Calculation of zeta functions with Maass wave forms associated with (SL_{2} × SL_{2} × GL_{2}, M_{2} ⊕ M_{2})

Fumihiro Sato (Rikkyo University, Tokyo)

December 3, 1998

In [S2], I gave a general theory of zeta functions with automorphic forms associated with prehomogeneous vector spaces. Today I talk about an example of the general theory, namely, the zeta functions associated with the space mentioned in the title. In this talk, the emphsis will be laid on explicit description of the zeta functions.

First I explain the motivation of the calculation given in this talk. Recently considerable progress has been made in the explicit calculation of zeta functions associated with prehomogeneous vector spaces, mainly by Ibukiyama, Saito and Katsurada. Typical examples of their results are explicit formulas for the zeta functions associated with the spaces

(GL_{n}, Sym_{n}), (SO(k, k) × GL_{n}, M_{2k,n}).

Their results are quite explicit, however several questions arise naturally. For example,

Question 1. Why does the Mellin transform of the Cohen Eisenstein series appear as a non-trivial factor of the zeta functions?

2. Why does the convolution of the Dirichlet series appear in Example 2?

3. Can one predict the formulas theoretically?

In this talk I can not give any satisfactory answers to these questions. What I want to explain through a case study of the space (SO(2, 2) × GL_{2}, M_{4,2}) is the following:

1. Appropriate generalizations of this kind of calculations to zeta functions with automorphic forms will be very suggestive.

2. Geometric structure underlying the explicit form should be made clear.

I would like to add a further remark to the first point: Recall that the zeta function is of the form

∑_{x∈L∩V_{i}} μ(x) |P (x)|^{−s} , μ(x) = ∫_{G_{x}/Γ_{x}} dμ_{x}.

If we want to extend it to zeta functions with an automorphic form ϕ, then the coefficients μ(x) should be replaced by the period of ϕ, namely, by the integral of the form

μ(ϕ, x) = ∫_{G_{x}/Γ_{x}} ϕ(g_{x}^{−1}h)dμ_{x}(h),

where g_{x} is an element in G_{ℝ} that sends x_{0}, a base point fixed in advance for each real open orbit, to x. To obtain non-vanishing zeta functions, one has to assume that the non-vanishing of the period of ϕ. In the case of usual zeta functions, since ϕ is a constant function, the non-vanishing of the period is trivial; however, the period may vanish in general. It is believed that the non-vanishing of the period is related to the Langlands functoriality of automorphic forms. Therefore we can formulate the problem of the calculation of explicit formulas for zeta functions with automorphic forms as follows:

Problem : (Assuming that the non-vanishing of the period of ϕ implies that ϕ is a lifting (or a descent) of an automorphic form ψ on some grouop (possibly different from the original one), ) describe the zeta function attached to ϕ in terms of ψ.

In this talk I solve this problem for the space (SO(2, 2) × GL_{2}, M_{4,2}).

1 The space (Spin(2, 2)×GL_{2}, M_{4,2}) and the associated zeta functions

We identify Spin(2, 2), the twofold covering group of SO(2, 2) with SL_{2} × SL_{2} and consider the prehomogeneous vector space

G = SL_{2} × SL_{2} × GL_{2}, V = M_{2} ⊕ M_{2} ≅ M_{2,4}, g · v = (g_{1}, g_{2}, h) · (v_{1}, v_{2}) = (g_{1}v_{1}g_{2}^{−1}, g_{1}v_{2}g_{2}^{−1})^{t}h.

For x = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ M_{2}, put x^{∗} = ( \begin{matrix}d & −b \\ −c & a \end{matrix} ) and define a symmetric bilinear forms on M_{2} by

⟨x_{1}, x_{2}⟩ = tr x_{1}x_{2}^{∗}.

Let π : V → Sym_{2} be the mapping given by

π(x) = π(x_{1}, x_{2}) = ( \begin{matrix}⟨x_{1}, x_{1}⟩ & ⟨x_{1}, x_{2}⟩ \\ ⟨x_{2}, x_{1}⟩ & ⟨x_{1}, x_{2}⟩ \end{matrix} ) .

Then, the polynomial P (x) = P (x_{1}, x_{2}) = det π(x) is the basic relative invariant. Put Γ = SL_{2}(ℤ) and Γ_{G} = SL_{2}(ℤ) × SL_{2}(ℤ) × SL_{2}(ℤ).

Now we introduce automorphic forms on SL_{2}. Let ℌ = {z ∈ ℂ | ℑz > 0}. Then the group SL_{2}(ℝ) acts on ℌ by linear fractional transformation. We put Γ = SL_{2}(ℤ) as in §1. Let L^{2}(Γ\ℌ) be the space of measurable functions on Γ\ℌ square integrable with respect to the invariant measure \frac{dx dy}{y^{2}} . Put

𝔖_{0}^{+}(Γ\ℌ, λ) = \{ϕ ∈ L^{2}(Γ\ℌ) | \begin{matrix}Δ_{0}ϕ + λ(1 − λ)ϕ = 0, ϕ(−z̄) = ϕ(z) \\ ∫_{0}^{1} ϕ(x + iy) dx = 0 \end{matrix} \} ,

where Δ_{0} = y^{2} (\frac{∂^{2}}{∂x^{2}} + \frac{∂^{2}}{∂y^{2}} ), the non-Eucledean Laplacian. A function in 𝔖_{0}^{+}(Γ\ℌ, λ) is called an even Maass wave form (of weight 0).

For Φ_{1} ∈ 𝔖_{0}^{+}(Γ\ℌ, λ_{1}), Φ_{2} ∈ 𝔖_{0}^{+}(Γ\ℌ, λ_{2}) and f ∈ S(V (ℝ)), we define the zeta integral Z(f; Φ_{1}, Φ_{2}; s) by setting

Z(f; Φ_{1}, Φ_{2}; s) = ∫_{SL_{2}(ℝ)/SL_{2}(ℤ)} ϕ_{1}(g_{1}) dg_{1} ∫_{SL_{2}(ℝ)/SL_{2}(ℤ)} ϕ_{2}(g_{2}) dg_{2}

× ∫_{GL_{2}(ℝ)/GL_{2}(ℤ)} |det h|^{2s} ∑_{x∈V (ℤ)−S} f(ρ(g_{1}, g_{2}, h)x) dh,

where ϕ_{i}(g_{i}) = Φ_{i}(g_{i}^{−1} · \sqrt{−1}) (i = 1, 2).

Lemma A The zeta integral Z(f; Φ_{1}, Φ_{2}; s) is absolutely convergent for Re(s) > \frac{7}{4}.

By the general theory, the zeta integral decomposes into a product of zeta functions and local zeta functios:

Z(f; Φ_{1}, Φ_{2}; s) = ζ_{+}(Φ_{1}, Φ_{2}; s)ζ_{+}^{(∞)}(f; λ_{1}, λ_{2}; s) + ζ_{−}(Φ_{1}, Φ_{2}; s)ζ_{−}^{(∞)}(f; λ_{1}, λ_{2}; s).

Our aim is to give an explicit description of the local and global zeta functions.

2 Description of the local zeta functions

Let σ : V → V/({1} × {1} × SL_{2}) be the quotient mapping. The the quotient can be identified with the space

X = {(Y_{1}, Y_{2}) ∈ Sym_{2} × Sym_{2} | det Y_{1} = det Y_{2}}  (1)

and σ is then given by

σ(x) = (σ_{1}(x), σ_{2}(x)), σ_{1}(x) = Jx_{1}x_{2}^{∗} +^{t}(Jx_{1}x_{2}^{∗}), σ_{2}(x) = −(Jx_{1}^{∗}x_{2} +^{t}(Jx_{1}^{∗}x_{2})).

Moreover we have

P (x) = det σ_{1}(x) = det σ_{2}(x)

and G_{x}^{∘} ≅ SO(σ_{1}(x)) × SO(σ_{2}(x)).

We put

V_{±} = {x ∈ V_{ℝ} = M_{2}(ℝ) ⊕ M_{2}(ℝ) | sgn det π(x) = ±} .

For a Y ∈ Sym_{2}(ℝ) with nonzero determinat, we put

Ȳ = \{ \begin{matrix}− |det Y |^{−1/2} Y & if Y is negative definite, \\ |det Y |^{−1/2} Y & otherwise. \end{matrix}

Then, we have

ζ_{±}^{(∞)}(f; λ_{1}, λ_{2}; s) = ∫_{V_{±}} |P (x)|^{s−2} ω_{λ_{1}}^{±}(\overline{σ_{1}(x)})ω_{λ_{2}}^{±}(\overline{σ_{2}(x)})f(x) dx,

where

ω_{λ}^{+}(Ȳ) = P_{−λ}(cosh(log |α|)) P_{−λ} = the Legendre fucntion
ω_{λ}^{−}(Ȳ) = cosh(log |α|) × {P_{−1/2}^{λ−1/2}(tanh(log |α|)) + P_{−1/2}^{λ−1/2}(− tanh(log |α|))}
P_{−1/2}^{λ−1/2} = the associated Legendre fucntion

for Ȳ with eigenvalues α, ±α^{−1}.

3 Description of the global zeta functions

First we recall the Maass correspondence which associates a Maass form of weight 1/2 to a Maass cusp form with non-vanishing period.

3.1 The Maass correspondence

For γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{0}(4), we put

J(γ, z) = ϵ_{d}^{−1} (\frac{c}{d}) (\frac{cz + d}{|cz + d|})^{1/2} , ϵ_{d} = \{ \begin{matrix}1 & d ≡ 1 (mod 4) \\ \sqrt{−1} & d ≡ 3 (mod 4) \end{matrix}

where the Legendre symbol (\frac{c}{d}) has the same meaning as in [Shm].

Put

Δ_{1/2} = y^{2} (\frac{∂^{2}}{∂x^{2}} + \frac{∂^{2}}{∂y^{2}} ) − \frac{iy}{2} \frac{∂}{∂x}.

Let

𝔖_{1/2}^{+}(Γ_{0}(4)\ℌ, μ) = \{F : ℌ → ℂ | \begin{matrix}F (γ · z) = J(γ, z)F (z) (∀γ ∈ Γ_{0}(4)) \\ Δ_{1/2}F + μ(1 − μ)F = 0, LF = F \\ ∫_{0}^{1} F (x + iy) dx = 0, ∫_{Γ_{0}(4)\ℌ} |F (z)|^{2} \frac{dx dy}{y^{2}} < ∞ \end{matrix} \} ,

where

LF (z) = \frac{1}{4}e^{iπ/4} (\frac{z}{|z|})^{−1/2} ∑_{ν mod 4} F (\frac{−1 + 4νz}{16z} ) .

We call an F ∈ 𝔖_{1/2}^{+}(Γ_{0}(4)\ℌ, μ) a Maass cusp form of weight \frac{1}{2}. A Maass cusp form F in 𝔖_{1/2}^{+}(Γ_{0}(4)\ℌ, μ) has a Fourier expansion of the form

F (z) = ∑_{n≠0} ρ(n)W_{\frac{1}{4} sgn(n),μ− \frac{1}{2}} (4π |n| y)e[nx].

Then there exixts a linear mapping, the Maass correspondence,

Θ : 𝔖_{0}^{+}(Γ\ℌ, λ) ⟶ 𝔖_{1/2}^{+}(Γ_{0}(4)\ℌ, μ), μ = \frac{2λ + 1}{4}

such that the kernel of Θ is the subspace of Maass cusp forms with vanishing period with respect to SO(2) (see, e.g., [KS], Proposition 2.3).

For Φ ∈ 𝔖_{0}^{+}(Γ\ℌ, λ), let

Θ(Φ)(z) = ∑_{n≠0} ρ(Θ(Φ); n)W_{\frac{1}{4} sgn(n),μ− \frac{1}{2}} (4π |n| y)e[nx].

be the Fourier expansion of the corresponding half-integral weight form. Then, under a suitable normalization of the Haar measure dg on SL_{2}(ℝ), we have

ρ(Θ(Φ); n) = |n|^{−3/4} ∑_{{x∈Γ\Sym_{2}^{∗}(ℤ) \atop det x=n}} ∫_{SO(x)_{ℝ}/SO(x)_{ℤ}} Φ(g_{x}^{−1}h) dμ_{x}(h) × \{ \begin{matrix}1/2\sqrt{π} & (n > 0), \\ 1 & (n < 0), \end{matrix}  (2)

where g_{x} is an element in SL_{2}(ℝ) such that

x = \{ \begin{matrix}±\sqrt{n}g_{x} ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} )^{t}g_{x} & if x is definite, \\ \sqrt{|n|}g_{x} ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} )^{t}g_{x} & if x is indefinite \end{matrix}

(see [KS], [M]). In [KS], a correspondence of the inverse direction (an analogue of the Shimura correspondnence) is also given.

3.2 Main result

Theorem A We have

ζ_{ϵ}(Φ_{1}, Φ_{2}; s) = ζ(2s − 1) ∑_{n=1}^{∞} \frac{ρ(Θ(Φ_{1}); ϵn)ρ(Θ(Φ_{2}); ϵn)}{n^{s−3/2}} × \{ \begin{matrix}π & for ϵ = +, \\ \frac{Γ(1− \frac{λ_{1}}{2} )^{2}Γ(1− \frac{λ_{2}}{2} )^{2}}{2^{λ_{1}+λ_{2}−1}π} & for ϵ = −. \end{matrix}

This result is compatible with the result of [IK] for k = n = 2 and ϵ = +. The appearence of D_{2}^{∗}(s) in their formula is explained by the fact that the image of the constant function Φ ≡ 1 under the Maass correspondnence is the Cohen Eisenstein series whose Mellin transform is D_{2}^{∗}(s). The origin of the convolution is the defining equation det Y_{1} = det Y_{2} of X (see (1)).

Theorem B The functions ζ_{±}(Φ_{1}, Φ_{2}; s) have analytic continuations to meromorphic functions on ℂ and satisfy the following functional equation:

( \begin{matrix}ζ_{+} \\ ζ_{−} \end{matrix} ) (Φ_{1}, Φ_{2}; 2 − s) = Γ(λ_{1}, λ_{2}; s) ( \begin{matrix}ζ_{+} \\ ζ_{−} \end{matrix} ) (Φ_{1}, Φ_{2}; s) .

Here the gamma matrix Γ(λ_{1}, λ_{2}; s) is given by

Γ(λ_{1}, λ_{2}; s)

= 2^{−2}π^{−4s+2}Γ (s + \frac{λ_{1} + λ_{2}}{2} − 1) Γ (s + \frac{λ_{1} − λ_{2} − 1}{2}) Γ (s + \frac{λ_{2} − λ_{1} − 1}{2}) Γ (s − \frac{λ_{1} + λ_{2}}{2} )

× ( \begin{matrix}γ_{++}(λ_{1}, λ_{2}; s) & γ_{+−}(λ_{1}, λ_{2}; s) \\ γ_{−+}(λ_{1}, λ_{2}; s) & γ_{−−}(λ_{1}, λ_{2}; s) \end{matrix} ) ,

where

γ_{++}(λ_{1}, λ_{2}; s) = sin 2πs + sin πλ_{1} + sin πλ_{2}

γ_{+−}(λ_{1}, λ_{2}; s) = \frac{Γ(1 − λ_{1})Γ(1 − λ_{2})}{2π}

× \{− sin π (s − \frac{λ_{1} + λ_{2}}{2} ) + sin π (s + \frac{λ_{1} − λ_{2}}{2} ) + sin π (s + \frac{λ_{2} − λ_{1}}{2} ) − sin π (s + \frac{λ_{1} + λ_{2}}{2} )\}

γ_{−+}(λ_{1}, λ_{2}; s) = \frac{2π}{Γ(1 − λ_{1})Γ(1 − λ_{2})}

× \{sin π (s − \frac{λ_{1} + λ_{2}}{2} ) + sin π (s + \frac{λ_{1} − λ_{2}}{2} ) + sin π (s + \frac{λ_{2} − λ_{1}}{2} ) + sin π (s + \frac{λ_{1} + λ_{2}}{2} )\}

γ_{−−}(λ_{1}, λ_{2}; s) = − sin 2πs + sin πλ_{1} + sin πλ_{2}.

Remark. Theorems A and B can be extended to holomorphic modular forms.

3.3 Outline of the proof

Since the results in [S2] apply to the present case, Theorem B (except the explicit form of the gamma matrix) is a consequences of the general theory. As is explained in [S2, §5], the calculation of the gamma matrix is reduced to the case of (B_{2} × B_{2} × GL_{2}, M_{4,2}), which has already been treated in [S1].

The key to the proof of Theorem A is the following:

1. Let V_{ℤ}^{prim} be the set of all matrices x = (x_{1}, x_{2}) ∈ M_{2}(ℤ)⊕M_{2}(ℤ) primitive as matrices in M_{2,4}(ℤ) satisfying P (x) ≠ 0. Denote by ζ_{±}^{prim}(Φ_{1}, Φ_{2}; s) the subseries of the zeta functions obtained by restricting the summation to primitive matrices. Then we have

ζ_{±}(Φ_{1}, Φ_{2}; s) = ζ(2s)ζ(2s − 1)ζ_{±}^{prim}(Φ_{1}, Φ_{2}; s).

2. Let X_{ℤ}^{prim} be the set of all (Y_{1}, Y_{2}) ∈ X such that det Y_{i} ≠ 0, Y_{1} and Y_{2} are even-integral, and, for any positive integer n ≥ 2, at least one of \frac{1}{n}Y_{1} and \frac{1}{n}Y_{2} is not even-integral. Then σ(V_{ℤ}^{prim}) is contained in X_{ℤ}^{prim} and σ induces a bijection

Γ_{G}\V_{ℤ}^{prim} ⟷ (Γ × Γ)\X_{ℤ}^{prim}.

Combining these two observation with (2), we can prove Theorem A rather easily.

References

[IK] T. Ibukiyama and H. Katsurada, Koecher-Maass series for Siegel Eisenstein series, Preprint.

[IS] T. Ibukiyama and H. Saito, On zeta functions associated to symmetric matrices I: An explicit form of zeta functions, Amer. J. Math. 117(1995), 1097–1155.

[KS] S.Katok and P.Sarnak, Heegner points, cycles and Maass forms, Israel J. Math. 34(1993), 193–227.

[M] H. Maass, Über die räumliche Verteilung der Punkte in Gittern mit indefiniter Metrik. Math. Ann. 138(1959), 287–315.

[S1] F. Sato, On functional equations of zeta distributions, Adv. Studies in pure Math. 15(1989), 465– 508.

[S2] F. Sato, Zeta functions of prehomogeneous vector spaces with coefficients related to periods of automorphic forms, Proc. Ind. Acad. (K.G.Ramanathan memorial issue) 104(1994), 99–135.

[Shm] G.Shimura, On modular forms of half-integral weight, Ann. Math. 97(1973), 440–481.

Examples of explicit formulas for zeta functions

Notation

For δ = ±1, put

D_{n}^{∗}(s, δ) = (−1)^{n/4} ∑_{(−1)^{n/2}δd_{K}>0} 2(2π)^{−n/2} (\frac{n}{2} − 1)! |d_{K}|^{(n−1)/2} L (\frac{n}{2}, χ_{K})

×\frac{ζ (2s) ζ (2s − n + 1)}{L (2s − \frac{n}{2} + 1, χ_{K})} |d_{K}|^{−s} ,

where d_{K} runs through all the discriminant of quadratic fields K or ℚ ⊕ ℚ such that (−1)^{n/2}δd_{K} > 0. Expand D_{n}(s, δ) as

D_{n}^{∗}(s, δ) = ∑_{n=1}^{∞} H (\frac{n}{2}, d, δ) d^{−s}

and put

D_{n}(s, δ) = ∑_{n=1}^{∞} H (\frac{n}{2}, 4d, δ) d^{−s}.

Example 1 : (GL_{n}, Sym_{n}) ([IS])

Let L = Sym_{n} ∩ M(n; ℤ) and

L_{i} = {x ∈ L | sgn x = (i, n − i)} (1 ≤ i ≤ n).

We put

δ = (−1)^{n−i}, ϵ = (−1)^{(n−i)(n−i+1)/2}.

If n is odd, then

ζ_{i}(L; s) = \frac{|∏_{i=1}^{[n/2]} B_{2i}|}{2^{n−1} (\frac{n−1}{2})!} (2^{(n−1)/2}ζ (s − \frac{n − 1}{2}) ∏_{i=1}^{[n/2]} ζ (2s − (2i − 1))

+ ϵδ^{(n+1)/2}(−1)^{(n^{2}−1)/8}ζ (s) ∏_{i=1}^{[n/2]} ζ (2s − 2i)) ,

and, if n is even, then

= \frac{|∏_{i=1}^{n/2−1} B_{2i}|}{2^{n−1} (\frac{n−2}{2})!} ((−1)^{[n/4]}D_{n}(s, δ) ∏_{i=1}^{n/2−1} ζ (2s − 2i)

+ ϵ κ(n, δ)(−1)^{n(n+2)/8} \frac{2^{(n+2)/2} |B′_{n/2}|}{n} ∏_{i=1}^{n/2} ζ (2s − (2i − 1))) ,

where κ(n, δ) = 1 or 0 according as δ ≡ (−1)^{n/2} mod 4 or otherwise, and

B′_{n/2} = 2 (\frac{n}{2})!(2π)^{−n/2}ζ (\frac{n}{2}) × \{ \begin{matrix}(−1)^{1+n/4} & if 4|n, \\ 1 & otherwise. \end{matrix}

Example 2 : (SO(k, k) × GL_{n}, M_{2k,n}) ([IK])

Let SO(k, k) be the special orthogonal group of the matrix

S = ( \begin{matrix}0_{k} & 1_{k} \\ 1_{k} & 0_{k} \end{matrix} ) .

Consider the zeta function ζ_{k,n}(S, s) associated with

M_{2k,n}(ℤ) ∩ {x ∈ M_{2k,n}(ℝ) | ^{t}xSx > 0} .

Then, if n is odd,

ζ_{k,n}(S, s) = some constant
× {ζ (s) ζ (s − k + 1) ∏_{i=1}^{(n−1)/2}(ζ (2s − 2i) ζ (2s − 2k + 2i + 1))
+ (−1)^{(n^{2}−1)/8}ζ (s − \frac{n − 1}{2}) ζ (s − k − \frac{n + 1}{2}) ∏_{i=1}^{(n−1)/2}(ζ (2s − 2i + 1) ζ (2s − 2k + 2i))\} ,

and, if n is even,

= some constant
× {(−1)^{n+k)/2}(D_{n}^{∗} ⊗ D_{2k−n}^{∗})(s) ∏_{i=1}^{n/2−1}(ζ (2s − 2i) ζ (2s − 2k + 2i + 1))
+ δ_{4,n}(−1)^{n(n+2)/8} · \frac{|B_{n/2}B_{k−n/2}|}{(n/2)(k − n/2)} ∏_{i=1}^{n/2}(ζ (2s − 2i + 1) ζ (2s − 2k + 2i))\} ,

where δ_{4,n} = 1 or 0 according as n ≡ 0 mod 4 or otherwise, and the convolution D_{n}^{∗} ⊗ D_{2k−n}^{∗} of the Dirichlet series

D_{n}^{∗}(s) = D_{n}^{∗}(s, 1), D_{2k−n}^{∗}(s) = D_{2k−n}^{∗}(s, 1)

is given by

D_{n}^{∗} ⊗ D_{2k−n}^{∗}(s) = ζ (2s − k + 1) ∑_{d=1}^{∞} \frac{H (\frac{n}{2}, d, 1) H (\frac{2k−n}{2}, d, 1)}{d^{s}}

(for the notation, see Example 1).