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Poincaré Theta Series and Singular Sets of Schottky Groups

Published online by Cambridge University Press:  22 January 2016

Tohru Akaza
Tohru Akaza*
Affiliation:
Mathematical Institute, Kanazawa University
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In the theory of automorphic functions for a properly discontinuous group G of linear transformations, the Poincaré theta series plays an essential role, since the convergence problem of the series occupies an important part of the theory. This problem was treated by many mathematicians such as Poincaré, Burnside [2], Fricke [4], Myrberg [6], [7] and others. Poincaré proved that the (-2m)-dimensional Poincaré theta series always converges if m is a positive integer greater than 2, and Burnside treated the problem and conjectured that ( -2)-dimensional Poincaré theta series always converges if G is a Schottky group. This conjecture was solved negatively by Myrberg. As is shown later (Theorem A), the convergence of Poincaré theta series gives an information on a metrical property of the singular set of the group.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 24 , June 1964 , pp. 43 - 65
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1964

References

[1] Akaza, T., Length of the singular set of Schottky group, Ködai Math. Sem. Report, 15 (1963), 62–66.Google Scholar
[2] Burnside, W., On a class of automorphic functions, Proc. London Math. Soc, 23 (1891), 49–88.CrossRefGoogle Scholar
[3] Ford, L. R., Automorphic Functions, 2nd Ed., New York (1951).Google Scholar
[4] Fricke, R., Über die Poincaré’schen Reihen der (— 1)-ten Dimension, Dedekind Festschrift (1901), 1–36.Google Scholar
[5] Maurer, L., Über die Schottkysche Gruppe von linearen Substitutionen, Tubingen naturwissensch. Abh., Heft 3 (1921), 5–21.Google Scholar
[6] Myrberg, P. J., Zur Theorie der Konvergenz der Poincaréschen Reihen, Ann. Acad. Sci. Fennicae, (A) 9, No. 4 (1916), 1–75.Google Scholar
[7] Myrberg, P. J., Zur Theorie der Konvergenz der Poincaréschen Reihen, (zweite Abh.) ibid., (A) 11, No. 4 (1917), 1–29.Google Scholar
[8] Myrberg, P. J., Die Kapazität der singuläre Menge der linearen Gruppen, Ann. Acad. Sci. Fennicae, A. I., 10 (1941), 1–19.Google Scholar
[9] Petersson, H., Über der Bereich absoluter Konvergenz der Poincaré’schen Reihen. Acta Math., 80 (1948), 23–49.Google Scholar
[10] Sario, L., Über Riemannsche Flächen mit hebbarem Rand, Ann. Acad. Sci. Fennicae, A. I, 50 (1948), 1–79.Google Scholar
[11] Schottky, F., Über eine specialle Function, welche bei einer bestimmten linearen Transformation ihres Arguments unverändert bleibt, Crelle’s J., 101 (1887), 227–272.CrossRefGoogle Scholar
[12] Tsuji, M., Theory of Meromorphic Functions on an open Riemann surface with null boundary, Nagoya Math. J., 6 (1953), 137–150.CrossRefGoogle Scholar