Abstract
Siegel defined zeta functions associated with indefinite quadratic forms, and proved their analytic properties such as analytic continuations and functional equations. Coefficients of these zeta functions are called measures of representations, and play an important role in the arithmetic theory of quadratic forms. In a 1938 paper, Siegel made a comment to the effect that the modularity of his zeta functions would be proved with the help of a suitable converse theorem. In the present paper, we accomplish Siegel’s original plan by using a Weil-type converse theorem for Maass forms, which has appeared recently. It is also shown that “half” of Siegel’s zeta functions correspond to holomorphic modular forms.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
In 1903, Epstein [3] defined the zeta function \(\zeta _0(s)\) associated with a positive definite symmetric matrix S of degree m by
and studied their analytic properties such as analytic continuations and functional equations. (For a modern treatment of Epstein’s zeta functions, we refer to Terras [31, §1.4.2].) In a 1938 paper [23], Siegel defined and investigated the zeta functions associated with quadratic forms of signature \((1,m-1)\), and in a 1939 paper [24], those for general indefinite quadratic forms. Although Siegel’s calculations were rather involved, Siegel’s results are now well understood in the framework of the theory of prehomogeneous vector spaces. Let Y be a non-degenerate half-integral symmetric matrix of degree m with p positive eigenvalues and \(m-p\) negative eigenvalues (\(0<p<m\)). Let SO(Y) be the special orthogonal group of Y and denote by \(SO(Y)_{\mathbb {Z}}\) its arithmetic subgroup. We put \(V_{\pm }=\{v\in \mathbb {R}^m\, ; \textrm{sgn}Y[v]=\pm \}\). Then Siegel’s zeta functions are Dirichlet series associated with the prehomogeneous vector space \((GL_1(\mathbb {C})\times SO(Y), \, \mathbb {C}^m)\), and are defined by
where the sum runs over a complete set of representatives of \(SO(Y)_{\mathbb {Z}}\backslash (\mathbb {Z}^m \cap V_{\pm })\), and \(\mu (v)\) is a certain volume of the fundamental domain related to the isotropy subgroup \(SO(Y)_v\) of SO(Y) at v. In the positive definite case, the modularity of Epstein’s zeta function \(\zeta _0(s)\) is almost obvious; \(\zeta _0(s)\) is obtained by taking the Mellin transform of (the restriction to the imaginary axis of) the theta series
which is a modular form for a subgroup of \(SL_2(\mathbb {Z})\). (cf. Miyake [15, §4.9], Terras [31, §3.4.4].) On the contrary, it is not clear from the definition whether or not Siegel’s zeta function arises as an integral transform of some infinite series with modular properties. Rather, in the preface to a 1938 paper [23], Siegel wrote that such theta series would be constucted from his zeta functions, citing the work of Hecke [7], in which Hecke derived the transformation formula for the theta series associated with indefinite binary quadratic forms from the functional equation of zeta functions of real quadratic fields. Furthermore, Siegel made the following remark in the last section of [23]:
Will man die Transformationstheorie von \(f(\mathfrak {S}, x)\) für beliebige Modulsubstitutionen entwickeln, so hat man außer \(\zeta _1(\mathfrak {S}, s)\) auch analog gebildete Zetafunktionen mit Restklassen-Chrakteren zu untersuchen. Die zum Beweise der Sätze 1, 2,3 führenden Überlegungen lassen sich ohne wesentiche Schwierigkeit auf den allgemeinen Fall übertragen. Vermöge der Mellinschen Transformation erhält man dann das wichtige Resultat, daß die durch (53) definierte Funktion \(f(\mathfrak {S}, x)\) eine Modulform der Dimension \(\frac{n}{2}\) und der Stufe 2D ist; dabei wird vorausgesetzt, daß n ungerade und \(\mathfrak {x}'\mathfrak {S}\mathfrak {x}\) keine ternäre Nullform ist.
(A translation) If one wants to develop the transformation theory of \(f(\mathfrak {S},x)\) for arbitrary modular substitutions, then in addition to \(\zeta _1(\mathfrak {S},s)\) one also has to investigate zeta functions formed analogously with residual class characters. The considerations leading to the proof of Theorems 1, 2, 3 can be transferred to the general case without any major difficulty. By virtue of the (inverse) Mellin transformation, one then obtains an important result that the function \(f(\mathfrak {S}, x)\) defined by (53) is a modular form of weight \(\frac{n}{2}\) and level 2D, provided that n is odd and \(\mathfrak {x}'\mathfrak {S}\mathfrak {x}\) is not a ternary zero form.
As of 1938, Siegel seemed to have noticed the possibility that by considering the twists of zeta functions by Dirichlet characters, one can prove modularity for congruence subgroups. In the holomorphic case, this fact is known as Weil’s converse theorem [34]. It was 1967 when Weil’s paper [34] appeared! Revisiting Siegel’s prediction in the light of recent developments is one motivation for the present study.
We should note that in the quotation above, Siegel mentioned the parity of n, the number of variables of quadratic forms. This is related to the fact that the concept of non-holomorphic modular forms was not yet in place at that time. In a celebrated paper [12], Maaß introduced the notion of the so-called Maass forms and established a Hecke correspondence for Maass forms. Further, in [13], as its application of his theorem, Maaß proved that in a very special case (when Y is diagonal of even degree with \(\det Y=1\)), Siegel’s zeta functions can be expressed as the product of two standard Dirichlet series such as the Riemann zeta function \(\zeta (s)\) and the Dirichlet L-function \(L(s, \chi )\). On the other hand, it is only recently that papers on Weil-type converse theorems for Maass forms have emerged (cf. [16, 17]). It would be a very natural idea for us to accomplish Siegel’s original plan to prove the modularity via converse theorem including the case of non-holomorphic forms.
Siegel’s zeta functions are closely related to the so-called Siegel’s main theorem (Siegelsche HauptSatz). In a 1951 paper [26], Siegel proved the transformation formula for some theta series arising from indefinite quadratic forms, and the equality between an integral of the indefinite theta series over fundamental domains and some Eisenstein series (cf. [26, Satz 1]). It was shown in [26, Hilfssatz 4] that the coefficients
of \(\zeta _{\pm }(s)\) coincide with the Fourier coefficients of the non-holomorphic modular forms appearing in Siegel’s formula. Here we ignore the differences in the definitions of \(\mu (v)\); the definitions of measures are different for each of the papers [23, 24, 26]. Siegel called M(Y; n) the measures of representations (Darstellungsmaß). The measure M(Y; n) of representations is an analogue of the representation number
for a positive symmetric matrix S, and Siegel’s formula can be reformulated as an arithmetic identity that M(Y, n) is equal to the product of local representation densities over all primes. Weil [33] generalized Siegel’s result by using the language of adeles, and it is the Siegel-Weil formula—a cornerstone in the modern number theory.
Now we explain the main results of the present paper. First, along the Sato-Shintani theory [21] of prehomogeneous vector spaces, we define Siegel’s zeta functions and prove their analytic properties. Here, to treat twisted zeta functions as well as the original Siegel’s zeta functions, we first consider Siegel’s zeta functions with congruence conditions, which are defined using Schwartz-Bruhat functions on \(\mathbb {Q}^m\). This idea is due to Sato [20]. Then the converse theorem in [16] is applied to the zeta functions, and the following result is obtained:
Mainresult 1
(Theorem 2) Let \(m\ge 5\). Assume that at least one of m or p is an odd integer. Take an integer \(\ell \) with \(\ell \equiv 2p-m \pmod {4}\), and put \(D=\det (2Y)\). Let N be the level of 2Y. Define \(C^{\infty }\)-function \(F(z)= F(x+iy)\) on the Poincaré upper half-plane \(\mathcal {H}\) by
where \(d^{1}g\) is a suitably normalized Haar measure on \(SO(Y)_{\mathbb {R}}\), \(\alpha (0)\) is some constant determined by the residues of \(\zeta _{\pm }(s)\), and \(W_{\mu , \nu }(y)\) denotes the Whittaker function. Then, F(z) is a Maass form of weight \(\ell /2\) with respect to the congruence subgroup \(\Gamma _0(N)\).
The above formula can be compared with Siegel’s calculation [26, Hilfssatz 4] of the Fourier expansions of non-holomorphic modular forms. Our F(z) is essentially the same as the modular form given by Siegel [26]. See Remark 4. The theorem above excludes the case where both m and p are even. Our second result states that if one of \(m-p\) and p are even, we can construct holomorphic modular forms from \(M(Y; \pm n)\).
Mainresult 2
(Theorem 3) Let \(m\ge 5\). Assume that \(m-p\) is even. We define a holomorphic function F(z) on \(\mathcal {H}\) by
Then, F(z) is a holomorphic modular form of weight m/2 with respect to \(\Gamma _0(N)\). (In the case that p is even, we can construct holomorphic modular forms from \(M(Y; -n)\; (n=1,2,3,\dots )\).)
The theorem above is consistent with a result of Siegel that was published in a 1948 paper [25]. In this paper, Siegel calculated the action of certain differential operators on indefinite theta series, and proved that in the case of \(\det Y>0\), we can construct holomorphic modular forms from indefinite theta series associated with Y.
Before closing Introduction, we give some remarks on related researches, future problems, and possible applications. Special values of Siegel’s zeta functions associated with Y of signature \((1, m-1)\) appear in the dimension formula for automorphic forms on orthogonal groups of signature (2, m) (cf. Ibukiyama [8]), and it is important to investigate their arithmetic aspects. Ibukiyama [9] proved an explicit formula expressing Siegel zeta functions (with m even) as linear combinations of products of two shifted Dirichlet L-functions and certain elementary factors. His proof is given by direct calculations using Siegel’s main theorem in [26] and not by converse theorems. Ibukiyama’s explicit formula is quite general and includes the above-mentioned result of Maaß [13]. We also mention the work [6] of Hafner-Walling, in which they carried out extensive calculations to make Siegel’s formula more explicit in terms of standard Eisenstein series. This work is also restricted to the case where m is even. It is worthwhile to investigate the case where m is odd. Finally, in a good situation, the method of converse theorems can be used to prove lifting theorems. In [29], a Shintani-Katok-Sarnak type correspondence is derived from analytic properties of a certain prehomogeneous zeta function whose coefficients involve periods of Maass cusp forms. In [14], Maaß studied a generalization of Siegel’s zeta functions, which can be regarded as prehomogeneous zeta functions whose coefficients involve periods of automorphic forms on orthogonal groups. It is quite probable that our method can be applied to these zeta functions, and some lifting theorems will be obtained. We hope to discuss this topic elsewhere.
The present paper is organized as follows. In Sect. 1, we recall a Weil-type converse theorem for Maass forms, and in Sect. 2, we define our prehomogeneous vector spaces and give the local functional equations. Section 3 is devoted to define Siegel’s zeta functions with congruence conditions, and analytic properties of Siegel’s zeta functions are proved in Sect. 4. We prove our main theorems in Sects. 5 and 6.
Notation. We denote by \(\mathbb {Z}, \mathbb {Q},\mathbb {R}\), and \(\mathbb {C}\) the ring of integers, the field of rational numbers, the field of real numbers, and the field of complex numbers, respectively. The set of non-zero real numbers and the set of positive real numbers are denoted by \(\mathbb {R}^{\times }\) and \(\mathbb {R}_{+}\), respectively. The set of positive integers and the set of non-negative integers are denoted by \(\mathbb {Z}_{>0}\) and \(\mathbb {Z}_{\ge 0}\), respectively. The real part and the imaginary part of a complex number s are denoted by \(\Re (s)\) and \(\Im (s)\), respectively. For complex numbers \(\alpha , z\) with \(\alpha \ne 0\), \(\alpha ^z\) always stands for the principal value, namely, \(\alpha ^z = \exp ((\log |\alpha |+i\,\textrm{arg}\, \alpha )z)\) with \(-\pi <\textrm{arg}\, \alpha \le \pi \). We use \(\textbf{e}[x]\) to denote \(\exp (2\pi i x)\). The quadratic residue symbol \(\left( \frac{*}{*}\right) \) has the same meaning as in Shimura [22, p. 442]. For a meromorphic function f(s) with a pole at \(s=\alpha \), we denote its residue at \(s=\alpha \) by \(\displaystyle \mathop {\textrm{Res}}_{s=\alpha } f(s)\).
2 A Weil-type converse theorem for Maass forms
In this section, we define Maass forms on the Poincaré upper half-plane \(\mathcal {H}=\{z\in \mathbb {C}\, | \, \Im (z)>0\}\) of integral and half-integral weight, and recall a Weil-type converse theorem for Maass forms that is proved in [16]. We refer to Cohen-Strömberg [2] for an overview of the theory of Maass forms. Let \(\Gamma =SL_{2}(\mathbb {Z})\) be the modular group, and for a positive integer N, we denote by \(\Gamma _0(N)\) the congruence subgroup defined by
As usual, \(\Gamma \) acts on \(\mathcal {H}\) by the linear fractional transformation
We put \(j(\gamma , z)=cz+d\), and define \(\theta (z)\) and \(J(\gamma , z)\) by
Then it is well-known that
where
For an integer \(\ell \), the hyperbolic Laplacian \(\Delta _{\ell /2}\) of weight \(\ell /2\) on \(\mathcal {H}\) is defined by
Let \(\chi \) be a Dirichlet character mod N. Then we use the same symbol \(\chi \) to denote the character of \({\Gamma }_0(N)\) defined by
Definition 1
(Maass forms) Let \(\ell \in \mathbb {Z}\), and N be a positive integer, with 4|N when \(\ell \) is odd. A complex-valued \(C^{\infty }\)-function F(z) on \(\mathcal {H}\) is called a Maass form for \({\Gamma }_0(N)\) of weight \(\ell /2\) with character \(\chi \), if the following three conditions are satisfied;
-
(i)
for every \(\gamma \in \Gamma _0(N)\),
$$\begin{aligned} F(\gamma z) = {\left\{ \begin{array}{ll} \chi (\gamma ) j(\gamma , z)^{\ell /2} \cdot F(z) &{} (\ell \ \text {is even}) \\ \chi (\gamma ) J(\gamma , z)^{\ell } \cdot F(z) &{} (\ell \ \text {is odd}) \end{array}\right. }, \end{aligned}$$ -
(ii)
\(\Delta _{\ell /2} F= \Lambda \cdot F\) with some \(\Lambda \in \mathbb {C}\),
-
(iii)
F is of moderate growth at every cusp, namely, for every \(A \in SL_2(\mathbb {Z})\), there exist positive constants C, K and \(\nu \) depending on F and A such that
$$\begin{aligned} |F(A z)| \cdot |j(A, z)|^{-\ell /2} < C y^{\nu } \quad \text {if}\;\; y= \Im (z)>K. \end{aligned}$$
We call \(\Lambda \) the eigenvalue of F.
Let \(\lambda \) be a complex number with \(\lambda \not \in 1-\frac{1}{2}\mathbb {Z}_{\ge 0}\). Let \(\alpha =\{\alpha (n)\}_{n\in \mathbb {Z}\setminus \{0\}}\) and \(\beta =\{\beta (n)\}_{n\in \mathbb {Z}\setminus \{0\}}\) be complex sequences of polynomial growth. For \(\alpha ,\beta \), we can define the L-functions \(\xi _{\pm }(\alpha ;s), \xi _{\pm }(\beta ;s)\) and the completed L-functions \(\Xi _{\pm }(\alpha ;s), \Xi _{\pm }(\beta ;s)\) by
In the following, for simplicity, we put
Now we assume the following conditions [A1] – [A4]:
- [A1]:
-
The L-functions \(\xi _{\pm }(\alpha ;s), \xi _{\pm }(\beta ;s)\) have meromorphic continuations to the whole s-plane, and \((s-1)(s-2+2\lambda )\xi _\pm (\alpha ;s)\) and \((s-1)(s-2+2\lambda )\xi _\pm (\alpha ;s)\) are entire functions, which are of finite order in any vertical strip.
Here a function f(s) on a vertical strip \(\sigma _1 \le \Re (s) \le \sigma _2\) \((\sigma _1,\sigma _2 \in \mathbb {R}, \sigma _1<\sigma _2)\) is said to be of finite order on the strip if there exist some positive constants \(A,B,\rho \) such that
- [A2]:
-
The residues of \(\xi _{\pm }(\alpha ;s)\) and \(\xi _{\pm }(\beta ;s)\) at \(s=1\) satisfy
$$\begin{aligned} \mathop {\textrm{Res}}_{s=1} \xi _+(\alpha ;s) = \mathop {\textrm{Res}}_{s=1} \xi _-(\alpha ;s), \quad \mathop {\textrm{Res}}_{s=1} \xi _+(\beta ;s) = \mathop {\textrm{Res}}_{s=1} \xi _-(\beta ;s). \end{aligned}$$ - [A3]:
-
The following functional equation holds:
$$\begin{aligned}{} & {} \gamma (s)\left( \begin{array}{c} \Xi _{+}(\alpha ;s) \\ \Xi _{-}(\alpha ;s) \end{array}\right) \\{} & {} \quad = N^{2-2\lambda -s}\cdot \Sigma (\ell )\cdot \gamma (2-2\lambda -s)\left( \begin{array}{c} \Xi _{+}(\beta ;2-2\lambda -s) \\ \Xi _{-}(\beta ;2-2\lambda -s) \end{array}\right) , \end{aligned}$$where \(\gamma (s)\) and \(\Sigma (\ell )\) are defined by
$$\begin{aligned} \gamma (s) = \begin{pmatrix} e^{\pi s i/2} &{} e^{-\pi s i/2} \\ e^{-\pi s i/2} &{} e^{\pi s i/2} \end{pmatrix}, \qquad \Sigma (\ell ) = \begin{pmatrix} 0 &{} i^{\ell } \\ 1 &{} 0 \end{pmatrix}. \end{aligned}$$(4) - [A4]:
-
If \(\lambda = \frac{q}{2}\) \((q \in \mathbb {Z}_{\ge 0},\ q \ge 4)\), then
$$\begin{aligned} \xi _+(\alpha ;-k)+(-1)^k\xi _-(\alpha ;-k)=0 \quad (k = 1,2,\ldots ,q-3). \end{aligned}$$
Under the assumptions [A1] – [A4], we define \(\alpha (0)\), \(\beta (0)\), \(\alpha (\infty )\), \(\beta (\infty )\) by
For an odd prime number r with \((N,r)=1\) and a Dirichlet character \(\psi \) mod r, the twisted L-functions \(\xi _{\pm }(\alpha , \psi ;s), \Xi _{\pm }(\alpha , \psi ;s), \xi _{\pm }(\beta , \psi ;s), \Xi _{\pm }(\beta , \psi ;s)\) are defined by
where \(\tau _{\psi }(n)\) is the Gauss sum defined by
We put
and denote by \(\psi _{r,0}\) the principal character modulo r. Recall that the Gauss sums are calculated as follows:
Let \(\mathbb {P}_N\) be a set of odd prime numbers not dividing N such that, for any positive integers a, b coprime to each other, \(\mathbb {P}_N\) contains a prime number r of the form \(r=am+b\) for some \(m \in \mathbb {Z}_{>0}\). For an \(r \in \mathbb {P}_N\), denote by \(X_r\) the set of all Dirichlet characters mod r (including the principal character \(\psi _{r,0}\)). For \(\psi \in X_r\), we define the Dirichlet character \(\psi ^*\) by
We put
(For the definition of \(\varepsilon _r\), see (1).)
In the following, we fix a Dirichlet character \(\chi \) mod N that satisfies \(\chi (-1)=i^{\ell }\) (resp. \(\chi (-1)=1\)) when \(\ell \) is even (resp. odd).
For an \(r \in \mathbb {P}_N\) and a \(\psi \in X_r\), we consider the following conditions \([\textrm{A1}]_{r,\psi }\) – \([\textrm{A5}]_{r,\psi }\) on \(\xi _{\pm }(\alpha ,\psi ;s)\) and \(\xi _{\pm }(\beta ,\psi ^*;s)\).
- [A1]\(_{r,\psi }\):
-
\(\xi _{\pm }(\alpha ,\psi ;s), \xi _{\pm }(\beta ,\psi ^*;s)\) have meromorphic continuations to the whole s-plane, and \((s-1)(s-2+2\lambda )\xi _{\pm }(\alpha ,\psi ;s), (s-1)(s-2+2\lambda )\xi _{\pm }(\beta ,\psi ^*;s)\) are entire functions, which are of finite order in any vertical strip.
- [A2]\(_{r,\psi }\):
-
The residues of \(\xi _{\pm }(\alpha ,\psi ;s)\) and \(\xi _{\pm }(\beta ,\psi ^*;s)\) satisfy
$$\begin{aligned} \mathop \textrm{Res}_{s=1} \xi _+(\alpha ,\psi ;s) = \mathop \textrm{Res}_{s=1} \xi _-(\alpha ,\psi ;s), \, \mathop \textrm{Res}_{s=1} \xi _+(\beta ,\psi ^*;s) = \mathop \textrm{Res}_{s=1} \xi _-(\beta ,\psi ^*;s). \end{aligned}$$ - [A3]\(_{r,\psi }\):
-
\(\Xi _{\pm }(\alpha ,\psi ;s)\) and \(\Xi _{\pm }(\beta ,\psi ^*;s)\) satisfy the following functional equation:
$$\begin{aligned} \gamma (s)\left( \begin{array}{c} \Xi _{+}(\alpha ,\psi ;s) \\ \Xi _{-}(\alpha ,\psi ;s) \end{array}\right)&=\chi (r) \cdot C_{\ell ,r} \cdot \psi ^*(-N) \cdot r^{2\lambda -2}\cdot (Nr^2)^{2-2\lambda -s} \\&\qquad \cdot \Sigma (\ell )\cdot \gamma (2-2\lambda -s) \begin{pmatrix} \Xi _{+}\left( \beta , {\psi }^*;2-2\lambda -s\right) \\ \Xi _{-}\left( \beta , {\psi }^*;2-2\lambda -s\right) \end{pmatrix}, \end{aligned}$$where \(\gamma (s)\) and \(\Sigma (\ell )\) are the same as (4) in [A3].
- [A4]\(_{r,\psi }\):
-
If \(\lambda = \frac{q}{2}\) \((q \in \mathbb {Z}_{\ge 0},\ q \ge 4)\), then
$$\begin{aligned} \xi _+(\alpha ,\psi ;-k)+(-1)^k\xi _-(\alpha ,\psi ;-k)=0 \quad (k = 1,2,\ldots ,q-3). \end{aligned}$$ - [A5]\(_{r,\psi }\):
-
The following four relations between residues and special values hold:
-
\(\xi _e(\alpha ,\psi ;0) = \tau _\psi (0) \xi _e(\alpha ;0)\).
-
\({\chi (r)}\cdot \psi ^*(-N)\cdot C_{\ell ,r}\cdot r^{2\lambda } {\displaystyle \mathop {\textrm{Res}}_{s=1}}\,\xi _{\pm }(\beta , \psi ^{*};s) =\tau _{\psi }(0){\displaystyle \mathop {\textrm{Res}}_{s=1}}\,\xi _{\pm }(\beta ;s)\).
-
\(\xi _e(\beta , \psi ^{*};0) = \tau _{\psi ^*}(0) \xi _e(\beta ;0)\).
-
\({\displaystyle \mathop {\textrm{Res}}_{s=1}}\,\xi _{\pm }(\alpha ,\psi ;s) = {\chi (r)}\cdot \psi ^*(-N)\cdot C_{\ell ,r}\cdot r^{-2\lambda } \cdot \tau _{\psi ^*}(0)\cdot {\displaystyle \mathop {\textrm{Res}}_{s=1}} \,\xi _{\pm }(\alpha ;s)\).
Lemma 1
(Converse Theorem) We assume that \(\xi _\pm (\alpha ;s)\) and \(\xi _\pm (\beta ;s)\) satisfy the conditions [A1] – [A4], and define \(\alpha (0), \alpha (\infty ), \beta (0), \beta (\infty )\) by (5), (6), (7), (8), respectively. We assume furthermore that, for any \(r \in \mathbb {P}_N\) and \(\psi \in X_r\), \(\xi _{\pm }(\alpha ,\psi ;s)\) and \(\xi _{\pm }(\beta ,\psi ^*;s)\) satisfy the conditions \([\textrm{A1}]_{r,\psi }\) – \([\textrm{A5}]_{r,\psi }\). Define the functions \(F_\alpha (z)\) and \(G_\beta (z)\) on the upper half plane \(\mathcal {H}\) by
Here \(W_{\mu , \nu }(x)\) denotes the Whittaker function. Then \(F_\alpha (z)\) (resp. \(G_\beta (z)\)) gives a Maass form for \({\Gamma }_0(N)\) of weight \(\frac{\ell }{2}\) with character \(\chi \) (resp. \(\chi _{N,\ell }\)), and eigenvalue \((\lambda -\ell /4)(1-\lambda -\ell /4)\), where
Moreover, we have
Remark 1
In [16], we proved the converse theorem under a weaker condition \(\lambda \not \in \frac{1}{2} -\frac{1}{2} \mathbb {Z}_{\ge 0}\). In the present paper, since the case of \(\lambda =1\) will not be treated, we assume \(\lambda \not \in 1-\frac{1}{2}\mathbb {Z}_{\ge 0}\), which simplifies the description of the converse theorem.
3 Prehomogeneous vector spaces
Let Y be a non-degenerate half-integral symmetric matrix of degree m, and let p be the number of positive eigenvalues of Y. Throughout the present paper, we assume that \(m\ge 5\) and \(p(m-p)>0\). We denote by SO(Y) the special orthogonal group of Y defined by \(SO(Y)=\{g\in SL_m(\mathbb {C})\; |\; {}^{t}g Y g=Y\}\). We define the representation \(\rho \) of \(G=GL_1(\mathbb {C})\times SO(Y)\) on \(V=\mathbb {C}^{m}\) by
Let P(v) be the quadratic form on V defined by
where we use Siegel’s notation. Then, for \(\tilde{g}=(t, g)\in G\) and \(v\in V\), we have
and \(V-S\) is a single \(\rho (G)\)-orbit, where S is the zero set of P:
That is, \((G, \rho , V)\) is a reductive regular prehomogeneous vector space. (We refer to [11, 21] for the basics of the theory of prehomogeneous vector spaces.) We identify the dual space \(V^*\) of V with V itself via the inner product \(\langle v, v^*\rangle = {}^{t}v v^*\). Then the dual triplet \((G, \rho ^*, V^*)\) is given by
We define the quadratic form \(P^*(v^*)\) on \(V^*\) by
Then, for \(\tilde{g}=(t, g)\in G\), \(v^*\in V^*\), we have
and \(V-S^*\) is a single \(\rho ^*({G})\)-oribit, where \(S^*\) is the zero set of \(P^*\):
For \(\epsilon , \eta = \pm \), we put
We denote by \(dv= dv_1\cdots dv_m\) the Lebesgue measure on \(V_{\mathbb {R}}\), and by \(\mathcal {S}(V_{\mathbb {R}})\) the space of rapidly decreasing functions on \(V_{\mathbb {R}}\). Then, for \(f, f^*\in \mathcal {S}(V_{\mathbb {R}})\) and \(\epsilon , \eta = \pm \), we define the local zeta functions \(\Phi _{\epsilon }(f; s)\) and \(\Phi ^*_{\eta }(f^*;s)\) by
For \(\Re (s)>\frac{m}{2}\), the integrals \(\Phi _{\epsilon }(f; s)\) and \(\Phi ^*_{\eta }(f^*;s)\) converge absolutely, and as functions of s, they can be continued analytically to the whole s-plane as meromorphic functions. Further, we define the Fourier transform \(\widehat{f}(v^*)\) of \(f\in \mathcal {S}(V_{\mathbb {R}})\) by
The following lemma is due to Gelfand-Shilov [5]; a detailed proof is given in Kimura [11, § 4.2].
Lemma 2
(Local Functional Equation) Let p be the number of positive eigenvalues of Y, and put \(D=\det (2Y)\). Then the following functional equation holds:
In the rest of this section, we investigate singular distributions whose supports are contained in the real points \(S_{\mathbb {R}}\) of S; these distributions play an important role in the calculation of residues of Siegel’s zeta functions. We decompose \(S_{\mathbb {R}}\) as
A measure on \(S_{\mathbb {R}}\) that is \(SO(Y)_{\mathbb {R}}\)-invariant is constructed as follows. Since P(v) is a non-degenerate quadratic forms, we have
For \(i=1,\dots , m\), we define an \((m-1)\)-dimensional differential form \(\omega _i\) on \(U_i\) by
It is easy to see that there exists an \((m-1)\)-dimensional differential form \(\omega \) on \(S_{\mathbb {R}}\) that satisfies
and
Since \(P(gv)=P(v)\) for \(g\in SO(Y)_{\mathbb {R}}\), we have
Further, \(\omega (tv)= t^{m-2}\omega (v)\) for \(t>0\). Now let \(|\omega (v)|_{\infty }\) denote the measure on \(S_{1, \mathbb {R}}\) defined by \(\omega \). Then we have
for \(g\in SO(Y)_{\mathbb {R}}\) and \(t>0\). Similary, for the zero set \(S^*\) of \(P^*\), we decompose the real points \(S_{\mathbb {R}}^*\) as
The same argument as above ensures the existence of an \((m-1)\)-dimensional differential form \(\omega ^*\) on \(S_{1, \mathbb {R}}^*\) such that the restriction of \(\omega ^*\) on
is given by
We have
for \(g\in SO(Y)_{\mathbb {R}}\) and \(t>0\), where \(|\omega ^*|_{\infty }\) denotes the measure on \(S_{1, \mathbb {R}}^*\) defined by \(\omega ^*\). We refer to [5, Chap. III] for further details on the measures \(|\omega |_{\infty }, |\omega ^*|_{\infty }\). Then we have the following
Lemma 3
-
(1)
If \(f\in C_0^{\infty }(V_{\mathbb {R}}-S_{\mathbb {R}})\), then we have
$$\begin{aligned} \int _{S_{1,\mathbb {R}}^*}\widehat{f}(v^*) |\omega ^*(v^*)|_{\infty }&= \Gamma \left( \frac{m}{2}-1\right) |D|^{\frac{1}{2}} \cdot 2^{2-\frac{m}{2}}\cdot \pi ^{1-\frac{m}{2}} \\&\quad \times \begin{pmatrix} \sin \dfrac{\pi }{2}(m-p)&\sin \dfrac{\pi p}{2} \end{pmatrix} \begin{pmatrix} \displaystyle \int _{V_{+}} f(v) |P(v)|^{1-\frac{m}{2}} dv \\ \displaystyle \int _{V_{-}} f(v) |P(v)|^{1-\frac{m}{2}} dv \\ \end{pmatrix}. \end{aligned}$$ -
(2)
If \(\widehat{f}\in C_0^{\infty }(V_{\mathbb {R}}-S_{\mathbb {R}}^*)\), then we have
$$\begin{aligned} \int _{S_{1,\mathbb {R}}} f(v) |\omega (v)|_{\infty }&= \Gamma \left( \frac{m}{2}-1\right) |D|^{-\frac{1}{2}} \cdot 2^{2-\frac{m}{2}}\cdot \pi ^{1-\frac{m}{2}} \\&\quad \times \begin{pmatrix} \sin \dfrac{\pi }{2}(m-p)&\sin \dfrac{\pi p}{2} \end{pmatrix} \begin{pmatrix} \displaystyle \int _{V_{+}^*} \widehat{f}(v^*)|P^*(v^*)|^{1-\frac{m}{2}} dv^* \\ \displaystyle \int _{V_{-}^*} \widehat{f}(v^*)|P^*(v^*)|^{1-\frac{m}{2}} dv^* \\ \end{pmatrix}. \end{aligned}$$
This is stated, without proof, on p. 156 of Sato-Shintani [21] where Siegel’s zeta function is picked up as an example of their theory. Since the details cannot be found in other literature, we give a proof of the lemma for convenience of readers.
Proof
For \(f\in C_0^{\infty }(V_{\mathbb {R}}-S_{\mathbb {R}})\), we consider the integral
We may replace \(S_{1,\mathbb {R}}^*\) by \(S_{\mathbb {R}}^*=\{v^*\in V_{\mathbb {R}}\,|\, P^*(v^*)=0\}\), since \(S_{2,\mathbb {R}}^*=\{0\}\) has measure 0 in \(S_{\mathbb {R}}^*\). From the identity (19) (or the first formula on p. 257) in Gelfand-Shilov [5, Chap III, §2.2], we have
By the shift \(s\mapsto s-\frac{m}{2}\), we have
It then follows from the local functional equation (Lemma 2) that
which proves the first assertion of Lemma 3. The second assertion can be proved in a similar fashion. \(\square \)
4 Siegel’s zeta functions with congruence conditions
In this section, followig Sato-Shintani [21], we define Siegel’s zeta functions associated with \((G, \rho , V)\), and give their integral representations. Moreover, we calculate the singular parts of the zeta integrals. For this calculation, we also refer to Kimura [11]. Furthermore, following Sato [20], we slightly generalize Siegel’s zeta functions with using Schwartz-Bruhat functions on \(\mathbb {Q}^m\) in order to treat the twisted zeta functions simultaneously. Let dx be the measure on \(GL_{m}(\mathbb {R})\) defined by
and \(d\lambda \) the measure on the space \(\text {Sym}_{m}(\mathbb {R})\) of symmetric matrices of degree m defined by
Then we normalize a Haar measure \(d^{1}g\) on the Lie group \(SO(Y)_{\mathbb {R}}\) in such a way that the integration formula
holds for all integrable functions \(F(x)\in L^{1}(GL_{m}(\mathbb {R}))\). Further, let dt be the Lebesgue measure on \(\mathbb {R}\) and put
By (18), \(|P(v)|^{-\frac{m}{2}} dv\) is an \(\mathbb {R}_{+}\times SO(Y)_{\mathbb {R}}\)-invariant measure on \(V_{\epsilon }\) and the isotropy subgroup
at \(v\in V-S\) is a reductive algebraic group. Hence, for \(v\in V_{\epsilon }\), there exists a Haar measure \(d\mu _{v}\) on \(SO(Y)_{v, \mathbb {R}}\) such that the integration formula
holds for all integrable functions \(H(t, g)\in L^{1}(G_{\mathbb {R}})\). Similarly, for \(v^*\in V_{\eta }^*\), we write
and fix a Haar measure \(d\mu _{v^*}^*\) on \(SO(Y)_{v^*, \mathbb {R}}\) such that the integration formula
holds for all integrable functions \(H(t, g)\in L^{1}(G_{\mathbb {R}})\).
We call a function \(\phi :V_{\mathbb {Q}}\rightarrow \mathbb {C}\) a Schwartz-Bruhat function if the following two conditions are satisfied:
-
(1)
there exists a positive integer M such that \(\phi (v)=0\) for \(v\not \in \frac{1}{M}V_{\mathbb {Z}}\), and
-
(2)
there exists a positive integer N such that if \(v, w\in V_{\mathbb {Q}}\) satisfy \(v-w\in NV_{\mathbb {Z}}\). then \(\phi (v)=\phi (w)\).
The totality of Schwartz-functions on \(V_{\mathbb {Q}}\) is denoted by \(\mathcal {S}(V_{\mathbb {Q}})\). We define the Fourier transform \(\widehat{\phi }\in \mathcal {S}(V_{\mathbb {Q}})\) of a Schwartz-Bruhat function \(\phi \in \mathcal {S}(V_{\mathbb {Q}})\) by
where r is a sufficiently large positive integer such that the value \(\phi (v)\textbf{e}[-\langle v, v^*\rangle ]\) depends only on the residue class \(v\bmod {r V_{\mathbb {Z}}}\). Though r is not unique, the value \(\widehat{\phi }(v^*)\) does not depend on the choice of r. The following lemma is essentially an adelic version of Poisson summation formula.
Lemma 4
(Poisson summation formula) For \(\phi \in \mathcal {S}(V_{\mathbb {Q}})\) and \(f\in \mathcal {S}(V_{\mathbb {R}})\),
For \(\tilde{g}=(t, g)\in G_{\mathbb {R}}=\mathbb {R}^{\times }\times SO(Y)_{\mathbb {R}}\), we put \(f_{\tilde{g}}(v)=f(\rho (t, g)v)=f(tgv)\). Since
we have the following
Lemma 5
For \(\tilde{g}=(t, g)\in G_{\mathbb {R}}\), \(\phi \in \mathcal {S}(V_{\mathbb {Q}})\), \(f\in \mathcal {S}(V_{\mathbb {R}})\),
In the following, we assume that \(\phi \in \mathcal {S}(V_{\mathbb {Q}})\) is \(SO(Y)_{\mathbb {Z}}\)-invariant. That is, \(\phi \) is assumed to satisfy
Then we define the zeta integral \(Z(f, \phi ; s)\) by
Since \(V_{\mathbb {Q}}-S_{\mathbb {Q}}\) can be decomposed as
we have, by a formal calculation,
and further, by applying (27) to
we have
In the following, for \(v\in V_{\mathbb {Q}}-S_{\mathbb {Q}}\), we put
Since it is assumed that \(m\ge 5\), the generic isotropy subgroup \(SO(Y)_{v}\) is a semisimple algebraic group, and thus we have \(\mu (v)<+\infty \). (cf. [11, p. 184].) We further put \(\rho (t, \dot{g})v=x\) in the right hand side above. Then, since \(\mathbb {R}_{+}\times SO(Y)_{\mathbb {R}}/SO(Y)_{v, \mathbb {R}}\cong V_{\epsilon }\), we have
The Dirichlet series
converges absolutely for \(\Re (s)> \frac{m}{2}\), as will be explained in Remark 2 shortly. Hence the interchange of summation and integration, which leads to (33), can be justified under this condition. Similary, for \(f^*\in \mathcal {S}(V_{\mathbb {R}})\) and \(\phi ^*\in \mathcal {S}(V_{\mathbb {Q}})\) that satisfies
we define the zeta ingegral \(Z^*(f^*, \phi ^*;s)\) by
Furthermore, for \(v^*\in V_{\mathbb {Q}}-S_{\mathbb {Q}}^*\), we put
where \(d\mu _{v^*}^*\) is the Haar measure on \(SO(Y)_{v^*, \mathbb {R}}\) defined by (28).
Definition 2
(Siegel’s zeta functions with congruence conditions) Let \(\epsilon , \eta =\pm \) and assume that \(\phi , \phi ^*\in \mathcal {S}(V_{\mathbb {Q}})\) satisfy (30), (34), respectively. Then we define \(\zeta _{\epsilon }(\phi ; s)\) and \(\zeta _{\eta }^*(\phi ^*; s)\) by
We can summarize our argument as the following
Lemma 6
(Integral representations of the zeta functions) Let \(f, f^*\in \mathcal {S}(V_{\mathbb {R}})\) and assume that \(\phi , \phi ^*\in \mathcal {S}(V_{\mathbb {Q}})\) are \(SO(Y)_{\mathbb {Z}}\)-invariant. For \(\Re (s)> \frac{m}{2}\), we have
Remark 2
-
(1)
The original Siegel’s zeta functions are obtained by letting \(\phi =\phi _0\), where \(\phi _0\) is the characteristic function \(\text {ch}_{V_{\mathbb {Z}}}\) of \(V_{\mathbb {Z}}\). To apply Weil-type converse theorems, we need to examine the case where \(\phi (v)= \psi (P(v)) \phi _0(v)\) with Dirichlet character \(\psi \). Since each \(\phi (v)\) is a linear combination of characteristic functions of subsets of the form \(a + NV_{\mathbb {Z}} \; (a\in V_{\mathbb {Q}}, N\in \mathbb {Z}_{\ge 1})\), we call \(\zeta _{\epsilon }(\phi ; s)\), \(\zeta _{\eta }^*(\phi ^*; s)\) Siegel’s zeta functions with congruence conditions.
-
(2)
The absolute convergence of Siegel’s zeta functions is not at all obvious, though Siegel wrote just “Die Konvergents der Reihe entnimmt man der Reduktiontheorie”. A detailed proof of the convergence can be found in Tamagawa [30]. It also follows from the general theory of prehomogeneous vector spaces (Saito [18], F. Sato [19]).
-
(3)
We can write \(\zeta _{\pm }(\phi ;s)\) as
$$\begin{aligned} \zeta _{\pm }(\phi ;s) = \sum _{r\in \mathbb {Q}_{>0}} \frac{M(P, \phi ; \pm r)}{r^s} \end{aligned}$$with
$$\begin{aligned} M(P, \phi ; \pm r):= \sum _{\begin{array}{c} v\in SO(Y)_{\mathbb {Z}}\backslash V_{\pm }\cap \textrm{supp}(\phi ) \\ P(v) =\pm r \end{array}} \phi (v) \mu (v). \end{aligned}$$Since \(\phi (v)= 0\) for \(v\not \in \frac{1}{L}V_{\mathbb {Z}}\) with some integer L, we see that that the sum in the definition of \(M(P, \phi ;\pm r)\) is a finite sum (cf. Kimura [11, p.184]). In the case of \(\phi = \phi _0\), we have \(\textrm{supp}(\phi _0)=V_{\mathbb {Z}}\) and \(P(v)\in \mathbb {Z}\setminus \{0\}\) for \(v\in V_{\pm }\cap V_{\mathbb {Z}}\). For \(n=1,2,\dots \), we put
$$\begin{aligned} M(P; \pm n) =\sum _{\begin{array}{c} v\in SO(Y)_{\mathbb {Z}}\backslash V_{\pm }\cap V_{\mathbb {Z}} \\ P(v)= \pm n \end{array}} \mu (v). \end{aligned}$$(39)Siegel called M(P; n) the measures of representation (Darstellungsmaß). We have
$$\begin{aligned} \zeta _{\pm }(\phi _0;s) = \sum _{n=1}^{\infty } \frac{M(P; \pm n)}{n^s}. \end{aligned}$$
To investigate analytic properties of the zeta integrals, we define measures on isotropy subgroups at singular points. We fix an arbitrary point v of \(S_{1, \mathbb {R}}\). Recall that in the previous section, we have defined an \(SO(Y)_{\mathbb {R}}\)-invariant measure \(|\omega |_{\infty }\) on \(S_{1, \mathbb {R}}\cong SO(Y)_{\mathbb {R}}/SO(Y)_{v, \mathbb {R}}\). We can normalize a measure \(d\sigma _{v}\) on the isotropy subgroup \(SO(Y)_{v, \mathbb {R}}\) in such a way that the integration formula
holds for all integrable functions \(\psi (g)\in L^{1}(SO(Y)_{\mathbb {R}})\). Similarly, for \(v^*\in S_{1, \mathbb {R}}^*\), we take a measure \(d\sigma _{v^*}^*\) on the isotropy subgroup \(SO(Y)_{v^*, \mathbb {R}}\) such that the integration formula
holds for all integrable functions \(\psi (g)\in L^{1}(SO(Y)_{\mathbb {R}})\). Now we put
It is obvious that
The four integrals above converges absolutely for \(\Re (s)>\frac{m}{2}\), and further, two integrals \(Z_{+}(f, \phi ; s)\) and \(Z_{+}^*(f^*, \phi ^*; s)\) are absolutely convergent for any \(s\in \mathbb {C}\) and define entire functions of s. Let us calculate \(Z_{-}(f, \phi ;s)\) formally by using Lemma 5, the Poisson summation formula; the interchange of integral and summation will be justified later in Remark 3. Since \(\chi (t, g)= \chi ^*(t, g)^{-1}= t^2\), it follows from Lemma 5 that
The first term of the most right hand side is
Using (40) and (41), we calculate the second and third terms following the method of Sato-Shintani [21, Theorem 2]. Put
By the interchange of summation and integration, the third term above becomes
By applying (40) to \(\psi (g)=f(\rho (t, g)v)=f(tgv)\), we have
Here we have used (23) in the third equality. Hence the integral (42) is calculated as
Similarly, by term-by-term integration, we have
and by using (24) and (41), we obtain
Hence we see that
Now we put
Then we have the first assertion of the following lemma; the second assertion can be proved similarly as the first assertion, and then the third assertion follows immediately from the first and second assertions.
Lemma 7
-
(1)
For \(\Re (s)>\frac{m}{2}\), we have
$$\begin{aligned} Z(f, \phi ; s)&= Z_{+}(f, \phi ; s)+ Z_{+}^{*}\left( \widehat{f}, \widehat{\phi }; \frac{m}{2}-s\right) \\&\quad + \frac{1}{s-1}\int _{S_{1,\mathbb {R}}^{*}} \widehat{f}(z^*) | \omega ^*(z^*)|_{\infty } \sum _{v^*\in SO(Y)_{\mathbb {Z}}\backslash S_{1, \mathbb {Q}}^{*}} \widehat{\phi }(v^*) \sigma ^*(v^*) \\&\quad +\frac{\widehat{\phi }(0) \widehat{f}(0)}{s-\frac{m}{2}} \int _{SO(Y)_{\mathbb {R}}/SO(Y)_{\mathbb {Z}}} d^{1}g \\&\quad -\frac{1}{s+1-\frac{m}{2}} \int _{S_{1,\mathbb {R}}} f(z) |\omega (z)|_{\infty } \sum _{v\in SO(Y)_{\mathbb {Z}}\backslash S_{1, \mathbb {Q}}}\phi (v) \sigma (v) \\&\quad - \frac{\phi (0) f(0)}{s} \int _{SO(Y)_{\mathbb {R}}/SO(Y)_{\mathbb {Z}}} d^{1}g. \end{aligned}$$ -
(2)
For \(\Re (s)>\frac{m}{2}\), we have
$$\begin{aligned} Z^*(\widehat{f}, \widehat{\phi }; s)&= Z_{+}^* (\widehat{f}, \widehat{\phi }; s) +Z_{+}\left( f, \phi , \frac{m}{2}-s\right) \\&\quad +\frac{1}{s-1} \int _{S_{1,\mathbb {R}}} f(z) |\omega (z)|_{\infty } \sum _{v\in SO(Y)_{\mathbb {Z}}\backslash S_{1, \mathbb {Q}}}\phi (v) \sigma (v) \\&\quad + \frac{\phi (0) f(0)}{s-\frac{m}{2}} \int _{SO(Y)_{\mathbb {R}}/ SO(Y)_{\mathbb {Z}}} d^{1}g\\&\quad - \frac{1}{s+1-\frac{m}{2}}\int _{S_{1,\mathbb {R}}^{*}} \widehat{f}(z^*) |\omega ^*(z^*)|_{\infty } \sum _{v^*\in SO(Y)_{\mathbb {Z}}\backslash S_{1, \mathbb {Q}}^{*}} \widehat{\phi }(v^*) \sigma ^*(v^*) \\&\quad -\frac{\widehat{\phi }(0) \widehat{f}(0)}{s} \int _{SO(Y)_{\mathbb {R}}/SO(Y)_{\mathbb {Z}}} d^{1}g. \end{aligned}$$ -
(3)
As functions of s, the integrals \(Z(f, \phi , s)\) and \(Z^*(\widehat{f}, \widehat{\phi }; s)\) can be continued analytically to the whole s-plane, and satisfy the following functional equation:
$$\begin{aligned} Z^*(\widehat{f}, \widehat{\phi }; s) = Z\left( f, \phi ; \frac{m}{2}-s\right) . \end{aligned}$$
Remark 3
In [10], Igusa studied the so-called admissible representations related to the Siegel-Weil formula [33]. According to his classification, our prehomogeneous vector space \((GL_1(\mathbb {C})\times SO(Y), \mathbb {C}^m)\) gives an admissible representation if \(m\ge 5\), and this implies that the integrals
are absolutely convergent for all Schwartz-Bruhat functions \(f, f^*\in \mathcal {S}(V_{\mathbb {R}})\) and \(\phi , \phi ^*\in \mathcal {S}(V_{\mathbb {Q}})\). Hence the integrals
which appear in Lemma 7, are absolutely convergent, and the interchange of integral and summation can be justified by Fubini’s theorem.
5 Analytic properties of Siegel’s zeta functions
Theorem 1
Assume that \(\phi \in \mathcal {S}(V_{\mathbb {Q}})\) is \(SO(Y)_{\mathbb {Z}}\)-invariant.
-
(1)
The zeta functions \(\zeta _{\epsilon }(\phi ; s)\) and \(\zeta _{\eta }^*(\widehat{\phi };s)\) have analytic continuations of s in \(\mathbb {C}\), and the zeta functions multiplied by \((s-1)(s-\frac{m}{2})\) are entire functions of s of finite order in any vertical strip.
-
(2)
Th zeta functions \(\zeta _{\epsilon }(\phi ; s)\) and \(\zeta _{\eta }^*(\widehat{\phi };s)\) satisfy the following functional equation:
$$\begin{aligned} \begin{pmatrix} \zeta _{+}\left( \phi ; \frac{m}{2}-s\right) \\ \zeta _{-}\left( \phi ; \frac{m}{2}-s\right) \end{pmatrix}&=\Gamma \left( s+1-\frac{m}{2}\right) \Gamma (s) |D|^{\frac{1}{2}} \cdot 2^{-2s+\frac{m}{2}}\cdot \pi ^{-2s+\frac{m}{2}-1} \nonumber \\&\quad \times \begin{pmatrix} \sin \pi \left( \frac{p}{2}-s\right) &{} \sin \frac{\pi (m-p)}{2} \\ \sin \frac{\pi p}{2} &{} \sin \pi \left( \frac{m-p}{2}-s\right) \end{pmatrix} \begin{pmatrix} \zeta _{+}^*(\widehat{\phi };s) \\ \zeta _{-}^*(\widehat{\phi };s) \end{pmatrix}. \end{aligned}$$(46) -
(3)
The residues of \(\zeta _{\epsilon }(\phi ; s), \zeta _{\eta }^*(\widehat{\phi };s)\) at \(s=1\) and \(s=\frac{m}{2}\) are given by
$$\begin{aligned} \mathop {\textrm{Res}}_{s=\frac{m}{2}}\zeta _{\epsilon }(\phi ;s)&=\widehat{\phi }(0) \int _{SO(Y)_{\mathbb {R}}/SO(Y)_{\mathbb {Z}}} d^{1}g, \end{aligned}$$(47)$$\begin{aligned} \mathop {\textrm{Res}}_{s=\frac{m}{2}}\zeta _{\eta }^*(\widehat{\phi };s)&=\phi (0) \int _{SO(Y)_{\mathbb {R}}/SO(Y)_{\mathbb {Z}}} d^{1}g, \end{aligned}$$(48)$$\begin{aligned} \mathop {\textrm{Res}}_{s=1}\zeta _{\epsilon }(\phi ;s)&= \Gamma \left( \frac{m}{2}-1\right) |D|^{\frac{1}{2}} \cdot 2^{2-\frac{m}{2}}\cdot \pi ^{1-\frac{m}{2}} \sum _{v^*\in SO(Y)_{\mathbb {Z}}\backslash S_{1, \mathbb {Q}}^{*}} \widehat{\phi }(v^*) \sigma ^*(v^*) \nonumber \\&\quad \quad \times \left\{ \begin{array}{ll} \sin \dfrac{\pi }{2}(m-p) &{} (\epsilon = +) \\ \sin \dfrac{\pi p}{2} &{} (\epsilon = -) \end{array}\right. , \end{aligned}$$(49)$$\begin{aligned} \mathop {\textrm{Res}}_{s=1}\zeta _{\eta }^*(\widehat{\phi };s)&= \Gamma \left( \frac{m}{2}-1\right) |D|^{-\frac{1}{2}} \cdot 2^{2-\frac{m}{2}}\cdot \pi ^{1-\frac{m}{2}} \sum _{v\in SO(Y)_{\mathbb {Z}}\backslash S_{1, \mathbb {Q}}} {\phi }(v) \sigma (v) \nonumber \\&\quad \times \left\{ \begin{array}{ll} \sin \dfrac{\pi }{2}(m-p) &{} (\eta = +) \\ \sin \dfrac{\pi p}{2} &{} (\eta = -) \end{array}\right. . \end{aligned}$$(50) -
(3)
The following relations hold:
$$\begin{aligned} \zeta _{+}\left( \phi ;\frac{m}{2}-1\right) + \zeta _{-}\left( \phi ;\frac{m}{2}-1\right)&= - \sum _{v\in SO(Y)_{\mathbb {Z}}\backslash S_{1, \mathbb {Q}}} {\phi }(v) \sigma (v), \end{aligned}$$(51)$$\begin{aligned} \zeta _{+}^*\left( \widehat{\phi }\, ;\frac{m}{2}-1\right) + \zeta _{-}^*\left( \widehat{\phi }\, ;\frac{m}{2}-1\right)&= -\sum _{v^*\in SO(Y)_{\mathbb {Z}}\backslash S_{1, \mathbb {Q}}^{*}} \widehat{\phi }(v^*) \sigma ^*(v^*). \end{aligned}$$(52)
Proof
Let \(f\in C_{0}^{\infty }(V_{\mathbb {R}})\) in Lemma 7 (1). Then we see that
and thus the integral \(Z(f, \phi ;s)\) can be continued to a meromorphic function on the whole \(\mathbb {C}\), and (\(s-1)(s-\frac{m}{2})Z(f, \phi ;s)\) is an entire function of s. Further, for any \(s\in \mathbb {C}\), we take \(f_{\epsilon }\in C_{0}^{\infty }(V_{\epsilon })\) such that \(\Phi _{\epsilon }(f;s) \ne 0\). Then Lemma 6 implies that
and hence \(\zeta _{\epsilon }(\phi ; s)\) also can be continued to a meromorphic function on the whole \(\mathbb {C}\), and (\(s-1)(s-\frac{m}{2})\zeta _{\epsilon }(\phi ;s)\) is an entire function of s. The analytic continuation of \(\zeta _{\eta }^*(\widehat{\phi };s)\) can be proved in a similar fashion. Further, one can prove the boundedness of \(\zeta _{\epsilon }(\phi ; s)\) and \(\zeta _{\eta }^*(\widehat{\phi };s)\) in the same method as in Ueno [32, § 4]. By Lemma 6 and Lemma 7 (3), we have
and by Lemma 2, we have
where A(s) is given by
This implies that the vector
is orthogonal to the vector
for arbitrary \(f\in \mathcal {S}(V_{\mathbb {R}})\). For any \(s\in \mathbb {C}\), there exists an \(f_{\epsilon }\in C_{0}^{\infty }(V_{\epsilon })\) such that \(\Phi _{\epsilon }(f; \frac{m}{2}-s)\ne 0\), and hence (53) is the zero vector. This proves the functional equation (46). Next we calculate the residues. For the simple pole at \(s=\frac{m}{2}\), we have
by Lemma 7 (1). For \(f\in C_0^{\infty }(V_{\epsilon })\), Lemma 6 implies \(Z(\phi , f;s)=\zeta _{\epsilon }(\phi ;s) \cdot \Phi _{\epsilon }\left( f; s\right) \), and \(\Phi _{\epsilon }\left( f; \tfrac{m}{2}\right) \) is meaningful:
Hence we have
and similarly
By Lemma 7 (1), it is easy to pick up the residue of \(Z(f, \phi ; s)\) at the simple pole \(s=1\), and together with Lemma 6, it implies that for \(f\in C_0^{\infty }(V_{\epsilon })\),
Here the value \(\Phi _{\epsilon }\left( f; 1\right) \) is meaningful, and
Furthermore, by Lemma 3 (1), we have
and hence we obtain the residue formula (49). Similarly, the residue formula (50) can be proved with Lemma 3 (2); the detail is omitted. To prove the relation (51), we let \(s=1\) in the functiona equation (46):
Since
we have
By using (50) and
we see that
Since
we obtain the desired relation
Finally, let \(s=\frac{m}{2}-1\) in the functional equation (46). We have
and by using (49), we obtain
\(\square \)
Let N be the level of 2Y. By definition, N is the smallest positive integer such that \(N(2Y)^{-1}\) is an even matrix (a matrix whose entries are integers and even along the diagonal). We normalize the zeta functions \(\zeta _{\epsilon }(\phi ; s), \zeta _{\eta }^*(\widehat{\phi };s)\) as follows:
Lemma 8
The normalized zeta functions \(\widetilde{\zeta }_{\epsilon }(\phi ;s)\), \(\widetilde{\zeta }_{\eta }^*(\widehat{\phi };s)\) satisfy the following functional equation:
where \(\gamma (s)\) and \(\Sigma (\ell )\) are matrices defined by (4).
Proof
Let \(s\mapsto 1-s\) in the functional equation (46):
By (54), (55) and \(\displaystyle \Gamma (1-s) =\frac{\pi }{\Gamma (s)\sin \pi s}\), we see that this relation can be written as
An elementary calculation with \(\det \gamma (s) =2i \sin \pi s\) shows that
which completes the proof of the lemma. \(\square \)
The functional equation (56) is quite the same as the functional equation of the condition [A3] in Sect. 1 with \(\frac{m}{2}=2\lambda \), \(\ell \equiv 2p-m \pmod {4}\). Hence it is reasonable to expect that our converse theorem (Lemma 1) can apply to the normalized zeta functions \(\widetilde{\zeta }_{\epsilon }(\phi ;s)\), \(\widetilde{\zeta }_{\eta }^*(\widehat{\phi };s)\) to obtain Maass forms. The following lemma is indispensable for the application.
Lemma 9
-
(1)
If m is odd, then we have
$$\begin{aligned} \widetilde{\zeta }_{+}(\phi ; -k) + (-1)^{k} \cdot \widetilde{\zeta }_{-}(\phi ; -k)=0 \end{aligned}$$for \(k=1,2,3, \dots \).
-
(2)
Assume that m is even and p is odd. Let \(q=\frac{m}{2}\). Then we have
$$\begin{aligned} \widetilde{\zeta }_{+}(\phi ; -k) + (-1)^{k} \cdot \widetilde{\zeta }_{-}(\phi ; -k)=0 \end{aligned}$$for \(k=1, 2, \dots , q-2\).
Proof
By a little calculation, we obtain
Let us consider the values of both sides at \(s=-k \; (k\in \mathbb {Z}_{> 0})\). On the left hand side, \(\zeta _{\eta }^*\left( \widehat{\phi }\, ; 1-s\right) \) is holomorphic at \(s=-k\) except when m is even and \(k=\frac{m}{2}-1=q-1\). On the right hand side, if m is odd, then \(\Gamma (s)\Gamma \left( s+\frac{m}{2}-1\right) \) has a simple pole at \(s=-k \; (k\in \mathbb {Z}_{>0})\), and if m is even, then \(\Gamma (s)\Gamma \left( s+\frac{m}{2}-1\right) =\Gamma (s)\Gamma (s+q-1)\) has a simple pole at \(s=-k\) (\(1\le k\le q-2\)). We assume that \(1\le k\le q-2\) in the case of even m. Then, since \(\Gamma (s)\Gamma \left( s+\frac{m}{2}-1\right) \) has a simple pole at \(s=-k\), we see that
becomes the zero vector at \(s=-k\). Since \(\sin \pi (-k)=0\), \(\cos \pi (-k)=(-1)^{k}\), we have
If m is odd, then either p or \(m-p\) is odd, and thus we have
In the case of even m, if p is odd, then the relation above should hold. In the case that both of p and \(m-p\) are even, this argument can not apply since \(\sin \frac{\pi p}{2}= \sin \frac{\pi (m-p)}{2}=0\). \(\square \)
The following lemma follows immediately from the relations (51) and (52).
Lemma 10
We have the following relations:
In the rest of this section, we discuss the invariance of volumes with respect to scalar multiplications.
Lemma 11
-
(1)
For \(v\in V_{\mathbb {Q}}-S_{\mathbb {Q}}, v^*\in V_{\mathbb {Q}}-S_{\mathbb {Q}}^*\), we define the volumes \(\mu (v)\) and \(\mu ^*(v^*)\) by (32) and (36), respectively. For \(r>0\), we have
$$\begin{aligned} \mu (r v) = \mu (v), \qquad \mu ^*(r v^*) = \mu ^*(v^*). \end{aligned}$$ -
(2)
For \(v\in S_{1, \mathbb {Q}}, v^*\in S_{1, \mathbb {Q}}^*\), we define the volumes \(\sigma (v)\) and \(\sigma ^*(v^*)\) by (44) and (45), respectively. For \(r>0\), we have
$$\begin{aligned} \sigma (r v)= r^{2-m} \cdot \sigma (v), \qquad \sigma ^*(r v^*) = r^{2-m} \cdot \sigma ^*(v^*). \end{aligned}$$
Proof
(1) We prove the second formula \(\mu ^*(r v^*) = \mu ^*(v^*)\), which will be used later. Let \(F\in C_{0}^{\infty }(V_{\eta }^*)\). Then, by (28) and (36), we have
By the substitution \(v^*\mapsto r v^*\), we have
Put \(F_{r}(v^*):=F(r v^*)\). Since \(SO(Y)_{r v^*}= SO(Y)_{v^*}\), we have
This proves \(\mu ^{*}(r v^*)= \mu ^*(v^*)\). The first formula can be proved similarly.
(2) Let us show that \(\sigma (r v)= r^{2-m} \cdot \sigma (v)\). Take an \(f\in \mathcal {S}(V_{\mathbb {R}})\) and put \(\psi (g)= f(gv)\). By using (40), we have
By the substitution \(v\mapsto r v\), we have
Put \(f_{r}(v):= f(rv)\). Since \(SO(Y)_{rv}=SO(Y)_{v}\), we have
where we have used (23) on the fifth equality. This proves \(\sigma (rv) = r^{2-m}\cdot \sigma (v)\). The second formula can be proved in a similar fashion. \(\square \)
6 The main theorem
To prove the functional equation of twisted zeta functions, we quote a result of Stark [28]. Let Y be a non-degenerate half-integral symmetric matrix of degree m. Let \(D=\det (2Y)\) and N be the level of 2Y. We define a half-integral symmetric matrix \(\widehat{Y}\) by
We define the quadratic form P(v) on V by \(P(v)=Y[v]= {}^{t} v Yv\), and the quadratic form \(\widehat{P}(v^*)\) on \(V^*\) by
where \(P^*\) is defined by (19). For this \(\widehat{P}\), we define the measure \(M^*(\widehat{P};n)\) of representation by
For an odd prime r with \((r, N)=1\) and a Dirichlet character \(\psi \) of modulus r, we define the function \(\phi _{\psi ,P}(v)\) on \(V_{\mathbb {Q}}\) by
where \(\tau _{\psi }(P(v))\) is the Gauss sum defined by (9), and \(\phi _0(v)\) is the characteristic function of \(\mathbb {Z}^m\). It is easy to see that \(\phi _{\psi , P}(v)\) is a Schwartz-Bruhat function on \(V_{\mathbb {Q}}\). We define a field K by
and \(\chi _{K}\) be the Kronecker symbol associated to K. (If \(K=\mathbb {Q}\), we regard \(\chi _K\) as the principal character.) Furthermore, we define a Dirichlet character \(\psi ^*\bmod {r}\) by
and put
as (13). Then the following lemma follows from Stark [28, Lemmas 5 and 6].
Lemma 12
Let \(\widehat{\phi _{\psi , P}}(v^*)\) be the Fourier transform of \(\phi _{\psi , P}\) defined by (29). Then the support of \(\widehat{\phi _{\psi , P}}(v^*)\) is contained in \(r^{-1}\mathbb {Z}^{m}\), and for \(v^{*}\in \mathbb {Z}^m\), we have
Let \(\phi =\phi _0\) in the normalized zeta function \(\widetilde{\zeta }_{\pm }(\phi ;s)\) of (54). For \(v\in V_{\epsilon }\cap V_{\mathbb {Z}}\), we have \(P(v)=\epsilon n\) for some \(n=1,2,3,\dots \), and hence \(\widetilde{\zeta }_{\pm }(\phi _0;s)\) can be transformed as
where \(\textbf{a}(\pm n) \; (n=1,2,3,\dots )\) is defined by
where M(P; n) is the measure of representation defined as (39). Further, by plugging \(\phi _{\psi , P}(v) = \tau _{\psi }(P(v)) \cdot \phi _0(v)\) in (54), we have
On the other hand, let \(\phi =\phi _0\) in the normalized zeta function \(\widetilde{\zeta }_{\eta }^*(\widehat{\phi };s)\) of (55). Since \(\widehat{\phi _0}= \phi _0\), we have
By the definition (61), we have \(\widehat{P}(v^*)= NP^*(v^*) \in \mathbb {Z}\setminus \{0\}\) for \(v^*\in V_{\pm }^{*}\cap V_{\mathbb {Z}}\) and hence
where \(\textbf{b}(\pm n) \; (n=1,2,3,\dots )\) is defined by
where \(M^*(\widehat{P}; n)\) is defined as (62). Finally, let \(\phi =\phi _{\psi , P}(v) = \tau _{\psi }(P(v)) \cdot \phi _0(v)\) in (55). It then follows from Lemmas 11 (1) and 12, and also \(\widehat{P}(r^{-1} v^*)= r^{-2} \cdot \widehat{P}(v^*)\) that
We thus obtain the first assertion of the following
Lemma 13
For \(n=1,2,3,\dots \), we define \(\textbf{a}(\pm n)\) and \(\textbf{b}(\pm n)\) by (63) and (64) respectively, and let
-
(1)
We have
$$\begin{aligned} \widetilde{\zeta }_{\pm }(\phi _0;s)&= \zeta _{\pm }(\textbf{a};s), \\ \widetilde{\zeta }_{\pm }(\phi _{\psi , P};s)&=\zeta _{\pm }(\textbf{a}, \psi ; s), \\ \widetilde{\zeta }_{\eta }^*(\widehat{\phi _0};s)&=\zeta _{\pm }(\textbf{b}; s), \\ \widetilde{\zeta }_{\eta }^*(\widehat{\phi _{\psi , P}};s)&=r^{2s+\frac{m}{2}-2}\chi _{K}(r)\cdot C_{2p-m, r}\cdot \psi ^*(-N) \cdot \zeta _{\pm }(\textbf{b}, \psi ^*; s). \end{aligned}$$ -
(2)
On residues and special values of zeta functions, the following four relations hold:
$$\begin{aligned}&{\zeta }_{+}(\textbf{a}, \psi ; 0)+ {\zeta }_{-}(\textbf{a}, \psi ; 0) = \tau _{\psi }(0)\cdot \left( {\zeta }_{+}(\textbf{a}; 0)+ {\zeta }_{-}(\textbf{a}; 0)\right) , \end{aligned}$$(65)$$\begin{aligned}&r^{\frac{m}{2}} \cdot \chi _{K}(r)\cdot C_{2p-m, r}\cdot \psi ^*(-N) \cdot \mathop {\textrm{Res}}_{s=1} \zeta _{\pm }(\textbf{b}, \psi ^*; s) =\tau _{\psi }(0) \mathop {\textrm{Res}}_{s=1}\zeta _{\pm }(\textbf{b}; s), \end{aligned}$$(66)$$\begin{aligned}&{\zeta }_{+}(\textbf{b}, \psi ^*; 0) +{\zeta }_{-}(\textbf{b}, \psi ^*; 0) =\tau _{\psi ^*}(0) \cdot \left( {\zeta }_{+}(\textbf{b} ; 0)+ {\zeta }_{-}(\textbf{b}\, ; 0)\right) , \end{aligned}$$(67)$$\begin{aligned}&\mathop {\textrm{Res}}_{s=1} \zeta _{\pm }(\textbf{a}, \psi ; s) =r^{-\frac{m}{2}}\cdot \chi _{K}(r)\cdot C_{2p-m, r}\cdot \psi ^*(-N)\cdot \mathop {\textrm{Res}}_{s=1}\zeta _{\pm }(\textbf{a}; s). \end{aligned}$$(68) -
(3)
Assume that at least one of m or p is an odd integer. Let \(\lambda =\frac{m}{4}\) and take an integer \(\ell \) with \(\ell \equiv 2p-m \pmod {4}\). Then \(\zeta _{\pm }(\textbf{a}; s)\) and \(\zeta _{\pm }(\textbf{b};s)\) satisfy the assumptions [A1]–[A4] of Sect. 1, and further, \(\zeta _{\pm }(\textbf{a}, \psi ; s)\) and \(\zeta _{\pm }(\textbf{b}, \psi ^*;s)\) satisfy the assumptions [A1]\({}_{r,\psi }\)–[A5]\({}_{r, \psi }\) of Sect. 1.
Proof
(2) By letting \(\phi =\phi _0\) in (59), we have
and by letting \(\phi =\phi _{\psi , P}\) in (59), we have
which proves (65). By (55) and Theorem 1 (3), we have
and we thus obtain
Let us consider the residues at \(s=1\) of the both sides of
The residue at \(s=1\) of the left hand side is
and that of the right hand side is
by which we obtain (66). Next let \(\phi =\phi _0\) in the relation (60). Then we have
By letting \(\phi =\phi _{\psi , P}\) in (60) and using Lemmas 12 and 11 (2), we have
and this proves
which is the relation (67). By (54) and Theorem 1 (3), we have
and we thus obtain
Furthermore, it follows from Lemma 12 that the residue of \(\zeta _{\pm }(\textbf{a}, \psi ; s)= \widetilde{\zeta }_{\pm }(\phi _{\psi , P};s)\) at \(s=1\) is given by
by which we obtain the relation (68).
(3) By Theorem 1 (1), (3), we see that our zeta functions satisfy the assumptions [A1], [A1]\({}_{r, \psi }\), [A2], and [A2]\({}_{r, \psi }\). The functional equation of [A3] is nothing but the equation (56) with \(\phi =\phi _0\). Let \(\phi =\phi _{\psi , P}\) in (56); then the first assertion of the lemma implies that
which shows that the functional equation of [A3]\({}_{r, \psi }\) holds. Lemma 9 implies that our zeta functions satisfy the assumptions [A4] and [A4]\({}_{r, \psi }\). Finally, the compatibility condition [A5]\({}_{r, \psi }\) on residues and special values follows from (65), (66), (67) and (68). \(\square \)
In general, \(SO(Y)_{\mathbb {Z}}\backslash S_{1, \mathbb {Z}}\) is always an infinite set, since for \(v\in S_{1, \mathbb {Z}}\), any two of \(v, 2v, 3v,\dots \) can not lie in the same \(SO(Y)_{\mathbb {Z}}\)-orbit. However, as is seen in the following lemma that is taken from [11, pp.188–189], the number of \(SO(Y)_{\mathbb {Z}}\)-orbits in primitive vectors in \(S_{1, \mathbb {Z}}\) and \(S_{1, \mathbb {Z}}^*\) is finite.
Lemma 14
-
(1)
We call a vector \(v=(v_1, \dots , v_{m})\in V_{\mathbb {Z}}\) primitive if the greatest common divisor of \(v_1,\dots , v_m\) is 1. Then
$$\begin{aligned} \{v\in SO(Y)_{\mathbb {Z}}\backslash S_{1,\mathbb {Z}}\, ;\, v \, \text {is primitive}\} \end{aligned}$$is a finite set. Let \(a_1, \dots , a_h\) be a complete system of representatives of this set. Then we have
$$\begin{aligned} \sum _{v\in SO(Y)_{\mathbb {Z}}\backslash S_{1, \mathbb {Z}}} \sigma (v) = \zeta (m-2) \sum _{i=1}^{h} \sigma (a_i). \end{aligned}$$ -
(2)
Let \(b_1, \dots , b_k\) be a complete system of the finite set
$$\begin{aligned} \{v^*\in SO(Y)_{\mathbb {Z}}\backslash S_{1,\mathbb {Z}}^*\, ;\, v^* \, \text {is primitive}\}. \end{aligned}$$Then we have
$$\begin{aligned} \sum _{v\in SO(Y)_{\mathbb {Z}}\backslash S_{1, \mathbb {Z}}^*} \sigma ^*(v^*) = \zeta (m-2) \sum _{i=1}^{k} \sigma ^*(b_i). \end{aligned}$$
Now we are in a position to state
Theorem 2
Assume that at least one of m or p is an odd integer. Take an integer \(\ell \) with \(\ell \equiv 2p-m \pmod {4}\). Define \(C^{\infty }\)-functions F(z) and G(z) on \(\mathcal {H}\) by
Then, F(z) (resp. G(z)) is a Maass form for \(\Gamma _0(N)\) of weight \(\ell /2\) with eigenvalue \((m-\ell )(4-m-\ell )/16\) and character \(\chi _{K}\) (resp. \(\chi _{K_N}\)). Here we denote by \(\chi _{K}\) and \(\chi _{K_{N}}\) the Kronecker characters associated to the fields
and
respectively. Further we have
Proof
We apply the converse theorem (Lemma 1) to the normalized zeta functions \(\zeta _{\pm }(\textbf{a};s)\) and \(\zeta _{\pm }(\textbf{b};s)\) of Lemma 13. It remains to calculate the constant terms \(\textbf{a}(0),\, \textbf{a}(\infty ), \, \textbf{b}(0), \, \textbf{b}(\infty )\) along with the definitions (5), (6), (7), (8). First, by (69) and Lemma 14 (1), we have
Second, by (70), we have
Third, by (71) and Lemma 14 (2), we have
Finally, by (72), we have
\(\square \)
Remark 4
One can verify that our M(P; n) is identical to \(M(\mathfrak {S}, \mathfrak {a}, t)\) (\(\mathfrak {a}=0\)), which is defined as the formula (14) of Siegel [26]. Moreover, up to a power of y, our F(z) coincides with the integral \(\int _{F} f_{\mathfrak {a}}(z, \mathfrak {P})dv \; (\mathfrak {a}=0)\) of the indefinite theta series \(f_{\mathfrak {a}}(z, \mathfrak {P})\) over some fundamental domain F. See Siegel [26, Hilfssatz 4], [27] for the detail. We also note that Funke [4] calculated the Mellin transform of some indefinite theta series and obtained Siegel’s zeta functions associated with ternary zero forms.
7 Holomorphic modular forms arising from Siegel’s zeta functions
Under some conditions, the \(\gamma \)-matrix in Siegel’s functional equation (46) can be an upper or lower triangular matrix. In such a case, we obtain a single functional equation. More precisely,
\(\bullet \) Assume that the number of negative eigenvalues of Y is even; that is, \(m-p\) is an even integer. Then the first row of (46) is of the following form:
This suggests that \(\zeta _{+}(\phi ; s)\) and \(\zeta _{+}^*(\phi ;s)\) satisfy the functional equation of Hecke type.
\(\bullet \) Assume that the number of positive eigenvalues of Y is even; that is, p is an even integer. Then the second row of (46) is of the following form:
This suggests that \(\zeta _{-}(\phi ; s)\) and \(\zeta _{-}^*(\phi ;s)\) satisfy the functional equation of Hecke type.
In the following, we assume that \(m-p\) is even; if p is even, we replace P with \(-P\). We introduce Dirichlet series L(M; s) and \(L(M^*; s)\) as follows:
Further, we put
and
Then Theorem 1 implies that the following lemma holds:
Lemma 15
Assume that \(m-p\) is even. Both \(\Lambda _{N}(s; M)\) and \(\Lambda _{N}(s; M^*)\) can be continued analytically to the whole s-plane, satisfy the functional equation
and the function
is holomorphic on the whole s-plane and bounded on any vertical strip.
Let r be an odd prime with \((N, r)=1\). We denote by \(\varphi =\left( \frac{*}{r}\right) \) the Dirichlet character defined by the quadratic residue symbol. For a primitive Dirichlet character \(\psi \bmod {r}\), we define Dirichlet series \(L(M;s, \psi )\) and \(L(M^*; s, \psi )\) by
where a(n) and b(n) are defined by (73) and (74), respectively. Furthermore, we set
Then, by using Lemma 12, the formulas (10) and (11), we can prove the following
Lemma 16
Assume that \(m-p\) is even.
-
(1)
In the case of even m, for any primitive Dirichlet character \(\psi \bmod {r}\), \(\Lambda _{N}(s;M, \psi )\) can be holomorphically continued to the whole s-plane, bounded on any vertical strip, and satisfies the following functional equation
$$\begin{aligned} \Lambda _{N}(s; M, \psi )= i^{\frac{m}{2}} C_{\psi } \Lambda _{N} \left( \frac{m}{2}-s; M^*, \overline{\psi }\right) \end{aligned}$$with the constant
$$\begin{aligned} C_{\psi }= \chi _{K}(r) \psi (-N)\tau _{\psi }/\tau _{\overline{\psi }}. \end{aligned}$$ -
(2)
In the case of odd m, for any primitive Dirichlet character \(\psi \mod r\) with \(\psi \ne \varphi =\left( \frac{*}{r}\right) \), \(\Lambda _{N}(s;M, \psi )\) can be holomorphically continued to the whole s-plane, bounded on any vertical strip, and satisfies the following functional equation
$$\begin{aligned} \Lambda _{N}(s; M, \psi )= i^{\frac{m}{2}} C_{\psi }^{(1)} \Lambda _{N} \left( \frac{m}{2}-s; M^*, \overline{\psi }\varphi \right) \end{aligned}$$with the constant
$$\begin{aligned} C_{\psi }^{(1)}= \left( \frac{-1}{m}\right) ^{\frac{m-1}{2}} \cdot \chi _{K}(r)\left( \frac{N}{r}\right) \psi (-N)\epsilon _{r}^{-1} \tau _{\psi \varphi }/\tau _{\overline{\psi }}. \end{aligned}$$ -
(3)
In the case that m is odd and \(\psi = \varphi =\left( \frac{*}{r}\right) \),
$$\begin{aligned} \Lambda _{N}(s; M, \psi )+ C_{\psi }^{(2)}\frac{(r^{1/2}-r^{-1/2})b(0)}{\frac{m}{2}-s} \end{aligned}$$can be holomorphically continued to the whole s-plane, bounded on any vertical strip, and satisfies the following functional equation
$$\begin{aligned} \Lambda _{N}(s; M, \varphi )= i^{\frac{m}{2}} C_{\psi }^{(2)} \Lambda _{N} \left( \frac{m}{2}-s; M^*, \varphi \right) \end{aligned}$$with the constant
$$\begin{aligned} C_{\psi }^{(2)}= \left( \frac{-1}{m}\right) ^{\frac{m-1}{2}} \cdot \chi _{K}(r). \end{aligned}$$
These lemmas show that Weil’s converse theorems for holomorphic modular forms can apply to L(M; s) and \(L(M^*;s)\). We refer to Miyake [15, Theorem 4.3.15] for Weil’s converse theorem for the case of integral weight. For the case of half-integral weight, Shimura [22] stated a similar converse theorem. Although the details were not given in [22], the proof is roughly identical to the case of integral weight, and can be found in Bruinier [1]. We therefore obtain the following
Theorem 3
Assume that \(m-p\) is even. We define holomorphic functions F(z) and G(z) on \(\mathcal {H}\) by
Then, F(z) (resp. G(z)) is a holomorphic modular form for \(\Gamma _0(N)\) of weight m/2 with character \(\chi _{K}\) (resp. \(\chi _{K_N}\)). Further we have
Remark 5
If p is even, we can prove the same assertion for \(M(P;-n)\). Theorem 2 excludes the case where both m and p are even, but Theorem 3 shows that both \(\zeta _+\) and \(\zeta _-\) correspond to holomorphic modular forms in this case.
Data Availability
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
References
Bruinier, J.H.: Modulformen halbganzen Gewichts und Beziehungen zu Dirichletreihen, Diplomarbeit. Universität Heidelberg, Heidelberg (1997)
Cohen, H., Strömberg, F.: Modular Forms, A Classical Approach. Graduate Studies in Mathematics, vol. 179. American Mathematical Society, Providence, RI (2017)
Epstein, P.: Zur Theorie allgemeiner Zetafunctionen. Math. Ann. 56, 615–644 (1903)
Funke, J.: Heegner divisors and nonholomorphic modular forms. Compositio Math. 133, 289–321 (2002)
Gel’fand, I.M., Shilov, G.E.: Generalized Functions, vol. I: Properties and Operations. Academic Press, New York (1964)
Hafner, J.L., Walling, L.: Indefinite quadratic forms and Eisenstein series. Forum Math. 11, 313–348 (1999)
Hecke, E.: Über einen neuen Zusammenhang zwischen elliptischen Modulfunktionen und indefiniten quadratischen Formen, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse 35-44 (1925)
Ibukiyama, T.: On dimensions of automorphic forms and zeta functions of prehomogeneous vector space. Sūrikaisekikenkyūsho Kōkyūroku 924, 127–133 (1995)
Ibukiyama, T.: Topics on Modular Forms (in Japanese). Kyoritu Shuppan, Bunkyo (2018)
Igusa, J.: On certain representations of semi-simple algebraic groups and the arithmetic of the corresponding invariants. I. Invent. Math. 12, 62–94 (1971)
Kimura, T.: Introduction to Prehomogeneous Vector Spaces, Translations of Mathematical Monographs, vol. 215 (2003). American Mathematical Society, Providence, RI, Translated by Makoto Nagura and Tsuyoshi Niitani and Revised by the Author
Maaß, H.: Über eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen. Math. Ann. 121, 141–183 (1949)
Maaß, H.: Automorphe Funktionen und indefinite quadratische Formen, Sitsungsberichte der Heidelberger Akademie der Wissenshaften (1949)
Maaß, H.: Über die räumliche Vertilung der Punkte in Gittern mit indefiniter Metrik. Math. Ann. 138, 287–315 (1959)
Miyake, T.: Modular Forms, Translated from the 1976 Japanese original by Yoshitaka Maeda. Springer Monographs in Mathematics (2006)
Miyazaki, T., Sato, F., Sugiyama, K., Ueno, T.: Converse theorems for automorphic distributions and Maass forms of level \(N\). Res. Number Theory 6(6) (2020)
Neururer, M., Oliver, T.: Weil’s converse theorem for Maass forms and cancellation of zeros. Acta Arithmetica 196, 387–422 (2020)
Saito, H.: Convergence of the zeta functions of prehomogeneous vector spaces. Nagoya Math. J. 170, 1–31 (2003)
Sato, F.: Zeta functions in several variables associated with prehomogeneous vector spaces II: a convergence criterion. Tohoku Math. J. 35, 77–99 (1983)
Sato, F.: On functional equations of zeta distributions. Adv. Stud. Pure Math. 15, 465–508 (1989)
Sato, M., Shintani, T.: On zeta functions associated with prehomogeneous vector spaces. Ann. Math. 2(100), 131–170 (1974)
Shimura, G.: On modular forms of half integral weight. Ann. Math. 97, 440–481 (1973)
Siegel, C.L.: Über die Zetafunktionen indefiniter quadratischer Formen. Math. Z. 43, 682–708 (1938)
Siegel, C.L.: Über die Zetafunktionen indefiniter quadratischer Formen, II. Math. Z. 44, 398–426 (1939)
Siegel, C.L.: Indefinite quadratische Formen und Modulfunktionen. In: Studies and Essays Presented to R. Courant on his 60th Birthday, January 8 1948, pp. 395–406 (1948)
Siegel, C.L.: Indefinite quadratische Formen und Funktionentheorie. I. Math. Ann. 124, 17–54 (1951)
Siegel, C.L.: Lectures on Quadratic Forms (Notes by K. G. Ramanathan), Vol. 7. Tata Institute of Fundamental Research (1957)
Stark, H.M.: \(L\)-functions and character sums for quadratic forms (I). Acta Arith. XIV, 35–50 (1968)
Sugiyama, K.: Shintani correspondence for Maass forms of level \(N\) and prehomogeneous zeta functions. Proc. Jpn. Acad. Ser. A Math. Sci. 98, 41–46 (2022)
Tamagawa, T.: On indefinite quadratic forms. J. Math. Soc. Jpn. 29, 355–361 (1977)
Terras, A.: Harmonic Analysis on Symmetric Spaces-Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane, 2nd edn. Springer, New York (2013)
Ueno, T.: Modular forms arising from zeta functions in two variables attached to prehomogeneous vector spaces related quadratic forms. Nagoya Math. J. 175, 1–37 (2004)
Weil, A.: Sur la formule de Siegel dans la théorie des groupes classiques. Acta Math. 113, 1–87 (1965)
Weil, A.: Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen. Math. Ann. 168, 149–156 (1967)
Acknowledgements
The author wishes to thank Professor Fumihiro Sato for stimulating discussion and helpful suggestions. The author also thanks Professor Tomoyoshi Ibukiyama for valuable comments; in particular, Professor Ibukiyama explained the relation of the results of this paper to the prior work of [8, 9, 13, 25]. Finally, the author would like to thank anonymous reviewers for their careful reading and helpful comments.
Funding
Open access funding provided by Chiba Institute of Technology. This research is supported by JSPS KAKENHI Grant Number 22K03251.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Sugiyama, K. The modularity of Siegel’s zeta functions. Res. number theory 10, 31 (2024). https://doi.org/10.1007/s40993-024-00516-7
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40993-024-00516-7