On Shimura, Shintani and Eichler-Zagier correspondences

Manickam, M.; Balasubramaniam, R.

doi:10.1090/s0002-9947-00-02423-5

Cited by 37 publications

(34 citation statements)

References 21 publications

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“…Among these, some explicit and useful formulas about the proportionality constant were formulated under assumption that f satisfies the multiplicity one theorem of Hecke operators. Manickam-Ramakrishnan-Vasudevan [13] and [14] also obtained related results in the case of elliptic modular forms of half integral weight. On the other hand, Kojima [8] and [9] generalized Kohnen-Zagier's results to the case of Kohnen's spaces of arbitrary odd level and of arbitrary character under the assumption of multiplicity two theorem of Hecke operators.…”

Section: Introductionmentioning

confidence: 86%

On the Fourier coefficients of modular forms of half integral weight belonging to Kohnen's spaces and the critical values of zeta functions

Kojima¹,

Tokuno²

2004

Tohoku Math. J. (2)

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The purpose of this paper is to derive a generalization of Kohnen-Zagier's results concerning Fourier coefficients of modular forms of half integral weight belonging to Kohnen's spaces, and to refine our previous results concerning Fourier coefficients of modular forms of half integral weight belonging to Kohnen's spaces. Employing kernel functions, we construct a correspondence Ψ from modular forms of half integral weight k + 1/2 belonging to Kohnen's spaces to modular forms of weight 2k. We explicitly determine the Fourier coefficients of Ψ (f ) in terms of those of f . Moreover, under certain assumptions about f concerning the multiplicity one theorem with respect to Hecke operators, we establish an explicit connection between the square of Fourier coefficients of f and the critical value of the zeta function associated with the image Ψ (f ) of f twisted with quadratic characters, which gives a further refinement of our results concerning Fourier coefficients of modular forms of half integral weight belonging to Kohnen's spaces.

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Section: Introductionmentioning

confidence: 86%

On the Fourier coefficients of modular forms of half integral weight belonging to Kohnen's spaces and the critical values of zeta functions

Kojima¹,

Tokuno²

2004

Tohoku Math. J. (2)

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“…For the arithmetic applications referenced above, one needs a classical construction of the Saito-Kurokawa lifting of square-free level. This lifting was claimed in a series of papers ( [19,17,18]). Unfortunately, there are many omitted proofs in these papers and the generalized Maass lifting used in these papers is known to be given incorrectly there.…”

Section: Introductionmentioning

confidence: 84%

Saito–Kurokawa lifts of square-free level

Agarwal

Brown²

2015

Kyoto J. Math.

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“…if we study the Fourier-Jacobi expansion for the case of a congruence subgroup of a higher level. There are results on the Eichler-Zagier correspondences of higher level cases (for instance see [36]). In this paper we have not tried the representation theoretic reformulation of the correspondence for such cases.…”

Section: Discussionmentioning

confidence: 99%

Fourier-Jacobi expansion of cusp forms on $Sp(2,{\Bbb R})$

Narita¹

2021

Preprint

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This paper develops a general theory of the Fourier-Jacobi expansion of cusp forms on the real symplectic group Sp(2, R) of degree two including generic cusp forms. An explicit description of such expansion is available for cusp forms generating discrete series representations, generalized principal series representations induced from the Jacobi parabolic subgroup and principal series representations, where we note that the latter two cases include non-spherical representations.As the archimedean local ingredients we need Fourier-Jacobi type spherical functions and Whittaker functions, whose explicit formulas are obtained by Hirano and by Oda, Miyazaki-Oda, Niwa and Ishii respectively. To realize these spherical functions in the Fourier-Jacobi expansion we use the spectral theory for the Jacobi group by Berndt-Böcherer and Berndt-Schmidt, which can be referred to as the global ingredient of our study. Based on the theory by Berndt-Böcherer we generalize the classical Eichler-Zagier correspondence in the representation theoretic context.

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On Shimura, Shintani and Eichler-Zagier correspondences

Cited by 37 publications

References 21 publications

On the Fourier coefficients of modular forms of half integral weight belonging to Kohnen's spaces and the critical values of zeta functions

On the Fourier coefficients of modular forms of half integral weight belonging to Kohnen's spaces and the critical values of zeta functions

Saito–Kurokawa lifts of square-free level

Fourier-Jacobi expansion of cusp forms on $Sp(2,{\Bbb R})$

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