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On Shimura, Shintani and Eichler-Zagier correspondences
Authors:
M. Manickam and B. Ramakrishnan
Journal:
Trans. Amer. Math. Soc. 352 (2000), 2601-2617
MSC (2000):
Primary 11F11, 11F37, 11F50; Secondary 11F25, 11F30
Posted:
March 7, 2000
MathSciNet review:
1637086
Full-text PDF Free Access
Abstract |
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Additional Information
Abstract: In this paper, we set up Shimura and Shintani correspondences between Jacobi forms and modular forms of integral weight for arbitrary level and character, and generalize the Eichler-Zagier isomorphism between Jacobi forms and modular forms of half-integral weight to higher levels. Using this together with the known results, we get a strong multiplicity 1 theorem in certain cases for both Jacobi cusp newforms and half-integral weight cusp newforms. As a consequence, we get, among other results, the explicit Waldspurger theorem.
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 , Math. Ann. 185 (1970), 134-160. MR 42:3022
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 series II, Math. Ann. 278 (1987), 497-562. MR 89i:11069
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- H. Iwaniec, Fourier coefficients of modular forms of half-integral weight, Invent. Math. 87 (1987), 385-401. MR 88b:11024
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- C. G. J. Jacobi, Fundamenta nova theories functionum ellipticarum, Königesberg, 1829.
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- W. Kohnen and D. Zagier, Values of -series of modular forms at the center of the critical strip, Invent. Math. 64 (1981), 175-198. MR 83b:11029
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- W. Kohnen, Newforms of half-integral weight, J. Reine Angew Math. 333 (1981), 32-72. MR 84b:10038
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- W. Kohnen, Fourier coefficients of modular forms of half-integral weight, Math. Ann. 271 (1985), 237-268. MR 86i:11018
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- W. -W. Li, Newforms and functional equations, Math. Ann. 212 (1975), 285-315. MR 51:5498
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- M. Manickam, Newforms of half-integral weight and some problems on modular forms, Ph. D. Thesis, Univ. of Madras, 1989.
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- M. Manickam, B. Ramakrishnan and T. C. Vasudevan, On the theory of newforms of half-integral weight, J. Number Theory 34 (1990), 210-224. MR 91b:11060
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- M. Manickam, B. Ramakrishnan and T. C. Vasudevan, On Saito-Kurokawa descent for congruence subgroups, Manuscripta Math. 81 (1993), 161-182. MR 94j:11048
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- B. Ramakrishnan, On Shimura and Shintani correspondences for modular forms and Jacobi forms and some results on Hecke operators, Ph.D. Thesis, Univ. of Madras 1989.
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- G. Shimura, On modular forms of half-integral weight, Ann. Math. 97 (1973), 440-481. MR 48:10989
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Additional Information
M. Manickam
Affiliation:
Department of Mathematics, RKM Vivekananda College, Mylapore, Chennai 600 004, India
B. Ramakrishnan
Affiliation:
Mehta Research Institute of Mathematics and Mathematical Physics, Chhatnag Rd., Jhusi, Allahabad 211 019, India
Email:
ramki@mri.ernet.in
DOI:
http://dx.doi.org/10.1090/S0002-9947-00-02423-5
PII:
S 0002-9947(00)02423-5
Keywords:
Jacobi forms,
modular forms of half-integral weight,
newforms
Received by editor(s):
November 5, 1997
Received by editor(s) in revised form:
May 12, 1998, and July 14, 1998
Posted:
March 7, 2000
Article copyright:
© Copyright 2000 American Mathematical Society
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