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Computing the arithmetic genus of Hilbert modular fourfolds

Authors: H. G. Grundman and L. E. Lippincott
Journal: Math. Comp. 75 (2006), 1553-1560
MSC (2000): Primary 11F41, 14E08; Secondary 14J10, 14J35
Published electronically: March 21, 2006
MathSciNet review: 2219045
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Abstract: The Hilbert modular fourfold determined by the totally real quartic number field $ k$ is a desingularization of a natural compactification of the quotient space $ \Gamma_k \backslash {\mathcal H}^4$, where $ \Gamma_k=$PSL$ _2({\mathcal O}_k)$ acts on $ {\mathcal H}^4$ by fractional linear transformations via the four embeddings of $ k$ into $ \bf R$. The arithmetic genus, equal to one plus the dimension of the space of Hilbert modular cusp forms of weight $ (2,2,2,2)$, is a birational invariant useful in the classification of these varieties. In this work, we describe an algorithm allowing for the automated computation of the arithmetic genus and give sample results.

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Additional Information

H. G. Grundman
Affiliation: Bryn Mawr College, 101 N. Merion Ave., Bryn Mawr, Pennsylvania 19010

L. E. Lippincott
Affiliation: Bryn Mawr College, 101 N. Merion Ave., Bryn Mawr, Pennsylvania 19010

Received by editor(s): April 23, 2004
Received by editor(s) in revised form: May 10, 2005
Published electronically: March 21, 2006
Additional Notes: The first author wishes to acknowledge the support of the Faculty Research Fund of Bryn Mawr College.
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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