Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



The cusp forms of weight $ 3$ on $ \Gamma\sb 2(2,4,8)$

Authors: Bert van Geemen and Duco van Straten
Journal: Math. Comp. 61 (1993), 849-872
MSC: Primary 11F55
MathSciNet review: 1181333
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The congruence subgroup $ {\Gamma _2}(2,4,8)$ of the group $ {\Gamma _2}$ of $ 4 \times 4$ integral symplectic matrices is contained in $ {\Gamma _2}(4)$ and contains $ {\Gamma _2}(8)$, with $ {\Gamma _2}(n)$ the principal congruence subgroup of level n. The Satake compactification of the quotient of the three-dimensional Siegel upper half space by $ {\Gamma _2}(2,4,8)$ is shown to be a complete intersection of ten quadrics in $ {\mathbb{P}^{13}}$. We determine the space of global holomorphic three forms on this space, which coincides with the space of cusp forms of weight 3 on $ {\Gamma _2}(2,4,8)$; it has dimension 2283. Finally, we study the action of the Hecke operators on this space and consider the Andrianov L-functions of some eigenforms.

References [Enhancements On Off] (What's this?)

  • [1] A. Ash, D. Mumford, M. Rapoport, and Y. Tai, Smooth compactification of locally symmetric varieties, Math. Sci. Press, Brookline, MA, 1975. MR 0457437 (56:15642)
  • [2] S. A. Evdokimov, A basis of eigenfunctions of Hecke operators in the theory of modular forms of genus n, Math. USSR-Sb. 43 (1982), 299-322.
  • [3] -, Euler products for congruence subgroups of the Siegel modular group of genus 2, Math. USSR-Sb. 28 (1976), 431-458.
  • [4] E. Freitag, Holomorphe Differentialformen zu Kongruenzgruppen der Siegelschen Modulgruppe zweiten Grades, Math. Ann. 216 (1975), 155-164. MR 0376540 (51:12715)
  • [5] -, Siegelsche Modulfunktionen, Die Grundlehren der mathematischen Wissenschaften, Band 254, Springer-Verlag, Berlin, 1983. MR 871067 (88b:11027)
  • [6] B. van Geemen and N. O. Nygaard, L-functions of some Siegel modular 3-folds, Preprint nr. 546, Dept. of Math., University of Utrecht, 1988.
  • [7] E. Hecke, Über Modulfunktionen und die Dirichletschen Reihen mit Eulerschen Produktentwicklung. II, Math. Ann. 114 (1937), 316-351. MR 1513142
  • [8] J. I. Igusa, Theta functions, Die Grundlehren der mathematischen Wissenschaften, Band 194, Springer-Verlag, Berlin, 1972. MR 0325625 (48:3972)
  • [9] D. Mumford, Tata Lectures of Theta. II, Birkhäuser, Boston, 1984. MR 742776 (86b:14017)
  • [10] I. Satake, Compactifications des espaces quotient de Siegel. II, Sem. H. Cartan, École Norm. Sup., 1957/58.
  • [11] B. Tessier and M. Merle, Conditions d'adjunction, d'apres Du Val, Séminaire sur les singularités des surfaces. Springer Lecture Notes in Math., vol. 777, 1980.
  • [12] R. Weissauer, On the cohomology of Siegel Modular Threefolds, Arithmetic of Complex Manifolds (W.-P. Barth and H. Lange, eds.), Springer Lecture Notes in Math., vol. 1399, 1989. MR 1034263 (91e:11051)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 11F55

Retrieve articles in all journals with MSC: 11F55

Additional Information

Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society