Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The Siegel modular variety of degree two and level three


Authors: J. William Hoffman and Steven H. Weintraub
Journal: Trans. Amer. Math. Soc. 353 (2001), 3267-3305
MSC (2000): Primary 11F75; Secondary 11F46, 14G35, 14J30
DOI: https://doi.org/10.1090/S0002-9947-00-02675-1
Published electronically: September 21, 2000
MathSciNet review: 1828606
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract:

Let $\mathcal{A}_{2}(n)$ denote the quotient of the Siegel upper half space of degree two by $\Gamma_{2}(n)$, the principal congruence subgroup of level $n$in $\mathbf{Sp}(4,\mathbf{Z})$. $\mathcal{A}_{2}(n)$ is the moduli space of principally polarized abelian varieties of dimension two with a level $n$structure, and has a compactification $\mathcal{A}_{2}(n)^{\ast}$ first constructed by Igusa. When $n\ge 3$ this is a smooth projective algebraic variety of dimension three.

In this work we analyze the topology of $\mathcal{A}_{2}(3)^{\ast}$ and the open subset $\mathcal{A}_{2}(3)$. In this way we obtain the rational cohomology ring of $\Gamma_{2}(3)$. The key is that one has an explicit description of $\mathcal{A}_{2}(3)^{\ast}$: it is the resolution of the 45 nodes on a projective quartic threefold whose equation was first written down about 100 years ago by H. Burkhardt. We are able to compute the zeta function of this variety reduced modulo certain primes.


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

  • 1. Artin, M. and Mumford, D. Some elementary examples of unirational varieties which are not rational, Proc. London Math. Soc. (3) 25 (1972), 75 -95. MR 48:299
  • 2. Ash, A. Mumford, D. Rappoport, M. and Tai, Y. ``Smooth compactifications of locally symmetric varieties'', Math. Sci. Press, 1975. MR 56:15642
  • 3. Baily, W. L. and Borel, A. Compactification of arithmetic quotients of bounded symmetric domains, Ann. of Math. 84 (1966) 442 - 528. MR 35:6870
  • 4. Baker, H. F. ``A locus with 25920 linear self-transformations'', Cambridge Tracts in Mathematics and Mathematical Physics, no. 39, Cambridge University Press, 1946. MR 8:400b
  • 5. Borel, A. Stable and real cohomology of arithmetic groups, Ann. Sci. Ec. Norm. Sup. 7 (1974), 235 - 272 = Collected Works of A. Borel, III, Springer - Verlag (1983), 315 - 352. (II). In: J. Hano et al. (ed.) Manifolds and Lie Groups (Notre Dame, IN, 1980; J. Hano et al., eds.), Progr. Math. 14, Birkhauser, 1981, pp. 21 - 55 = Collected Works of A. Borel, III, Springer - Verlag (1983), 650 - 684. MR 52:8338; MR 83h:22023; MR 85:01027c
  • 6. Borel, A., Carter, R., Curtis, C. W., Iwahori, N., Springer, T. A. and Steinberg, R. Seminar on Algebraic Groups and Related Finite Groups, Lecture Notes in Math. 131, Springer - Verlag 1970. MR 41:3186
  • 7. Borel, A. and Serre, J. - P. Corners and arithmetic groups, Comment. Math. Helv. 48 (1973) 436 - 491 = Collected Works of A. Borel, III, Springer - Verlag, 1983, 244 - 299. MR 52:8337; MR 85:01027c
  • 8. Borel, A. and Wallach, N. Continuous cohomology, discrete subgroups, and representations of reductive groups, Ann. of Math. Studies 94, Princeton U. Press, 1980. MR 83c:22018
  • 9. Burkhardt, H. Beiträge zur Theorie der hyperelliptischen Sigmafunctionen, Math. Ann. 32 (1888) 381 - 442; Grundzüge einer allgemeinen Systematik der hyperelliptischen Funktionen I. Ordnung. Nach Vorlesung von F. Klein, Math. Ann. 35, 1889, 198 - 296; Untersuchungen aus dem Gebiet der hyperelliptischen Modulfunctionen (I), Math. Ann. 36 (1890) 371 - 434; (II), Math. Ann. 38 (1891) 161 - 224; (III), Math. Ann. 41 (1893) 313 - 343; Ueber einen fundamentalen Satz der Lehre von den endlichen Gruppen linearer Substitutionen, Math. Ann. 41 (1893) 309 - 312.
  • 10. Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A., Wilson, R. A. Atlas of finite groups, Oxford University Press, Oxford, 1985. MR 88g:20025
  • 11. de Jong, A. J., Shepherd-Barron, N. I. and van de Ven, A. On the Burkhardt quartic, Math. Ann. 286 (1990) 309 - 328. MR 91f:14038
  • 12. Deligne, P. Théorie de Hodge II, III, Publ. IHES 40 (1972), 5 - 57 and 44 (1975), 6 - 77. MR 58:16653a; MR 58:16653b
  • 13. Deligne, P., Beilinson, A. A. and Bernstein, J. Faisceaux pervers, Astérisque 100 (1982). MR 86g:32015
  • 14. Digne, F., and Michel, J., Representations of Finite Groups of Lie Type, London Mathematical Society Student Texts 21, Cambridge University Press, 1991. MR 92g:20063
  • 15. Durfee, Alan H. Intersection homology Betti numbers, Proc. Amer. Math. Soc. 123 (1995), 989-993. MR 95e:14014
  • 16. Faltings, G. and Chai, C. L. Degenerations of abelian varieties, Springer-Verlag, 1990. MR 92d:14036
  • 17. Franke, H. Hilbert modulflächen in Siegelscher modulvarietäten, Bonner Math. Schrft.
  • 18. Harder, G. A Gauss-Bonnet formula for discrete arithmetically defined groups, Ann. Sci. Éc. Norm. Sup. (4) 4 (1971) 409 - 455. MR 46:8255
  • 19. Hartshorne, R. Algebraic Geometry, Springer - Verlag, 1977. MR 57:3116
  • 20. Heidrich, H. and Knöller, F. W. Über die Fundamentalgruppen Siegelscher Modulvarietäten vom Grade 2, Manus. Math. 57 (1987) 249 - 262. MR 88e:11033
  • 21. Hoffman, J. W. The zeta function of Burkhardt's quartic, (1995) .
  • 22. Hoffman J. W. and Weintraub, S. H. Cohomology of the Siegel modular group of degree two and level four, Mem. Amer. Math. Soc., 133 (1998), no. 631. MR 98j:11039
  • 23. Hoffman J. W. and Weintraub, S. H. Four-dimensional symplectic geometry over the field with three elements, and a moduli space of abelian surfaces, in preparation.
  • 24. Hoffman J. W. and Weintraub, S. H. Cohomology of the boundary of Siegel modular varieties of degree two, with applications, in preparation.
  • 25. Howlett, R. B., and Lehrer, G. I., Induced cuspidal representations and generalized Hecke rings, Invent. Math. 58 (1980), 37 - 64. MR 81j:20017
  • 26. Hulek, K., Kahn, C. and Weintraub, S. H. Moduli spaces of abelian surfaces: Compactifications, degenerations and theta functions, Walter de Gruyter, Berlin, New York, (1993). MR 95e:14034
  • 27. Hunt, B., The geometry of some special arithmetic quotients of bounded symmetric domains, Lecture Notes in Math 1637, 1996. MR 98c:14033
  • 28. Hunt, B. and Weintraub, S. H. Janus-like algebraic varieties, J. Differential Geom. 39 (1994) 509 - 557. MR 95e:14026
  • 29. Igusa, J. -I. A desingularization problem in the theory of Siegel modular functions, Math. Ann. 168 (1967), 228 - 260. MR 36:1439
  • 30. Kazhdan, D. Connection of the dual space of a group and the structure of its closed subgroups, Funct. Anal. Appl. 1 (1967), 63 - 65. MR 35:288
  • 31. Kottwitz, R. Points on some Shimura varieties over finite fields, J. Amer. Math. Soc. 5 (1992), 373 - 444. MR 93a:11053
  • 32. Lee, R. and Weintraub, S. H. Cohomology of a Siegel modular variety of degree 2, Groups Acting on Manifolds (Boulder, CO, 1983; R. Schultz, ed.), Contemp. Math. 36 (1985), 433 - 488. MR 87g:11056
  • 33. Lee, R. and Weintraub, S. H. Cohomology of $Sp_4 ({\mathbf Z})$ and related groups and spaces, Topology 24 (1985), 391 - 410. MR 87b:11044
  • 34. Lee, R. and Weintraub, S. H. Topology of the Siegel spaces of degree two and their compactifications, Topology Proceedings 11 (1986), 115 - 175 . MR 89c:32071
  • 35. Lee, R. and Weintraub, S. H. The Siegel modular variety of degree two and level four: a report, in Arithmetic of Complex Manifolds, W. - P. Barth and H. Lange, eds., Lecture Notes in Mathematics 1399, Springer Verlag 1989. MR 90k:11061
  • 36. Lee, R. and Weintraub, S. H. The Siegel modular variety of degree two and level four, Mem. Amer. Math. Soc., 133 (1998), no. 631. MR 98i:57002
  • 37. Looijenga, E. $L^2$-cohomology of locally symmetric varieties, Compos. Math. 67 (1987), 3 - 20. MR 90a:32044
  • 38. Lusztig, G. Representations of Finite Chevalley Groups, Conf. Board Math. Sci. Regional Conf. Ser. Math. 39, Amer. Math. Soc., Providence, RI, 1978. MR 80f:20045
  • 39. MacPherson, R. and McConnell. M. Explicit reduction theory for Siegel modular threefolds, Invent. Math. 111 (1993), 575 - 625. MR 94a:32052
  • 40. Maschke, H. Über die lineare Gruppe der Borchardt'schen Moduln, Math. Ann. 31 (1887), 496 - 515; Aufstellung des vollen Formensystems einer quaternären Gruppe von 51840 lineare Substitutionen, Math. Ann. 33 (1889), 317 - 344.
  • 41. Namikawa, Y. ``Toroidal compactifications of Siegel spaces'', Springer Lecture Notes 812, 1980. MR 82a:32034
  • 42. Oda, T. and Schwermer, J. Mixed Hodge structures and automorphic forms on Siegel modular varieties of degree two, Math. Ann. 286 (1990), 481 - 509. MR 90m:11072
  • 43. Ohsawa, T. On the $L^2 $ cohomology of complex spaces, Math. Z., 209 (1992), 519 - 530. MR 93d:32063
  • 44. Saito, M. Mixed Hodge modules and applications. Proceeding of the International Congress of Mathematicians, Kyoto, Springer - Verlag (1991), 725 - 734. MR 93d:32059
  • 45. Sankaran, G. K. Fundamental group of locally symmetric varieties, Manus. Math. 90 (1996), 39 - 48. MR 97b:14017
  • 46. Saper, L. and Stern M. $L^2$-cohomology of arithmetic varieties. Proc. Nat. Acad. Sci. USA 84 (1987), 5515 - 5519. MR 89g:32052
  • 47. Satake, I. On the compactification of the Siegel space, J. Indian Math. Soc. 20 (1956), 259 - 281. MR 18:934c
  • 48. Serre, J.-P. Facteurs locaux des fonctions zêta des variétés algebriques (définitions et conjectures), Séminaire Delange - Pisot - Poitou (1969/70) , Exposé 19, = Collected Papers II, 581 - 592. MR 53:5224; MR 89h:01009b
  • 49. Serre, J. -P. Motifs, Astérisque, 198 - 199 - 200 (1991), 333 - 349. MR 92m:14002
  • 50. Schwermer, J. On arithmetic quotients of the Siegel upper half space of degree two, Compositio Math., 58 (1986), 233 - 258. MR 87j:11040
  • 51. Shioda, T. On elliptic modular surfaces, J. Math. Soc. Japan, 24 (1972), 20 - 59. MR 55:2927
  • 52. Siegel, C. - L. Symplectic Geometry, Academic Press, New York, 1964. MR 29:1362
  • 53. Srinivasan, B. The characters of the finite symplectic group $\operatorname{Sp} (4, q)$, Trans. Am. Math. Soc. 131 (1963), 488 - 525. MR 36:3897
  • 54. van der Geer, G. Note on abelian schemes of level three, Math. Ann. 278 (1987), 401 - 408. MR 89a:14053
  • 55. Weintraub, S. H. Letter to Bruce Hunt, Aug. 16, 1987.
  • 56. Weissauer, R. Differentialformen zu Untergruppen der Siegelschen Modulgruppe zweiten Grades, J. Reine Angew. Math. 391 (1988), 100 - 156. MR 89i:32074
  • 57. Weissauer, R. On the cohomology of Siegel modular threefolds, In Arithmetic of Complex Manifolds, W. - P. Barth and H. Lange (Eds.), Lecture Notes in Math. 1399, Springer - Verlag, 1989, 155 - 170. MR 91e:11051
  • 58. Weissauer, R. The Picard Group of Siegel modular threefolds, J. Reine Angew. Math. 430 (1992), 179 - 211. MR 94e:11052
  • 59. Yamazaki, T. On Siegel modular forms of degree two, Amer. J. Math. 98 (1976), 39 - 53. MR 53:8517

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 11F75, 11F46, 14G35, 14J30

Retrieve articles in all journals with MSC (2000): 11F75, 11F46, 14G35, 14J30


Additional Information

J. William Hoffman
Affiliation: Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803
Email: hoffman@math.lsu.edu

Steven H. Weintraub
Affiliation: Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803
Email: weintr@math.lsu.edu

DOI: https://doi.org/10.1090/S0002-9947-00-02675-1
Received by editor(s): March 29, 1999
Published electronically: September 21, 2000
Additional Notes: The first named author would like to thank Meijo University in Nagoya, Japan, for its generous hospitality. Part of this work was done while visiting there.
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society