Modular forms arising from zeta functions in two variables attached to prehomogeneous vector spaces related to quadratic forms

Ueno, Toshiharu

doi:10.1017/s0027763000008874

Cited by 9 publications

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“…0.4 In §5, applying our converse theorem for Maass forms (Theorem 4.3), we show that the two-variable zeta functions related to quadratic forms studied by Peter [29] and the fourth author [46] independently can be viewed as L-functions associated with Maass forms for Γ 0 (N ) (Theorem 5.1). Let A = (a ij ) be a nondegenerate half-integral symmetric matrix of size m. Denote by D and N , respectively, the determinant and the level of 2A.Then the zeta functions considered in [29] and [46] areThe analytic properties of the zeta functions ζ ± (w, s), ζ * ± (w, s) and their twists by Dirichlet characters necessary for the application of the converse theorem are essentially obtained in [46], and it is shown that, if we put α(n) = e πi(p−q)/4 |D| −1/2 Z(n, w), β(n) = Z * (n, w) (n ∈ Z, n = 0),then the functions F α (z) and G β (z) in (III) are Maass forms of weight ℓ/2 with ℓ = p−q+4k for Γ 0 (N ) (Theorem 5.1). Here p and q are respectively the numbers of positive and negative eigenvalues of A, and k is an arbitrary integer.…”

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confidence: 90%

“…ζ − ) only. The specialized functional equation can be transformed into a functional equation of Hecke type, and it is proved in [29, Theorem 1.1] and [46,Theorem 4.5] that the specialized zeta functions are the Mellin transforms of holomorphic modular forms for Γ 0 (k|D|) (k = 1, 2). In the following we construct Maass forms from the original zeta functions ζ ± and ζ * ± by using Theorem 4.3.…”

Section: Zeta Functions In Two Variables Associated With Quadratic Formsmentioning

confidence: 99%

“…( The theorem above was proved in [46, Theorem 4.1, Corollary 4.2, Proof of Theorem 4.5] for special cases of φ (φ 0 and φ ψ with non-principal ψ defined in §5.3 below). The proof for general Schwartz-Bruhat functions φ is almost the same as that in [46], and is omitted.…”

Section: Prehomogeneous Zeta Functionsmentioning

confidence: 99%

“…The analytic properties of the zeta functions ζ ± (w, s), ζ * ± (w, s) and their twists by Dirichlet characters necessary for the application of the converse theorem are essentially obtained in [46], and it is shown that, if we put α(n) = e πi(p−q)/4 |D| −1/2 Z(n, w), β(n) = Z * (n, w) (n ∈ Z, n = 0),…”

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Converse theorems for automorphic distributions and Maass forms of level N

et al. 2019

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We investigate the relations for L-functions satisfying certain functional equation, summationa formulas of Voronoi-Ferrar type and Maass forms of integral and halfintegral weight. Summation formulas of Voronoi-Ferrar type can be viewed as an automorphic property of distribution vectors of non-unitary principal series representations of the double covering group of SL(2). Our goal is converse theorems for automorphic distributions and Maass forms of level N characterizing them by analytic properties of the associated L-functions. As an application of our converse theorems, we construct Maass forms from the two-variable zeta functions related to quadratic forms defines an automorphic distribution for the group ∆(N ) = ñ(1),ñ(N ) , and hence its Poisson transform, which is given by F α (z) in (III), is a Maass form automorphic for ∆(N ). The transformation formula (0.3) is derived from the comparison of the Poisson transforms of both sides of the summation formula.The group ∆(N ) is a subgroup of the lift Γ 0 (N ) of Γ 0 (N ) to G. In §4 we prove a converse theorem that gives a condition for the automorphic distribution for ∆(N ) associated with the Ferrar-Suzuki summation formula to be automorphic for the larger group Γ 0 (N ) in terms of twists of the corresponding L-functions by Dirichlet characters (Theorem 4.2). By the Poisson transformation, this yields immediately a converse theorem for Maass forms for Γ 0 (N ) (Theorem 4.3), which is an analogue of the converse theorem of Weil [49] for holomorphic modular forms of integral weight (in the case of even ℓ) and its generalization by Shimura [40, Section 5] to holomorphic modular forms of half-integral weight (in the case of odd ℓ). A merit of our approach is to keep calculations involving the Whittaker function to a minimum. The information on the Whittaker function we need is only the fact that the Whittaker function appears as the Fourier transform of the Poisson kernel (see (2.14)). We say now a few words about the assumptions in the converse theorem. In our converse theorem some analytic properties (functional equations, location of poles and so on) are assumed for the L-functions twisted by Dirichlet characters of prime modulus including the principal characters. In the original converse theorem of Weil for holomorphic modular forms ([49]), the assumptions are imposed for the L-functions twisted only by primitive Dirichlet characters of prime modulus. As pointed out in Gelbart-Miller [9, §3.4], it has been considered difficult to transfer the argument of Weil to the Maass form case. In [2] and [3], Diamantis and Goldfeld proved a converse theorem for double Dirichlet series by considering the twists of Dirichlet series by Dirichlet characters including imprimitive characters (the principal characters), namely by the method originally found by Razar [30] for holomorphic modular forms. We follow the approach of Razar and Diamantis-Goldfeld. In a paper [25] that appeared very recently, Nuerurer and Oliver succeeded in modifying the argument of Weil to get a c...

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confidence: 90%

Section: Zeta Functions In Two Variables Associated With Quadratic Formsmentioning

confidence: 99%

Section: Prehomogeneous Zeta Functionsmentioning

confidence: 99%

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Converse theorems for automorphic distributions and Maass forms of level N

et al. 2019

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“…We prove that the hypothesis holds for a fixed natural number k > 8 if it holds for the number k − 1.Quadratic forms are encountered in the solution of numerous problems in algebra, the theory of differential and integral equations, functional analysis, and other fields of mathematics (see, e.g., [1] and references therein; see also [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]). Among quadratic forms, an important role is played by Tits quadratic forms for various objects (graphs, posets, algebras, etc.).…”

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On one property of positive-definite quadratic forms corresponding to finite posets

Степочкина

2005

Nonlinear Oscill

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We consider the Bondarenko hypothesis about the structure of finite posets with positive-definite Tits quadratic form. We prove that the hypothesis holds for a fixed natural number k > 8 if it holds for the number k − 1.Quadratic forms are encountered in the solution of numerous problems in algebra, the theory of differential and integral equations, functional analysis, and other fields of mathematics (see, e.g., [1] and references therein; see also [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]). Among quadratic forms, an important role is played by Tits quadratic forms for various objects (graphs, posets, algebras, etc.). In the present paper, we study exactly these forms.In [18], Gabriel associated a quiver (an oriented graph) with a certain quadratic form and called it a Tits quadratic form. This form plays an important role in the theory of finite-dimensional representations of graphs. In particular, it was shown in [18] that a graph has a finite type (i.e., it has a finite, up to an isomorphism, number of indecomposable representations) if and only if the corresponding Tits form is positive definite. This paper originated a new trend in algebra that studies the relationship between the properties of representations of various objects and the properties of the corresponding quadratic forms.The next step in this direction was made in the works of Brenner [19] and Drozd [20], where Tits quadratic forms were determined for quivers with relations and for posets, respectively. In the general case, the Tits form for matrix problems without relations was introduced by Kleiner and Roiter in [21].It was shown in [20] that a poset has a finite type if and only if its Tits form is weakly positive (representations of posets were introduced in [22]). Positive-definite forms are not distinguished in this case, but in the further investigation of these representations positive-definite Tits forms play an important role [23]. For the first time, these forms were described in [1] (see also [24]), where infinite posets were considered. In [25], Bondarenko considered the case of finite sets and formulated a hypothesis concerning positive-definite Tits forms. This hypothesis is considered in the present paper. Formulation of a HypothesisIn the present paper, we consider finite posets. A subset X of a poset S is always understood as a subset complete with respect to the partial-order condition (i.e., if a, b ∈ X, then a ≥ b in X if and only if a ≥ b in S).If a poset S is the union of its pairwise disjoint subsets A 1 , . . . , A s , s ≥ 1, then we say that S is the sum of these subsets and write S = A 1 + . . . + A s . If elements of different summands are always incomparable, then S is called the direct sum of these subsets.We also recall several definitions from [1]. Let a poset S be the sum of subsets A 1 , . . . , A s . This sum is called one-sided if (up to reenumeration of summands) one has i < j whenever there exist elements b ∈ A i and c ∈ A j , i = j, such that b < c. Further, the sum S = A 1 + . . . + A s is ...

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On serial posets with positive-definite quadratic Tits form

Bondarenko

Степочкина

2006

Nonlinear Oscill

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We describe all serial posets with positive-definite quadratic Tits form and prove that any poset of order greater than 7 with positive-definite Tits form is serial.Quadratic forms are encountered in the solution of various problems in algebra, geometry, the theory of differential and integral equations, operator theory, and other fields of mathematics (see, e.g., ). Among them, an important role is played by quadratic Tits forms for oriented graphs, posets, algebras, etc. In the present paper, we consider exactly these forms. Formulation of the Main ResultFirst, recall some definitions. Let S be a finite or an infinite poset. We say that S is the sum of its subsets A 1 , . . . , A s and writeIf all elements of different terms are always incomparable, then S is called the direct sum of the indicated subsets. Further, according to [27], the sum S = A 1 + . . . + A s is called one-sided if (up to enumeration of terms) one has i < j whenever there exist elements b ∈ A i and c ∈ A j for i = j such that b < c. According to [27], the sum S = A 1 + . . . + A s is called minimax if it follows from the relation x < y, where x and y belong to different terms, that x and y are, respectively, minimal and maximal elements of the set S. Formally, a direct sum is minimax. Nevertheless, considering minimax sums in what follows, we always assume for convenience that they are not direct.A subset of a poset S is understood as a complete partially ordered subset, i.e., a partial order on it is induced by a partial order on S.The form q S (z) : Z S∪0 0 → Z defined by the equality

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Modular forms arising from zeta functions in two variables attached to prehomogeneous vector spaces related to quadratic forms

Cited by 9 publications

References 18 publications

Converse theorems for automorphic distributions and Maass forms of level N

Converse theorems for automorphic distributions and Maass forms of level N

On one property of positive-definite quadratic forms corresponding to finite posets

On serial posets with positive-definite quadratic Tits form

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