Computing the arithmetic genus of Hilbert modular fourfolds

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 $\Gamma _k=\mbox {PSL}_2({\mathcal O}_k)$ 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.