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Moduli spaces of Riemann surfaces of genus two with level structures. I

Authors: Ronnie Lee and Steven H. Weintraub
Journal: Trans. Amer. Math. Soc. 310 (1988), 217-237
MSC: Primary 32J05; Secondary 11F46, 11F75, 11G18, 14H15
MathSciNet review: 965750
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Abstract: The cohomology of modular varieties defined by congruence subgroups of $ {\operatorname{Sp} _4}({\mathbf{Z}})$ whose levels lie between $ 2$ and $ 4$ is studied. Using a counting argument and the techniques of zeta functions, the authors completely determine the cohomology of a particular variety of this type.

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  • [ArM] M. Artin and D. Mumford, Some elementary examples of unirational varieties which are not rational, Proc. London Math. Soc. (3) 25 (1972), 75–95. MR 0321934
  • [AMRT] 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
  • [BS] A. Borel and J.-P. Serre, Corners and arithmetic groups, Comment. Math. Helv. 48 (1973), 436–491. Avec un appendice: Arrondissement des variétés à coins, par A. Douady et L. Hérault. MR 0387495
  • [G] G. van der Geer, On the geometry of a Siegel modular threefold, Math. Ann. 260 (1982), no. 3, 317–350. MR 669299, 10.1007/BF01461467
  • [I] Jun-ichi Igusa, On the graded ring of theta-constants, Amer. J. Math. 86 (1964), 219–246. MR 0164967
  • [K] K. Kodaira, On compact analytic surfaces. II, Ann. of Math. 77 (1963), 563-626.
  • [LW$ _{0}$] R. Lee and S. H. Weintraub, Abstracts #83D-57-26, #83D-22-22, Abstracts Amer. Math. Soc. 4 (1983).
  • [LW$ _{1}$] Ronnie Lee and Steven H. Weintraub, Topology of the Siegel spaces of degree two and their compactifications, Proceedings of the 1986 topology conference (Lafayette, La., 1986), 1986, pp. 115–175. MR 898016
  • [LW$ _{2}$] -, Cohomology of a Siegel modular variety of degree two, Group Actions on Manifolds, R. Schultz, Ed., Amer. Math. Soc., Providence, R. I., 1985, pp. 433-488.
  • [LW$ _{3}$] Ronnie Lee and Steven H. Weintraub, Cohomology of 𝑆𝑝₄(𝑍) and related groups and spaces, Topology 24 (1985), no. 4, 391–410. MR 816521, 10.1016/0040-9383(85)90011-4
  • [Li] W. B. R. Lickorish, A representation of orientable combinatorial 3-manifolds, Ann. of Math. (2) 76 (1962), 531–540. MR 0151948
  • [M$ _{1}$] David Mumford, Stability of projective varieties, Enseignement Math. (2) 23 (1977), no. 1-2, 39–110. MR 0450272
  • [M$ _{2}$] David Mumford, Tata lectures on theta. I, Progress in Mathematics, vol. 28, Birkhäuser Boston, Inc., Boston, MA, 1983. With the assistance of C. Musili, M. Nori, E. Previato and M. Stillman. MR 688651
  • [N] Yukihiko Namikawa, Toroidal compactification of Siegel spaces, Lecture Notes in Mathematics, vol. 812, Springer, Berlin, 1980. MR 584625
  • [Sm] Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo; Princeton University Press, Princeton, N.J., 1971. Kan\cflex o Memorial Lectures, No. 1. MR 0314766
  • [So] Tetsuji Shioda, On elliptic modular surfaces, J. Math. Soc. Japan 24 (1972), 20–59. MR 0429918
  • [Tt] John T. Tate, Algebraic cycles and poles of zeta functions, Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963) Harper & Row, New York, 1965, pp. 93–110. MR 0225778

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Article copyright: © Copyright 1988 American Mathematical Society