On a relation between $\widetilde {\mathrm {SL}}_{2}$ cusp forms and cusp forms on tube domains associated to orthogonal groups
HTML articles powered by AMS MathViewer
- by S. Rallis and G. Schiffmann
- Trans. Amer. Math. Soc. 263 (1981), 1-58
- DOI: https://doi.org/10.1090/S0002-9947-1981-0590410-7
- PDF | Request permission
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
- Stephen Rallis and Gérard Schiffmann, Weil representation. I. Intertwining distributions and discrete spectrum, Mem. Amer. Math. Soc. 25 (1980), no. 231, iii+203. MR 567800, DOI 10.1090/memo/0231
- S. Rallis and G. Schiffmann, Automorphic forms constructed from the Weil representation: holomorphic case, Amer. J. Math. 100 (1978), no. 5, 1049–1122. MR 517145, DOI 10.2307/2373962
- S. Rallis and G. Schiffmann, Discrete spectrum of the Weil representation, Bull. Amer. Math. Soc. 83 (1977), no. 2, 267–270. MR 429753, DOI 10.1090/S0002-9904-1977-14299-7
- S. Rallis and G. Schiffmann, Automorphic cusp forms constructed from the Weil representation, Bull. Amer. Math. Soc. 83 (1977), no. 2, 271–275. MR 429754, DOI 10.1090/S0002-9904-1977-14301-2
- A. N. Andrianov, Dirichlet series with Euler product in the theory of Siegel modular forms of genus two, Trudy Mat. Inst. Steklov. 112 (1971), 73–94, 386 (Russian). Collection of articles dedicated to Academician Ivan Matveevič Vinogradov on his eightieth birthday, I. MR 0340178
- Tetsuya Asai, On the Doi-Naganuma lifting associated with imaginary quadratic fields, Nagoya Math. J. 71 (1978), 149–167. MR 509001, DOI 10.1017/S0027763000021693
- Armand Borel, Introduction aux groupes arithmétiques, Publications de l’Institut de Mathématique de l’Université de Strasbourg, XV. Actualités Scientifiques et Industrielles, No. 1341, Hermann, Paris, 1969 (French). MR 0244260
- Stephen S. Kudla, Theta-functions and Hilbert modular forms, Nagoya Math. J. 69 (1978), 97–106. MR 466025, DOI 10.1017/S0027763000017955
- Erik A. Lippa, Hecke eigenforms of degree two: Dirichlet series and Euler products, Math. Ann. 220 (1976), no. 3, 263–271. MR 480359, DOI 10.1007/BF01431096
- Hans Maass, Siegel’s modular forms and Dirichlet series, Lecture Notes in Mathematics, Vol. 216, Springer-Verlag, Berlin-New York, 1971. Dedicated to the last great representative of a passing epoch. Carl Ludwig Siegel on the occasion of his seventy-fifth birthday. MR 0344198, DOI 10.1007/BFb0058625
- Wilhelm Magnus, Fritz Oberhettinger, and Raj Pal Soni, Formulas and theorems for the special functions of mathematical physics, Third enlarged edition, Die Grundlehren der mathematischen Wissenschaften, Band 52, Springer-Verlag New York, Inc., New York, 1966. MR 0232968, DOI 10.1007/978-3-662-11761-3
- Shinji Niwa, Modular forms of half integral weight and the integral of certain theta-functions, Nagoya Math. J. 56 (1975), 147–161. MR 364106, DOI 10.1017/S0027763000016445
- Takayuki Oda, On modular forms associated with indefinite quadratic forms of signature $(2, n-2)$, Math. Ann. 231 (1977/78), no. 2, 97–144. MR 466026, DOI 10.1007/BF01361138
- Goro Shimura, On modular forms of half integral weight, Ann. of Math. (2) 97 (1973), 440–481. MR 332663, DOI 10.2307/1970831
- Goro Shimura, The special values of the zeta functions associated with cusp forms, Comm. Pure Appl. Math. 29 (1976), no. 6, 783–804. MR 434962, DOI 10.1002/cpa.3160290618
- Takuro Shintani, On construction of holomorphic cusp forms of half integral weight, Nagoya Math. J. 58 (1975), 83–126. MR 389772, DOI 10.1017/S0027763000016706
- Carl Ludwig Siegel, Über die Zetafunktionen indefiniter quadratischer Formen, Math. Z. 43 (1938), no. 1, 682–708 (German). MR 1545742, DOI 10.1007/BF01181113
- Don Zagier, Modular forms associated to real quadratic fields, Invent. Math. 30 (1975), no. 1, 1–46. MR 382174, DOI 10.1007/BF01389846
Bibliographic Information
- © Copyright 1981 American Mathematical Society
- 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