Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On a relation between $ \widetilde {\rm SL}\sb{2}$ cusp forms and cusp forms on tube domains associated to orthogonal groups


Authors: S. Rallis and G. Schiffmann
Journal: Trans. Amer. Math. Soc. 263 (1981), 1-58
MSC: Primary 10D40; Secondary 22E50, 32N10
DOI: https://doi.org/10.1090/S0002-9947-1981-0590410-7
MathSciNet review: 590410
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We use the decomposition of the discrete spectrum of the Weil representation of the dual reductive pair $ ({\tilde{SL}_2},\;O(Q))$ to construct a generalized Shimura correspondence between automorphic forms on $ O(Q)$ and $ \widetilde{S{L_2}}$. We prove a generalized Zagier identity which gives the relation between Fourier coefficients of modular forms on $ \widetilde{S{L_2}}$ and $ O(Q)$. We give an explicit form of the lifting between $ \widetilde{S{L_2}}$ and $ O(n,2)$ in terms of Dirichlet series associated to modular forms. For the special case $ n = 3$, we construct certain Euler products associated to the lifting between $ S{L_2}$ and $ {\text{S}}{{\text{p}}_2} \cong O(3,2)$ (locally).


References [Enhancements On Off] (What's this?)

  • [I] S. Rallis and G. Schiffmann, Weil representation. I, Intertwining distributions and discrete spectrum, no. 231, Mem. Amer. Math. Soc., 1980. MR 567800 (81j:22007)
  • [II] -, Automorphic forms constructed from the Weil representation: Holomorphic case, Amer. J. Math. 102 (1978), 1049-1122. MR 517145 (80d:10038)
  • [III] -, Discrete spectrum of the Weil representation, Bull. Amer. Math. Soc. 83 (1977), 267-270. MR 0429753 (55:2763)
  • [IV] -, Automorphic cusp forms constructed from the Weil representation, Bull. Amer. Math. Soc. 83 (1977), 271-275. MR 0429754 (55:2764)
  • [1] A. N. Andrianov, Dirichlet series with Euler products in the theory of Siegel modular forms of genus $ 2$, Trudy Mat. Inst. Steklov. 112 (1971), 73-94. MR 0340178 (49:4934)
  • [2] T. Asai, On the Doi-Naganuma lifting associated with imaginary quadratic fields, preprint. MR 509001 (80b:10034)
  • [3] A. Borel, Introduction aux groupes arithmétiques, Hermann, Paris, 1969. MR 0244260 (39:5577)
  • [4] S. Kudla, Theta-functions and Hilbert modular forms, Nagoya Math. J. 69 (1978), 97-106. MR 0466025 (57:5908)
  • [5] E. Lippa, Hecke eigenforms of degree $ 2$: Dirichlet series and Euler products, Math. Ann. 220 (1976), 263-271. MR 0480359 (58:527)
  • [6] H. Maass, Siegel's modular forms and Dirichlet series, Lecture Notes in Math., vol. 216, Springer-Verlag, Berlin and New York, 1971. MR 0344198 (49:8938)
  • [7] W. Magnus, F. Oberhettinger and R. Soni, Formulas and theorems for the special functions of mathematical physics, 3rd ed., Die Grundlehren der Math. Wissenschaften, Band 52, Springer-Verlag, New York, 1966. MR 0232968 (38:1291)
  • [8] S. Niwa, Modular forms of half integral weight and the integral of certain theta functions, Nagoya Math. J. 56 (1975), 147-163. MR 0364106 (51:361)
  • [9] Takayuki Oda, On modular forms associated with indefinite quadratic forms of signature $ (2, n - 2)$, Math. Ann. 231 (1978), 97-144. MR 0466026 (57:5909)
  • [10] G. Shimura, On modular forms of half integral weight, Ann. of Math. (2) 97 (1973), 440-481. MR 0332663 (48:10989)
  • [11] -, The special values of the zeta functions associated with cusp forms, Comm. Pure Appl. Math. 29 (6) (1976), 783-805. MR 0434962 (55:7925)
  • [12] T. Shintani, On construction of holomorphic cusp forms of half integral weight, Nagoya Math. J. 58 (1975), 83-126. MR 0389772 (52:10603)
  • [13] C. L. Siegel, Über die Zetafunktionen indefiniter quadratischen Formen, Math. Z. 43 (1938), 682-708. MR 1545742
  • [14] D. Zagier, Modular forms associated to real quadratic fields, Invent. Math. 30 (1975), 1-48. MR 0382174 (52:3062)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 10D40, 22E50, 32N10

Retrieve articles in all journals with MSC: 10D40, 22E50, 32N10


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1981-0590410-7
Article copyright: © Copyright 1981 American Mathematical Society

American Mathematical Society