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Transactions of the American Mathematical Society
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
Published electronically: September 21, 2000
MathSciNet review: 1828606
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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.


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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: http://dx.doi.org/10.1090/S0002-9947-00-02675-1
PII: S 0002-9947(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