Hostname: page-component-848d4c4894-2xdlg Total loading time: 0 Render date: 2024-07-06T22:22:23.706Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Doi-Naganuma lifting associated with imaginary quadratic fields

Published online by Cambridge University Press:  22 January 2016

Tetsuya Asai
Tetsuya Asai*
Affiliation:
Department of Mathematics, Faculty of General Education Ehime University
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.

Similarly to the real quadratic field case by Doi and Naganuma ([3], [9]) there is a lifting from an elliptic modular form to an automorphic form on SL2(C) with respect to an arithmetic discrete subgroup relative to an imaginary quadratic field. This fact is contained in his general theory of Jacquet ([6]) as a special case. In this paper, we try to reproduce this lifting in its concrete form by using the theta function method developed first by Niwa ([10]); also Kudla ([7]) has treated the real quadratic field case on the same line. The theta function method will naturally lead to a theory of lifting to an orthogonal group of general signature (cf. Oda [11]), and the present note will give a prototype of non-holomorphic case.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 71 , November 1978 , pp. 149 - 167
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1978

References

[1] 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
[2] Asai, T., On certain Dirichlet series associated with Hilbert modular forms and Rankin’s method, Math. Ann. 226 (1977), 81–94.Google Scholar
[3] Doi, K. and Naganuma, H., On the functional equation of certain Dirichlet series, Invent. Math. 9 (1969), 1–14.CrossRefGoogle Scholar
[4] Erdélyi, A. et al., Higher transcendental functions, Vol. 2, McGraw-Hill, 1953.Google Scholar
[5] Erdélyi, A. et al., Tables of integral transforms, Vol. 1, McGraw-Hill, 1954.Google Scholar
[6] Jacquet, H., Automorphic forms on GL(2), II, Lee. Notes in Math. 278, Springer, 1972.Google Scholar
[7] Kudla, S., Theta-functions and Hilbert modular forms, to appear.Google Scholar
[8] Magnus, W. et al., Formulas and theorems for the special functions of mathematical physics, 3rd ed., Springer, 1966.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), to appear.CrossRefGoogle Scholar
[12] Shimura, G., The special values of zeta functions associated with cusp forms, Comm. Pure appl. Math. 29 (1976), 783–804.Google Scholar
[13] Shintani, T., On construction of holomorphic cusp forms of half integral weight, Nagoya Math. J. 58 (1975), 83–126.Google Scholar
[14] Weil, A., Dirichlet series and automorphic forms, Lee. Notes in Math. 189, Springer, 1971.Google Scholar