Overview ======== In this short course we will study techniques to perform explicit computations the cohomology of arithmetic groups. The focus is on groups of Q-rank >= 1, with the chief example being finite index subgroups of SL(n,Z), n >= 3. The tools used generalize some topics discussed by Stein in his lectures, and complement some topics discussed by Dembele and Kohel. Course Topics ============= 1. Basics about cohomology of arithmetic groups and its relationship with modular forms and automorphic forms. 2. Topological tools to compute cohomology of arithmetic groups explicitly, and their relationship to modular symbols on SL(2,Z). 3. Tools to compute the action of the Hecke operators on the cohomology. Suggested Reading ================= Papers for which I'm a (co)author are available at my homepage: http://www.math.umass.edu/~gunnells/pubs/publications.html A general reference (start here): 1. P. E. Gunnells, _Computing in higher rank_, an appendix for the book _Computing with modular forms_, by W. Stein. References about computing Hecke operators in various contexts: 2. A. Ash and L. Rudolph, _The modular symbol and continued fractions in higher dimensions_, Invent. Math. 55 (1979), no. 3, 241-250. 3. P. E. Gunnells, _Computing Hecke eigenvalues below the cohomological dimension_, J. of Experimental Mathematics 9 (2000), no. 3, 351-367. 4. P. E. Gunnells and M. McConnell, _Hecke operators and Q-groups associated to self-adjoint homogeneous cones_, J. Number Theory 100 (2003), no. 1, 46-71. 5. P. E. Gunnells, _Symplectic modular symbols_, Duke Math. J. 102 (2000), no. 2, 329-350. References about computing cohomology groups with Hecke action: 6. A. Ash, P. E. Gunnells, and M. McConnell, _Cohomology of congruence subgroups of SL(4,Z)_, J. Number Theory 94 (2002), no. 1, 181-212. 7. A. Ash, D. Grayson, and P. Green, _Computations of cuspidal cohomology of congruence subgroups of SL(3, Z)_, J. Number Theory 19 (1984), no. 3, 412-436. References about relating cohomology to Galois representations: 8. Ash, Avner; McConnell, Mark, _ Experimental indications of three-dimensional Galois representations from the cohomology of SL(3,Z)_. Experiment. Math. 1 (1992), no. 3, 209-223. 9. Ash, Avner; Sinnott, Warren, _ An analogue of Serre's conjecture for Galois representations and Hecke eigenclasses in the mod p cohomology of GL(n,Z)_. Duke Math. J. 105 (2000), no. 1, 1-24. 10. B. van Geemen, W. van der Kallen, J. Top, and A. Verberkmoes, _Hecke eigenforms in the cohomology of congruence subgroups of SL(3, Z)_, Experiment. Math. 6 (1997), no. 2, 163-174. 11. B. van Geemen and J. Top, _A non-selfdual automorphic representation of GL(3) and a Galois representation_, Invent. Math. 117 (1994), no. 3, 391-401. Projects ======== See our section in the Problems Book for lots of projects.