Paul Gunnell's Course

MSRI 2006 Summer Graduate Workshop in Computational Number Theory

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
Suggested Reading 
Papers for which I'm a (co)author are available at my homepage:
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.
See our section in the Problems Book for lots of projects.