Mathematics of Computation
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)\H-4, where Gamma(k) = PSL2(O-k) acts on H-4 by fractional linear transformations via the four embeddings of k into 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.
First published in Mathematics of Computation in 2006, published by the American Mathematical Society.
Grundman, H.G., and L.E. Lippincott. "Computing the Arithmetic Genus of Hilbert Modular Fourfolds." Math. Comp. 75 (2006): 1553-1560.