Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-23T23:27:15.019Z Has data issue: false hasContentIssue false
  • English
  • Français

The Selberg trace formula for modular correspondences

Published online by Cambridge University Press:  22 January 2016

Shigeki Akiyama  and
Yoshio Tanigawa
Shigeki Akiyama
Affiliation:
Department of Mathematical Science, Graduate School of Science and Technology, Niigata University, Niigata, 950-21, Japan
Yoshio Tanigawa
Affiliation:
Department of Mathematics, School of Science, Nagoya University, Chikusa-ku, Nagoya 464-01, Japan
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 Selberg [11], he introduced the trace formula and applied it to computations of traces of Hecke operators acting on the space of cusp forms of weight greater than or equal to two. But for the case of weight one, the similar method is not effective. It only gives us a certain expression of the dimension of the space of cusp forms by the residue of the Selberg type zeta function. Here the Selberg type zeta function appears in the contribution from the hyperbolic conjugacy classes when we write the trace formula with a certain kernel function ([3J, [4], [7], [8], [9], [12]).

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 117 , March 1990 , pp. 93 - 123
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1990

References

[1] Akiyama, S., Selberg trace formula for odd weight I, II, Proc. Japan Acad., 64A, no. 9, 10, 341–344, 387–388.Google Scholar
[2] Akiyama, S., Tanigawa, Y., Hiramatsu, T., On traces of Hecke operators for the case of weight one (in Japanese), Automorphic Forms and Related Topics, ed. by Ihara, S., RIMS kokyuroku, 617 (1987), 49–65.Google Scholar
[3] Christian, U., Untersuchung Selbergscher Zetafunktionen, SFB Heft, 28 (1988).Google Scholar
[4] Christian, U., Zur Berechnung des Ranges der Schar der Elliptischen Spitzenformen, SFB Heft, 34 (1988).Google Scholar
[5] Hejhal, D. A., The Selberg trace formula for PSL (2, R) Vol. 1, Lecture Notes in Math., Springer, no. 548, (1976).Google Scholar
[6] Hejhal, D. A., The Selberg trace formula for PSL (2, R) Vol. 2, Lecture Notes in Math., Springer, no. 1001, (1983).Google Scholar
[7] Hiramatsu, T., On some dimension formula for automorphic forms of weight one II, Nagoya Math. J., 105 (1987), 169–186.CrossRefGoogle Scholar
[8] Hiramatsu, T. and Akiyama, S., On some dimension formula for automorphic forms of weight one III, Nagoya Math. J., 111 (1988), 157–163.Google Scholar
[9] Ishikawa, H., On the trace formula for Hecke operators, J. Fac. Sci. Univ. Tokyo, Sec. IA, 20, no. 2, (1973), 217–238.Google Scholar
[10] Kubota, T., Elementary theory of Eisenstein series, Kodansha and John Wiley, Tokyo and New York, 1973.Google Scholar
[11] Selberg, A., Harmonic analysis and discontinuous groups on weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc., 20 (1956), 47–87.Google Scholar
[12] Tanigawa, Y. and Ishikawa, H., The dimension formula of cusp forms of weight one for Γ 0(P), Nagoya Math. J., 111 (1988), 115–129.Google Scholar