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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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The Siegel modular variety of degree two and level three
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by J. William Hoffman and Steven H. Weintraub PDF
Trans. Amer. Math. Soc. 353 (2001), 3267-3305 Request permission

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
  • MR Author ID: 181515
  • ORCID: 0000-0002-3290-363X
  • Email: weintr@math.lsu.edu
  • 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.
  • © Copyright 2000 American Mathematical Society
  • 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
  • MathSciNet review: 1828606