Hostname: page-component-669899f699-7tmb6 Total loading time: 0 Render date: 2025-04-25T12:40:24.320Z Has data issue: false hasContentIssue false
  • English
  • Français

On the imaginary quadratic Doi-Naganuma lifting of modular forms of arbitrary level

Published online by Cambridge University Press:  22 January 2016

Solomon Friedberg
Solomon Friedberg*
Affiliation:
Department of Mathematics, Harvard University, Science Center, One Oxford Street, Cambridge, Massachusetts 02138, USA
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we use the theta function method [10] to give explicit Doi-Naganuma type maps associated to an imaginary quadratic field K, lifting cusp forms on any congruence subgroup of SL(2, Z) to forms on SL(2, C) automorphic with respect to an appropriate arithmetic discrete subgroup. The case of class number one, and form modular with respect to group Γ0(D) and character χ0 = (–D/*), where –D is the discriminant of K has been treated by Asai [1]. In order to complete his discussion, we must first introduce a more general theta function associated to an indefinite quadratic form (here of type (3, 1)), which we regard as a specialization of a symplectic theta function (see also [4]).

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 92 , December 1983 , pp. 1 - 20
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1983

References

[1] Asai, T., On the Doi-Naganuma lifting associated with imaginary quadratic fields, Nagoya Math. J., 71 (1978), 149–167.Google Scholar
[2] Asai, T., On the Fourier coefficients of automorphic forms at various cusps and some applications to Rankin’s convolution, J. Math. Soc. Japan, 28 (1976), 48–61.CrossRefGoogle Scholar
[3] Doi, K. and Naganuma, H., On the functional equation of certain Dirichlet series, Invent. Math., 9 (1969), 1–14.Google Scholar
[4] Friedberg, S., On theta functions associated to indefinite quadratic forms, preprint.Google Scholar
[5] Hecke, E., Mathematische Werke, Vandenhoeck & Ruprecht, Gottingen, 1970, no. 14.Google Scholar
[6] Howe, R., θ-series and invariant theory, in automorphic forms, representations, and L-functions, part 1, Proc. Symp. Pure Math., 33 (1979), 275–285.Google Scholar
[7] Jacquet, H., Automorphic forms on GL(2), II, Lecture Notes in Math., 278, Springer, 1972.Google Scholar
[8] Kudla, S., Theta-functions and Hilbert modular forms, Nagoya Math. J., 69 (1978), 97–106.Google Scholar
[9] Naganuma, H., On the coincidence of two Dirichlet series associated with cusp forms of Hecke’s “Neben”-type and Hilbert modular forms over a real quadratic field, J. Math. Soc. Japan, 25 (1973), 547–555.Google Scholar
[10] Niwa, S., Modular forms of half integral weight and the integral of certain theta-functions, Nagoya Math. J., 56 (1974), 147–161.Google Scholar
[11] Oda, T., On modular forms associated with indefinite quadratic forms of signature (2,n-2), Math. Ann., 231 (1977), 97–144.Google Scholar
[12] Ogg, A., On a convolution of L-series, Invent. Math., 7 (1969), 297–312.Google Scholar
[13] Rallis, S., On a relation between cusp forms and automorphic forms on orthogonal groups, in Automorphic forms, representations, and L-functions, part 1, Proc. Symp. Pure Math., 33 (1979), 297–314.CrossRefGoogle Scholar
[14] Saito, H., Automorphic forms and algebraic extensions of number fields, Lecture in Math., no. 8, Kinokuniya Book Store Co. Ltd., Tokyo Japan 1975.Google Scholar
[15] Shimura, G., The special values of the zeta functions associated with cusp forms, Comm. Pure Appl. Math., 29 (1976), 783–804.CrossRefGoogle Scholar
[16] Shintani, T., On construction of holomorphic cusp forms of half integral weight, Nagoya Math. J., 58 (1975), 83–126.Google Scholar
[17] Stark, H., On the transformation formula for the symplectic theta function and applications, J. Fac. Sci. Univ. Tokyo, 29 (1982), 1–12.Google Scholar
[18] Zagier, D., Modular forms associated to real quadratic fields, Invent. Math., 30 (1975), 1–46.Google Scholar