## On Shimura, Shintani and Eichler-Zagier correspondences

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- by M. Manickam and B. Ramakrishnan PDF
- Trans. Amer. Math. Soc.
**352**(2000), 2601-2617 Request permission

## 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.## References

- A. O. L. Atkin and J. Lehner,
*Hecke operators on $\Gamma _{0}(m)$*, Math. Ann.**185**(1970), 134–160. MR**268123**, DOI 10.1007/BF01359701 - Martin Eichler and Don Zagier,
*The theory of Jacobi forms*, Progress in Mathematics, vol. 55, Birkhäuser Boston, Inc., Boston, MA, 1985. MR**781735**, DOI 10.1007/978-1-4684-9162-3 - B. Gross, W. Kohnen, and D. Zagier,
*Heegner points and derivatives of $L$-series. II*, Math. Ann.**278**(1987), no. 1-4, 497–562. MR**909238**, DOI 10.1007/BF01458081 - Henryk Iwaniec,
*Fourier coefficients of modular forms of half-integral weight*, Invent. Math.**87**(1987), no. 2, 385–401. MR**870736**, DOI 10.1007/BF01389423 - C. G. J. Jacobi,
*Fundamenta nova theories functionum ellipticarum*, Königesberg, 1829. - Olga Taussky,
*An algebraic property of Laplace’s differential equation*, Quart. J. Math. Oxford Ser.**10**(1939), 99–103. MR**83**, DOI 10.1093/qmath/os-10.1.99 - Winfried Kohnen,
*Newforms of half-integral weight*, J. Reine Angew. Math.**333**(1982), 32–72. MR**660784**, DOI 10.1515/crll.1982.333.32 - Winfried Kohnen,
*Fourier coefficients of modular forms of half-integral weight*, Math. Ann.**271**(1985), no. 2, 237–268. MR**783554**, DOI 10.1007/BF01455989 - Wen Ch’ing Winnie Li,
*Newforms and functional equations*, Math. Ann.**212**(1975), 285–315. MR**369263**, DOI 10.1007/BF01344466 - M. Manickam,
*Newforms of half-integral weight and some problems on modular forms*, Ph. D. Thesis, Univ. of Madras, 1989. - M. Manickam, B. Ramakrishnan, and T. C. Vasudevan,
*On Shintani correspondence*, Proc. Indian Acad. Sci. Math. Sci.**99**(1989), no. 3, 235–247. MR**1032710**, DOI 10.1007/BF02864396 - M. Manickam, B. Ramakrishnan, and T. C. Vasudevan,
*On the theory of newforms of half-integral weight*, J. Number Theory**34**(1990), no. 2, 210–224. MR**1042494**, DOI 10.1016/0022-314X(90)90151-G - M. Manickam, B. Ramakrishnan, and T. C. Vasudevan,
*On Saito-Kurokawa descent for congruence subgroups*, Manuscripta Math.**81**(1993), no. 1-2, 161–182. MR**1247596**, DOI 10.1007/BF02567852 - 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. - Goro Shimura,
*On modular forms of half integral weight*, Ann. of Math. (2)**97**(1973), 440–481. MR**332663**, DOI 10.2307/1970831 - Takuro Shintani,
*On construction of holomorphic cusp forms of half integral weight*, Nagoya Math. J.**58**(1975), 83–126. MR**389772** - Nils-Peter Skoruppa,
*Über den Zusammenhang zwischen Jacobiformen und Modulformen halbganzen Gewichts*, Bonner Mathematische Schriften [Bonn Mathematical Publications], vol. 159, Universität Bonn, Mathematisches Institut, Bonn, 1985 (German). Dissertation, Rheinische Friedrich-Wilhelms-Universität, Bonn, 1984. MR**806354** - Nils-Peter Skoruppa and Don Zagier,
*Jacobi forms and a certain space of modular forms*, Invent. Math.**94**(1988), no. 1, 113–146. MR**958592**, DOI 10.1007/BF01394347 - Masaru Ueda,
*The decomposition of the spaces of cusp forms of half-integral weight and trace formula of Hecke operators*, J. Math. Kyoto Univ.**28**(1988), no. 3, 505–555. MR**965416**, DOI 10.1215/kjm/1250520402 - Masaru Ueda,
*The trace formulae of twisting operators on the spaces of cusp forms of half-integral weight and some trace relations*, Japan. J. Math. (N.S.)**17**(1991), no. 1, 83–135. MR**1115594**, DOI 10.4099/math1924.17.83 - Masaru Ueda,
*On twisting operators and newforms of half-integral weight*, Nagoya Math. J.**131**(1993), 135–205. MR**1238637**, DOI 10.1017/S002776300000458X - J.-L. Waldspurger,
*Correspondance de Shimura*, J. Math. Pures Appl. (9)**59**(1980), no. 1, 1–132 (French). MR**577010** - J.-L. Waldspurger,
*Sur les coefficients de Fourier des formes modulaires de poids demi-entier*, J. Math. Pures Appl. (9)**60**(1981), no. 4, 375–484 (French). MR**646366**

## 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
- MR Author ID: 256629
- Email: ramki@mri.ernet.in
- Received by editor(s): November 5, 1997
- Received by editor(s) in revised form: May 12, 1998, and July 14, 1998
- Published electronically: March 7, 2000
- © Copyright 2000 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**352**(2000), 2601-2617 - MSC (2000): Primary 11F11, 11F37, 11F50; Secondary 11F25, 11F30
- DOI: https://doi.org/10.1090/S0002-9947-00-02423-5
- MathSciNet review: 1637086