted by a character.

2 Eisenstein series

For a positive integer M, we denote by Γ⁽ⁿ⁾₀ (M) (resp. Γ^′(n)₀ (M)) the sub- group ofSp_n(Z) consisting of matrices whose lower left block (resp. upper right block) is congruent to 0 modulo M. For a Dirichlet character modulo M, we denote by ˜ϕ (resp. ϕ˜^′) the character of Γ⁽ⁿ⁾₀ (M) (resp. Γ^′₀⁽ⁿ⁾(M)) defined by ϕ(γ) =˜ ϕ(detD) (resp. ˜ϕ^′(γ) = ϕ(detA)) for γ =

( A B

C D

)

. We denote by H_n Siegel’s upper half space of degreen. For a Dirichlet character ϕ modulo M, We denote by M_k(Γ⁽ⁿ⁾₀ (M),ϕ) (resp.˜ M_k^∞(Γ⁽ⁿ⁾₀ (M),ϕ)) the space of holomorphic˜ (resp. C^∞-) modular forms of weight k and character ϕ belonging to Γ⁽ⁿ⁾₀ (M), and byS_k(Γ⁽ⁿ⁾₀ (M),ϕ) the subspace of˜ M_k(Γ⁽ⁿ⁾₀ (M),ϕ) consisting of cusp forms.˜ Furthermore, for a subgroup Γ of Sp_n(Z) we denote by Γ_∞ the subgroup of Γ consisiting of matrices whose lower left block is 0. For a function f on H_n we write f^c(Z) = f(−Z¯). For two C^∞-modular forms f and g of weight k and character ϕ with respect to a congruence subgroup Γ of Sp_n(Z), we define the

Petersson scalar product < f, g >_Γ by

< f, g >_Γ=

∫

Γ\Hn

f(Z)¯g(Z) det(Im(Z))^kdv, and the normalized Petersson scalar product

< f, g >=m(Φ_Γ)⁻¹ < f, g >_Γ,

where ΦΓis the fundamental domain for Hn modulo Γ,andm(ΦΓ) is the volume of Φ_Γ with respect to the standard invariant measure on H_n. From now on, we simply write ϕ as ˜ϕ if there is no confusion. For a positive integerM we define an Eisensetin series E_n,l^′ (Z;M, ϕ, s) by

E_n,l^′ (Z;M, ϕ, s) = detIm(Z)^sL(l+2s, ϕ) ^∑

γ∈Γ^′₀⁽ⁿ⁾(M)_∞\Γ^′₀⁽ⁿ⁾(M)

ϕ˜^′(γ)j(γ, Z)^−l|j(γ, Z)|^−2s,

where j(γ, Z) = det(CZ+D) forγ =

( A B

C D

)

, and

L(s, ϕ) = L(s, ϕ)

[n/2]∏

i=1

L(2s−2i, ϕ²).

We then define E_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 =

( 0_n −1_n 1_n 0_n

)

. Let Hn(Z) denote the set of half-integral matrices of degree n over Z, and we denote by Hn(Z)_>0 (resp. Hn(Z)_≥0) the subset of Hn(Z) consisting of positive definite (resp. semi-positive definite) matrices.

Then it is well known that E_n,l^∗ (Z;M, ϕ, s) belongs to M_l^∞(Γ⁽ⁿ⁾₀ (M), ϕ), and has the following Fourier exapnsion:

E_n,l^∗ (X+iY;M, ϕ, s) = ^∑

A∈Hn(Z)

c_n,l,s(A;Y, M, ϕ)exp(2πitr(AX)).

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

E_n,l^∗ (Z;M, ϕ, s) = ^∑

A∈Hⁿ(Z)_≥0

cn,l(A;M, ϕ, s)exp(2πitr(AZ)).

From now on letn= 2,andM >1.Furthermore assume thatl≥4 ands= 0,or l = 2 ands=−1/2. Then E_2,l^∗ (Z;M, ϕ, s) belongs to M_l(Γ⁽²⁾₀ (M), ϕ) (cf. [Sh].)

Let χ be a Dirichlet character modulo L, and m_χ be its conductor. Further- more, let χ⁽⁰⁾ be the primitive character associated with χ. Let W(χ) be the Gauss sum defined by

W(χ) = ^∑

XmodL

χ(X)exp(2πiX/L).

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

( a₁₁ a₁₂/2

a₁₂/2 a₂₂

)

∈ H2(Z), put e = e_A = GCD(a₁₁, a₁₂, a₂₂). For an element A ∈ H2(Z) such that rank A = 1 and for each prime number p define the polynomial F_p(A, X) as

F_p(A, X) =

ord∑p(eA) i=1

(pX)ⁱ.

For an element A=

( a₁₁ a₁₂/2

a₁₂/2 a₂₂

)

∈ H2(Z)_>0 write −4 detA as −4 detA = dAf_A² with dA the fundamental discriminant of Q(√

−detA) and fA a positive integer. Furthermore, let χ_A = (^d_∗^A) be the Kronecker character corresponding toQ(√

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

=

ord∑p(eA) i=0

(p²X)ⁱ

ordp∑(fA)−i j=0

(p³X²)^j−χ_A(p)pX

ord∑p(eA) i=0

(p²X)ⁱ

ordp(f∑A)−i−1 j=0

(p³X²)^j. Let ϕ be a character such that ϕ² = 1. Let l ≥ 4 be an even integer. Then for A∈ H2(Z)_≥0 put

˜

c_2,l(A,0) = ˜c_2,l(A;M, ϕ,0)

=













∏

p|fAF_p(A, ϕ(p)p^l−3)B_{l−1,(ϕχA)}⁽⁰⁾

×^dlA−3/2_m^Wl−1^((ϕχA)(0)) (ϕχA)(0)

∏

p|M(1−p^1−l(ϕχ_A)⁽⁰⁾(p)) A >0

0 otherwise,

whereB_{l−1,(ϕχA)}⁽⁰⁾ isl−1-th generalized Bernoulli number associated with (ϕχA)⁽⁰⁾. Letl = 2.Then for A∈ H2(Z)_≥0 put

˜

c_2,2(A,−1/2) = ˜c_2,2(A;M, ϕ,−1/2)

=









∏

p|fAF_p(A, ϕ(p)p⁻¹)L(0, ϕχ_A) A >0 L(−1, ϕ²)^∏_p|eAF_p(A, ϕ(p)p⁻¹) rank A= 1 1/2L(−1, ϕ²)L(−1, ϕ) A=O.

We define ˜E_2,l^∗ (Z;M, ϕ, s) by E˜_2,l^∗ (Z;M, ϕ, s) = ^∑

A∈H2(Z)_≥0

˜

c_2,l(A;M, ϕ, s)exp(2πitr(AZ)).

We note that this coincides with E_2,l^∗ (Z;M, ϕ, s) up to constant multiple (e.g.

[Ka1], [Sh].)

3 Pull-back formula

From now on put Γ₀(N) = Γ⁽¹⁾₀ (N), and in particular put Γ = Γ⁽¹⁾. Further- more, we simply write f|kγ as f|γ for γ ∈ GL₂(R)⁺ and f ∈ C^∞(H₁). For an even positive integer l and non-negative integer ν we define a polynomial G^2ν_l (u, v) in u, v by

G^2ν_l (u, v) =

∑ν µ=0

(−1)^µ (l+ 2ν−µ−2)!

(l−2)!(2ν−2µ)!µ!u^µv^2ν−2µ.

This was first defined by Zagier (cf. [Z], [I].) For positive even integers l < k,we define a function ˜E(z1, z2) = ˜E2,k(z1, z2;l, M, ϕ, s) on H1×H1 by

E˜2,k(z₁, z₂;l, M, ϕ, χ, s)

=

∑∞ m1=1

∑∞ m2=1

∑

r²≤4m1m2

˜ c_2,l(

( m₁ r/2

r/2 m2

)

, M, ϕχ, s)G^k−l_l (m₁m₂, r)χ(r)W(χ)q^m₁¹q₂^m2, where q₁ = exp(2πiz₁), q₂ = exp(2πiz₂). We note that ˜E2,k(z₁, z₂;l, M, ϕ, χ, s) belongs to S_k(Γ₀(M), ϕ)⊗S_k(Γ₀(M), ϕ). In addition to the above notation, let N ≥1 be a positive integr, andχa primitive Dirichlet character moduloN such that χ(−1) = ϕ(−1). For a Hecke eigenform f(z) = ^∑∞_m=1a(m)exp(2πimz) ∈ S_k(Γ₀(M), ϕ), we define the standard zeta function L^st(f, s, χ) twisted by χ as

L^st(f, s, χ)

=^∏

p

{(1−χ(p)α_pβ_pp^−s−k+1)(1−χ(p)α²_pp^−s−k+1)(1−χ(p)β_p²p^−s−k+1)}⁻¹,

whereα_p, β_pare complex numbers such thatα_p+β_p =a(p) andα_pβ_p =ϕ(p)p^k−1. For an odd integer m≥4 put

Λ(f, m, χ) = Γ(k−1)Γ(k+m−1) Γ(k−m)

L^st(f, m,χ)¯

2^2k+2m−2π^k+2m < f, f >, and

Λ(f,0, χ) = Γ(k−1) L^st(f,0,χ)¯ 2^2kπ^k < f, f >.

We note thatm(Φ_Γ0(N))) = π/3[Γ : Γ₀(N)].Then by [BS, Theorem 3.1], we have Theorem 3.1. In addition to the notation and the assumption as above, assume that M > 1, N² divides M, and that ϕ² =χ² = 1. Let f ∈S_k(Γ₀(M), ϕ) be a Hecke eigenform.

(1) Let l ≥4 be a positive even integer. Then

< f,E˜2,k(∗,−z;¯ l, M, ϕ, χ,0))>

= 3[Γ : Γ₀(M)]⁻¹N^k+l−2M^1−k/2Λ(f|W_M, l−1,χ)¯ < f|W_M, f|W_M > f|W_M|T(M/N²)(z), where W_M =

( 0 −1

M 0

)

, and T(M/N²) is the Hecke operator corresponding to M/N².

(2)

< f,E˜2,k(∗,−z; 2, M, ϕ, χ,¯ −1/2))>

= 3[Γ : Γ₀(M)]⁻¹N^k−1M^1−k/2Λ(f|W_M,0,χ)¯ < f|W_M, f|W_M > f|W_M|T(M/N²)(z).

For a positive integerr put δ_r =

( r 0 0 1

)

,and let S_k(Γ₀(M), ϕ)^(r) ={f|δ_r = r^k/2f(rz);f ∈ S_k(Γ₀(M), ϕ)}, and S_k(Γ₀(M), ϕ)^new the space of new forms in S_k(Γ₀(M), ϕ). Furthermore for a primitive form in S_k(Γ₀(M), ϕ) let c_f be the complex number such that f|W_M = c_ff^c. Let λ_f(m) be the eigenvalue of the Hecke operatorT(m) for a positive integer m. Then by Theorem 3.1 we have

Theorem 3.2. Under the above notation and the assumption, let l ≥ 4 and s= 0, or l = 2 and s=−1/2.

(1) Let ϕ be a primitive character mod p^r with r ≥ 1. Let f be a primitive form in S_k(Γ₀(p^r), ϕ)^new. Then

< f,E˜2,k(∗,−z;¯ l, p^r, ϕ,1, s))>

= 3[Γ : Γ0(p^r)]⁻¹p^(1−k/2)rΛ(f^c, l+ 2s−1,χ)¯ < f, f >λ¯f(p^r)cff^c(z).

(2) Letϕbe a primitive character modulop,andf a primitive form inS_k(Γ₀(p), ϕ)^new. Then for a primitive character χ modulo p we have

< f,E˜2,k(∗,−z;¯ l, p², ϕ, χ, s))>

= 3[Γ : Γ₀(p²)]⁻¹p^l+2sχ(−1)Λ(f^c|δ_p, l+ 2s−1,χ)¯ < f|δ_p, f|δ_p > c_ff^c|δ_p(z), and

< f|δp,E˜2,k(∗,−z;¯ l, p², ϕ, χ, s))>

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

(3) Let p₀ = 1 or a prime number different from p. Let ϕ be a primitive char- acter modulo p₀, and f a primitive form in S_k(Γ₀(p₀), ϕ)^new. Then for primitive character χ modulo p, we have

< f,E˜2,k(∗,−z;¯ l, p²p₀, ϕ, χ, s))>

= 3[Γ : Γ₀(p²p₀)]⁻¹p^l+2sp¹₀^−k/2Λ(f^c|δ_p², l+2s−1,χ)¯ < f|δ_p², f|δ_p² > c_fλ¯_f(p₀)f^c|δ_p²(z),

< f|δp,E2,k(∗,−z;¯ l, p²p0, ϕ, χ, s))>

= 3[Γ : Γ₀(p²p₀)]⁻¹p^l+2sp¹₀^−k/2Λ(f^c|δ_p, l+2s−1,χ)¯ < f|δ_p, f|δ_p > c_f¯λ_f(p₀)f^c|δ_p(z), and

< f|δ_p²,E˜2,k(∗,−z;¯ l, p²p₀, ϕ, χ, s))>

= 3[Γ : Γ₀(p²p₀)]⁻¹p^l+2sp¹₀^−k/2Λ(f^c, l+ 2s−1,χ)¯ < f, f > c_fλ¯_f(p₀)f^c(z).

4 Computation of L

(f, l, χ)

Letpbe prime a number such thatp≡1 mod 4.Letp₀ = 1 or a prime number such thatp₀ ̸=pand p₀ ≡1 mod 4.In this section we give formulas to compute L^st(f, m, χ) for a primitive form inf ∈S_k(Γ₀(N), ψ) in the following three cases:

(1) M =N =p, ψ is the quadratic character modulo p, and χ is trivial.

(2) M =p², N =p and χ=ψ is the quadratic character modulo p.

(3) M =p₀p², N =p₀ and χ is the quadratic character modulo p.

Letk ≥4 be a positive integer. As before letlbe an integer such that 4≤l < k and s= 0, orl = 2 ands=−1/2. For two positive integers m₁, m₂ put

ϵ(m₁, m₂;l, s) = ϵ(m₁, m₂;l, M, ϕ, χ, s)

= ^∑

r²≤4m1m2

˜ c_2,l(

( m₁ r/2

r/2 m₂

)

, M, ϕχ, s)G^k_l^−l(m₁m₂, r)χ(r)W(χ).

Take a basis {f_i}^di=1¹ ofS_k(Γ₀(N), ψ) consisiting of primitive forms. Let f_i|W_p = c_if_i^c with constant c_i, and write

f_i(z) =

∑∞ m=1

a_i(m)exp(2πimz).

Then by Theorem 3.2 we have the following. As for the proof, see [Kat 2].

Theorem 4.1. (1) In case (1), for any positive integer m₂ we have

ϵ(1, m₂;l, s) =

d1

∑

i=1

3(p+ 1)⁻¹p^k/2+l+2s−1Λ(f_i, l+ 2s−1,1)¯c_ia_i(p)a_i(m₂).

(2) In case (2), for any positive integer m₂ we have

ϵ(p, m₂;l, s) =

d1

∑

i=1

3p^−k/2(p+ 1)⁻¹Λ(f_i, l+ 2s−1,ψ)¯¯ a_i(p)a_i(m₂).

(3) In case (3), for any positive integer m₂ prime to pp₀ we have ϵ(p, p²m₂;l, s)−p^k−2ϵ(p, m₂;l, s)

= 3(p−1)p^k+l+2s−3t⁻_p0¹p¹₀^−k/2

d1

∑

i=1

Λ(fi, l+ 2s−1,χ)c¯ iai(p0)ai(p)ai(m2), where tp0 =p0+ 1 or 1 according as p0 is prime number or 1.

5 Numerical examples and comments

For a Hecke eigenform f ∈S_k(Γ₀(M), ψ),

L˜^st(f, s,1) = L^st(f, s,1)

∏

p|M(1−¯a(p)²p^−s−k+1), and

L˜^st(f, s, ψ) = L^st(f, s, ψ)

∏

p|M(1−p^−s). Put for an odd positive integer l

Λ(f, l, χ) =˜ Γ(k−1)Γ(k+l−1) Γ(k−l)

L˜^st(f, l, χ)

2^2k+2l−2π^k+2l < f, f >,

and

Λ(f,˜ 0, χ) = Γ(k−1) L˜^st(f,0, χ) 2^2k−1π^k< f, f >

for χ= 1 or ψ.Put

R(f, s, χ) =M^3(s+k−1)/2π^{−3/2(s+k−1)}Γ((s+k−1)/2)Γ((s+k)/2)Γ((s+ϵ)/2) ˜L^st(f, s, χ), where ϵ= 1 or 0 according as χ(−1) = 1 or −1.Similarly put

R(f, s, χ) =˜ M^(s+k−1)/2π^{−3/2(s+k−1)}Γ((s+k−1)/2)Γ((s+k)/2)Γ((s+ϵ)/2) ˜L^st(f, s, χ) for χ = 1 or ψ. Then by Asai [A] and Mizumoto [Mi], we have the following functional equation:

Proposition 5.1. In addition to the above notation and the assupmtion, suppose that χ is a quadartic character. Then we have

R(1−s, f, χ) =R(s, f, χ) Furthermore, for χ= 1 or ψ we have

R(1˜ −s, f, χ) = ˜R(s, f, χ).

By Theorem 4.1 combined with Proposition 5.1 we can compute the values Λ(f, m, χ) and ˜Λ(f, m, χ). Now we show some examples.

(1) Let M = 13, k= 8, ϕ= (¹³).Then dimS₈(Γ₀(13), ϕ) = 6.Take a primitive formf ∈S₈(Γ₀(13), ϕ). LetQ_f and Q⁺_f be the Hecke field of f and its maximal real subfield, respectively. Then

[Q_f :Q] = 6,[Q⁺_f :Q] = 3.

We have

|N_Q⁺

f/Q( ˜Λ(f,1,1))|= 13⁷×4357

2² ×3³×41×1429×25104281. This coincides with the result in [D-H-I]. Furtheremore,

|N_Q⁺

f/Q( ˜Λ(f,3,1))|= 2^∗×3^∗×13^∗×5 41×1429×25104281,

and

|N_Q⁺

f/Q( ˜Λ(f,3, ϕ))|= 2^∗×3^∗×13^∗×4583×10079 41×61×25104281 .

We note that the exact value ˜Λ(f, l,1) can also be obtained by the method in Zagier [Z].

(2) Let k = 24. Then dimS₂₄(SL₂(Z)) = 2, and the Hecke field Q_f of f is Q(√

144169). Let χ= (⁵). Then for a primitive formf ∈S₂₄(SL₂(Z)) we have N_Qf/Q(Λ(f,1, χ)) = 2²⁵×3⁹ ×7⁴×11⁴×13×17×19×109×54449

5³⁵×144169 .

This coincide with the result in Hiraoka [H] up to 2,3 and 5-factors. Similar results have been obtained by Goto [G] and Stopple [Sto].

Remark 1 Sturm [Str] gives a general algorithm to compute Λ(f, l, χ) for an eigenform f ∈ S_k(Γ₀(N), ϕ) and a Dirichlet character χ. However, even in the case N = 1 and χ is a quadratic character modulo a prime numberp,one needs the trace formula forS_k(Γ₀(p²)) to compute Λ(f, l, χ) by using his method. Thus it seems difficult to carry out his method directly for the computation of such a value.

Remark 2 By using the pull-back formula of Eisenstein series of degree 4, Ibukiyama’s differential operators in [I], and an explicit formula for the Siegel series in [Kat 1], we can obtain exact values of standard zeta function of cuspidal Hecke eigenform of degree 2. As for this, see [Kat 3].

MURORAN INSTITUTE OF TECHNOLOGY, 27-1 MIZUMOTO, MURO- RAN, 050-8585, JAPAN

Electronic mail: [email protected] References

[A] T. Asai, On the Fourier coefficients of automorphic forms at various cusps and some applications to Rankin’s convolution, J. Math. Soc. Japan, 28(1976), 48-61.

[B-S] S. B¨ocherer and C.G. Schmidt, p-adic measures attached to Siegel modular forms, Ann. Inst. Fourier, 50(2000), 1375-1443.

[D-H-I] K. Doi, H. Hida, and H. Ishii, Discriminant of Hecke fields and twisted adjoint L-values for GL(2), Invent. Math. 134(1998), 547-577.

[G] K. Goto, A twisted adjoint L-values of an elliptic modular form, J. Number Theory, 73(1998), 34-46

[H] Y. Hiraoka, Numerical calculation of twisted adjoint L-values attached to modular forms, Experimental Math. 9(2000), 67-73

[I] T. Ibukiyama, On Differential operators on automorphic forms and invariant pluri-harmonic polynomials, Comm. Math. Univ. St. Pauli, 48(1999), 103-118.

[Ka1] H. Katsurada, An explicit formula for Siegel series, Amer. J. Math.

121(1999), 415-452.

[Kat2] H. Katsurada, Special values of the twisted standard zeta function of an elliptic modular form, Preprint(2001)

[Kat3] H. Katsurada, Exact values of the standard zeta function of a Siegel modular form of degree two, Preprint(2001)

[Mi] S. Mizumoto, On the second L-functions attached to Hilbert modular forms, Math. Ann. 269(1984) 191-216.

[Sh] G. Shimura, Arithmeticity in the theory of automorphic forms, Mathemat- ical Surveys and Monographs 82(2000), Amer. Math. Soc.

[Sto] J. Stopple, Special values of twisted symmetric square L-functions and the trace formula, Compositio Math. 104(1996), 65-76.

[Str] J. Sturm, The critical values of zeta functions associated to the symplectic groups, Duke Math. J. 48(1981), 327-350.

[Z] D. Zagier, Modular forms whose Fourier coefficients involve zeta functions of quadratic fields, Lect. Notes in Math. 627(1977), 105-169.