Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-23T23:08:29.619Z Has data issue: false hasContentIssue false
  • English
  • Français

Rings of Modular Forms on Eichler’s Problem

Published online by Cambridge University Press:  22 January 2016

Shigeaki Tsuyumine
Shigeaki Tsuyumine*
Affiliation:
2856-235, Sashiogi, Omiya-shi, Saitama, 330, 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 his paper [4] or lecture note [3], Eichler asked the problem when the ring of modular forms is Cohen-Macaulay. We shall try to investigate it for the Hilbert or Siegel modular case.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 99 , September 1985 , pp. 31 - 44
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1985

References

[ 1 ] Ash, A., Mumford, D., Rapoport, M. and Tai, Y., Smooth compactification of locally symmetric varieties, Math. Sci. Press (1975).Google Scholar
[ 2 ] Cartan, H., Fonctions automorphes, Ecole Normale Supérieure Séminaire 1957/1958.Google Scholar
[ 3 ] Eichler, M., Projective varieties and modular forms, Lecture Notes in Math., 210 Springer-Verlag Berlin Heidelberg New York (1971).Google Scholar
[ 4 ] Eichler, M., On the graded ring of modular forms, Acta Arith., 18 (1971), 87–92.Google Scholar
[ 5 ] Freitag, E., Über die Struktur der Funktionenkörper zur hyperabbelschen Gruppen I, J. reine angew. Math., 247 (1971), 97–117.Google Scholar
[ 6 ] Freitag, E., Lokale und globale Invarianten der Hilbertschen Modulgruppen, Invent. math., 17 (1972), 106–134.Google Scholar
[ 7 ] Grauert, H. and Riemenschneider, O., Verschwindungssätze für analytische Kohomologiegruppen auf komplexen Raumen, Invent, math., 11 (1970), 263–292.Google Scholar
[ 8 ] Grothendieck, A., Sur quelques points d’algèbre homologique, Tohoku Math. J., 9 (1957), 119–221.Google Scholar
[ 9 ] Hartshorne, R., Algebraic geometry, G.T.M., 52 Springer-Verlag New York Heidelberg Berlin (1977).Google Scholar
[10] Hirzebruch, F., Hilbert modular surfaces, L’Enseigment math., 19 (1973).Google Scholar
[11] Hirzebruch, F., The ring of Hilbert modular forms for real quadratic fields of small discriminant, Lecture Notes in Math., 627 Springer-Verlag Berlin Heidelberg New York (1976), 287–323.Google Scholar
[12] Hochster, M. and Eagon, J. A., Cohen-Macaulay rings, invariant theory and the generic perfection of determinantal loci, Amer. J. Math., 93 (1971), 1020–1058.Google Scholar
[13] Hochster, M. and Robert, J. L., Rings of invariants of reductive groups acting on regular local rings are Cohen-Macaulay, Adv. in Math., 13 (1974), 115–175.Google Scholar
[14] Igusa, J., On Siegel modular forms of genus two (II), Amer. J. Math., 86 (1964), 392–412.Google Scholar
[15] Kleiman, S. L., Toward a numerical theory of ampleness, Ann. of Math., 84 (1966), 293–344.Google Scholar
[16] Serre, J. P., Algèbre locale Multiplicities, Lecture Notes in Math., 11 (1965), Springer-Verlag Berlin Heidelberg New York.Google Scholar
[17] Shimizu, H., On discontinuous groups operating on the product of upper half planes, Ann. of Math., 77 (1963), 33–71.CrossRefGoogle Scholar
[18] Stanley, R., Hilbert functions of graded algebras, Adv. in Math., 28 (1978), 57–83.Google Scholar
[19] Stanley, R., Invariants of finite groups and their applications to combinatrics, Bull. Amer. Soc. (New Series), 1 (1979), 475–511.Google Scholar
[20] Thomas, E. and Vasquez, A. T., Ring of Hilbert modular forms, Compositio Math., 48 (1983), 139–165.Google Scholar
[21] Tsushima, R., A formula for the dimension of the space of Siegel cusp forms of degree three, Amer. J. Math., 102 (1980), 937–977.Google Scholar
[22] Tsuyumine, S., On Kodaira dimensions of Hilbert modular varieties, Invent, math., 80 (1985), 269–282.Google Scholar
[23] Yamazaki, T., On Siegel modular forms of degree two, Amer. J. Math., 98 (1976), 39–53.Google Scholar