Here are some problems, both computational and theoretical, about computing with cohomology of arithmetic groups in -rank . Many of these are discussed in more detail in the appendix to the book [Ste07]. I thank Avner Ash for suggesting problems (2) and (3), and for many helpful discussions.

- Implement a robust user-friendly program to explore , where is a congruence subgroup of , and is the symmetric space . In particular your program should be able to compute a basis for the cohomology space and compute the action of the Hecke operators. One public version of such a program exists on the net at the homepage of Wilberd van der Kallen, but it's in Pascal and isn't maintained. Nevertheless it might be a good starting point.
- Beef up your program to include local coefficient systems.
- Beef it up even more to include integral and torsion coefficients.
- Distribute your tool to the world by incorporating it into SAGE [SJ05].

- Let and let be a congruence subgroup. Investigate the cohomology of the boundary of the Borel-Serre compactification of with coefficients or .
- Use the algorithm in [Gun00] to compute the Hecke action on of the boundary.
- Extend these computations to other s.

- Compute these spaces, perhaps using the tool you developed in Problem (1). Get data.
- Develop a combinatorial model for these spaces analogous to the modular symbol model for the top degree cohomology. (This might involve generalizing the Tits building in a nontrivial way.)
- Explain how to compute the Hecke action on the space spanned by the modular symbols using your model.
- Investigate generalized modular symbols on for .
- There's no reason to stop with . Investigate generalized modular symbols computationally on subgroups of , perhaps using the retract of MacPherson-McConnell [MM93], cf. [Ste07, A.6.4].

- Generalize the algorithm in [Gun00], which computes the
Hecke action on
(at least for
), to
deeper cohomology groups. Run tests similar to those in
[Gun00] to tweak and polish your algorithm.
- Perform computations with your algorithm. A natural place to start is of subgroups of with coefficients. By work of Ash [Ash92], such cohomology classes are connected to abelian Galois representations. Another check is of subgroups of . The results there could be compared to [AGM02]. For new results, you could apply your algorithm to of subgroups of .

- Investigate the connections between the different notions of perfect quadratic forms in the literature (cf. [Ste07, A.6.2]).
- Can the computational data from the work of Baeza-Coulangeon-Icaza-O'Ryan [BCIO01] be used to construct (real) dimensional deformation retracts of Hilbert modular varieties that can be used to compute cohomology?
- If not deformation retracts, can you use the data to construct cell complexes with actions of Hilbert modular groups that can then be used to compute cohomology?

- Study the geometry and combinatorics of the retract for [MM93,MM89]. Use the retract to compute cohomology of subgroups of with various coefficients.
- Can you characterize the sets of vectors that parameterize cells in the retract, analogous to Vorono's characterization for the retract?
- Can you define a retract for ?