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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
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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.

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  • [1] A. Ash, D. Mumford, M. Rapoport, and Y. Tai, Smooth compactification of locally symmetric varieties, Math. Sci. Press, Brookline, Mass., 1975. Lie Groups: History, Frontiers and Applications, Vol. IV. MR 0457437
  • [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] Eberhard Freitag, Holomorphe Differentialformen zu Kongruenzgruppen der Siegelschen Modulgruppe zweiten Grades, Math. Ann. 216 (1975), no. 2, 155–164 (German). MR 0376540,
  • [5] E. Freitag, Siegelsche Modulfunktionen, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 254, Springer-Verlag, Berlin, 1983 (German). MR 871067
  • [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 Eulerscher Produktentwicklung. II, Math. Ann. 114 (1937), no. 1, 316–351 (German). MR 1513142,
  • [8] Jun-ichi Igusa, Theta functions, Springer-Verlag, New York-Heidelberg, 1972. Die Grundlehren der mathematischen Wissenschaften, Band 194. MR 0325625
  • [9] David Mumford, Tata lectures on theta. II, Progress in Mathematics, vol. 43, Birkhäuser Boston, Inc., Boston, MA, 1984. Jacobian theta functions and differential equations; With the collaboration of C. Musili, M. Nori, E. Previato, M. Stillman and H. Umemura. MR 742776
  • [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] Rainer Weissauer, On the cohomology of Siegel modular threefolds, Arithmetic of complex manifolds (Erlangen, 1988) Lecture Notes in Math., vol. 1399, Springer, Berlin, 1989, pp. 155–171. MR 1034263,

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Article copyright: © Copyright 1993 American Mathematical Society

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