Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-26T07:45:52.907Z Has data issue: false hasContentIssue false
  • English
  • Français

Modular forms of degree n and representation by quadratic forms II

Published online by Cambridge University Press:  22 January 2016

Yoshiyuki Kitaoka
Yoshiyuki Kitaoka*
Affiliation:
Department of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464, 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.

Let S(m), T(n) be positive definite integral matrices and suppose that T is represented by S over each p-adic integer ring Zp. We proved arithmetically in [3] that T is represented by S over Z provided that m ≥ 2n + 3 and the minimum of T is sufficiently large. This guarantees the existence of at least one representation but does not give any asymptotic formula for the number of representations. To get an asymptotic formula we must employ analytic methods.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 87 , November 1982 , pp. 127 - 146
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1982

References

[1] Andrianov, A. N. and Maloletkin, G. N., Behavior of theta series of degree N under modular substitutions, Math. USSR Izvestija, 9 (1975), 227–241.Google Scholar
[2] Hsia, J. S., Recent developments in number theory, Arithmetic theory of integral quadratic forms, Proc. of the conference at Queen’s Univ. (to appear).Google Scholar
[3] Hsia, J. S., Kitaoka, Y. and Kneser, M., Representations of positive definite quadratic forms, J. reine angew. Math., 301 (1978), 132–141.Google Scholar
[4] Kitaoka, Y., Modular forms of degree n and representation by quadratic forms, Nagoya Math. J., 74 (1979), 95–122.Google Scholar
[5] Kneser, M., Quadratische Formen, Vorlesungs-Ausarbeitung, Göttingen (1973/4).Google Scholar
[6] Maaß, H., Siegel’s modular forms and Dirichlet series, Lecture Notes in Math. 216, Springer-Verlag (1971).Google Scholar
[7] O’Meara, O. T., Introduction to quadratic forms, Springer-Verlag (1963).Google Scholar
[8] Peters, M., Darstellungen durch definite ternäre quadratische Formen, Acta Arith., 34 (1977), 57–80.Google Scholar
[9] Raghavan, S., Modular forms of degree n and representation by quadratic forms, Ann. of Math., 70 (1959), 446–477.Google Scholar
[10] Siegel, C. L., Über die analytische Theorie der quadratischen Formen, Ann. of Math., 36 (1935), 527–606.Google Scholar
[11] Siegel, C. L., Einführung in die Theorie der Modulfunktionen n-ten Grades, Math. Ann., 116 (1939), 617–657.Google Scholar
[12] Siegel, C. L., Einheiten quadratischer Formen, Abh. Math. Sem. Univ. Hamburg, 13 (1940), 209–239.Google Scholar
[13] Siegel, C. L., On the theory of indefinite quadratic forms, Ann. of Math., 45 (1944), 577–622.CrossRefGoogle Scholar
[14] Tartakowskij, V., Die Gesamtheit der Zahlen, die durch eine positive quadratische Formen F(x1, …, xs ) (s ≥ 4) darstellbar sind, Izv. Akad. Nauk SSSR, 7 (1929), 111–122, 165–196.Google Scholar