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