On modular forms associated with indefinite quadratic forms of signature (2,n?2)

Oda, Takayuki

doi:10.1007/bf01361138

Cited by 98 publications

(76 citation statements)

References 17 publications

(14 reference statements)

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“…In particular, the Shimura lift [32] was realized as a theta lift by Niwa [25]. Niwa constructed this theta lift from an integral quadratic form of signature (2, 1), whereas Oda [26] extended this lift to quadratic forms of signature (2, n) for n ≥ 1. The theta lifts of interest in this paper may moreover be viewed in the framework of a much more general theta correspondence between automorphic forms associated to the two groups of a dual reductive pair [21].…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Theta lifts and local Maass forms

Bringmann

Kane

Viazovska

2013

Mathematical Research Letters

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Abstract. The first two authors and Kohnen have recently introduced a new class of modular objects called locally harmonic Maass forms, which are annihilated almost everywhere by the hyperbolic Laplacian operator. In this paper, we realize these locally harmonic Maass forms as theta lifts of harmonic weak Maass forms. Using the theory of theta lifts, we then construct examples of (non-harmonic) local Maass forms, which are instead eigenfunctions of the hyperbolic Laplacian almost everywhere.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Theta lifts and local Maass forms

Bringmann

Kane

Viazovska

2013

Mathematical Research Letters

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“…Specifically in [II,§7] a formula for the Fourier coefficients of ßn is given which involves an infinite sum of Bessel functions and certain types of trigonometric sums (such a sum is reminiscent of the Fourier coefficients of Eisenstein-Poincaré series). However we can express T£ in another way (this idea originated in [14] for the special case Q having type (2,2) and is extended to (k, 2) for k > 4 in [9]). Namely we can write (valid for any G G SL2 and certain g described below)…”

Section: On a Relation Between Sl2 Cusp Forms And Cusp Forms On Tube mentioning

confidence: 99%

“…Thus in §2 we have set up a rather elaborate technical machinery to deduce (1)(2)(3)(4). We have adopted this point of view in order to prove a more general formula than in [14] and [9] and to show the dependence of the formula on the cuspidal properties of the Weil representation.…”

Section: On a Relation Between Sl2 Cusp Forms And Cusp Forms On Tube mentioning

confidence: 99%

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Untitled

1981

Trans. Amer. Math. Soc.

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Abstract.We use the decomposition of the discrete spectrum of the Weil representation of the dual reductive pair (SL2, 0(0)) to construct a generalized Shimura correspondence between automorphic forms on 0(Q) and SL^. We prove a generalized Zagier identity which gives the relation between Fourier coefficients of modular forms on SL2 and O(Q). We give an explicit form of the lifting between SL2 and 0(n, 2) in terms of Dirichlet series associated to modular forms. For the special case n = 3, we construct certain Euler products associated to the lifting between SL-, and Sp, » 0(3, 2) (locally).

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“…In [9], Professor Asai has studied the case of signature (3,1) and has shown that forms of signature (3,1) can be used to produce a lifting of cusp forms of Neben type to modular forms on hyperbolic 3-space with respect to discrete subgroups of SL 2 (C). The case of signature (n -2,2) has been considered by Rallis and Schiffman [10], [11], and by Oda [12].…”

Section: Introductionmentioning

confidence: 99%

Theta-functions and Hilbert modular forms

Kudla

1978

Nagoya Mathematical Journal

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The purpose of this note is to show how the theta-functions attached to certain indefinite quadratic forms of signature (2, 2) can be used to produce a map from certain spaces of cusp forms of Nebentype to Hilbert modular forms. The possibility of making such a construction was suggested by Niwa [4], and the techniques are the same as his and Shintani’s [6]. The construction of Hilbert modular forms from cusp forms of one variable has been discussed by many people, and I will not attempt to give a history of the subject here. However, the map produced by the theta-function is essentially the same as that of Doi and Naganuma [2], and Zagier [7]. In particular, the integral kernel Ω(τ, z1, z2) of Zagier is essentially the ‘holomorphic part’ of the theta-function.

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On modular forms associated with indefinite quadratic forms of signature (2,n?2)

Cited by 98 publications

References 17 publications

Theta lifts and local Maass forms

Theta lifts and local Maass forms

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Theta-functions and Hilbert modular forms

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