|
On Shimura, Shintani and Eichler-Zagier correspondences
Author(s):
M.
Manickam;
B.
Ramakrishnan
Journal:
Trans. Amer. Math. Soc.
352
(2000),
2601-2617.
MSC (2000):
Primary 11F11, 11F37, 11F50;
Secondary 11F25, 11F30
Posted:
March 7, 2000
Retrieve article in:
PDF
This article is available free of charge
Abstract |
References |
Similar articles |
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.
References:
-
- 1.
- A. O. L. Atkin and J. Lehner, Hecke operators on
 , Math. Ann. 185 (1970), 134-160. MR 42:3022 - 2.
- M. Eichler and D. Zagier, The theory of Jacobi forms, Progress in Mathematics 55, Birkhaüser, Boston, 1985. MR 86j:11043
- 3.
- B. Gross, W. Kohnen and D. Zagier, Heegner points and derivatives of
 series II, Math. Ann. 278 (1987), 497-562. MR 89i:11069 - 4.
- H. Iwaniec, Fourier coefficients of modular forms of half-integral weight, Invent. Math. 87 (1987), 385-401. MR 88b:11024
- 5.
- C. G. J. Jacobi, Fundamenta nova theories functionum ellipticarum, Königesberg, 1829.
- 6.
- 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 - 7.
- W. Kohnen, Newforms of half-integral weight, J. Reine Angew Math. 333 (1981), 32-72. MR 84b:10038
- 8.
- W. Kohnen, Fourier coefficients of modular forms of half-integral weight, Math. Ann. 271 (1985), 237-268. MR 86i:11018
- 9.
- W. -W. Li, Newforms and functional equations, Math. Ann. 212 (1975), 285-315. MR 51:5498
- 10.
- M. Manickam, Newforms of half-integral weight and some problems on modular forms, Ph. D. Thesis, Univ. of Madras, 1989.
- 11.
- M. Manickam, B. Ramakrishnan and T. C. Vasudevan, On Shintani correspondence, Proc. Indian Acad. Sci. (Math. Sci.) 99 (1989), 235-247. MR 90m:11061
- 12.
- 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
- 13.
- M. Manickam, B. Ramakrishnan and T. C. Vasudevan, On Saito-Kurokawa descent for congruence subgroups, Manuscripta Math. 81 (1993), 161-182. MR 94j:11048
- 14.
- 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.
- 15.
- G. Shimura, On modular forms of half-integral weight, Ann. Math. 97 (1973), 440-481. MR 48:10989
- 16.
- T. Shintani, On construction of holomorphic cusp forms of half-integral weight, Nagoya Math. J. 58 (1975), 83-126. MR 52:10603
- 17.
- N.-P. Skoruppa, Über den Zussammenhang zwischen Jacobiformen und Modulformen halbganzen Gewichtz, Bonner Math. Schriften 159 (1995). MR 87a:11045
- 18.
- N.-P. Skoruppa and D. Zagier, Jacobi forms and a certain space of modular forms, Invent. Math. 94 (1988), 113-146. MR 89k:11029
- 19.
- M. Ueda, The decomposition of the spaces of cusp forms of half-integral weight and the trace formula of Hecke operators, J. Math. Kyoto U. 28 (1988), 505-555. MR 90a:11054
- 20.
- M. Ueda, The trace formula of the twisting operator on the spaces of cusp forms of half-integral weight and some trace relations, Japanese J. Math. 17 (1991), 83-135. MR 92g:11045
- 21.
- M. Ueda, On twisting operators and newforms of half-integral weight, Nagoya Math. J. 131 (1993), 135-205. MR 95a:11040
- 22.
- J. -L. Waldspurger, Correspondance de Shimura, J. Math. Pures Appl. (9) 59 (1980), 1-132. MR 83f:10029
- 23.
- J. -L. Waldspurger, Sur les coefficients de Fourier des formes modulaires de poids demi-entier, J. Math. Pures Appl. (9) 60 (1981), 375-484. MR 83h:10061
Similar Articles:
Retrieve articles in Transactions of the American Mathematical Society
with MSC
(2000):
11F11, 11F37, 11F50,
11F25, 11F30
Retrieve articles in all Journals with MSC
(2000):
11F11, 11F37, 11F50,
11F25, 11F30
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:
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
Copyright of article:
Copyright
2000,
American Mathematical Society
|
Comments: webmaster@ams.org