74 research outputs found
COHOMOLOGY OF CONGRUENCE SUBGROUPS OF SL4(Z). III
In two previous papers we computed cohomology groups for a range of levels , where is the congruence subgroup of consisting of all matrices with bottom row congruent to mod . In this note we update this earlier work by carrying it out for prime levels up to . This requires new methods in sparse matrix reduction, which are the main focus of the paper. Our computations involve matrices with up to 20 million nonzero entries. We also make two conjectures concerning the contributions to for prime coming from Eisenstein series and Siegel modular forms
Robert MacPherson and arithmetic groups
We survey contributions of Robert MacPherson to the theory of arithmetic
groups. There are two main areas we discuss: (i) explicit reduction theory for
Siegel modular threefolds, and (ii) constructions of compactifications of
locally symmetric spaces. The former is joint work with Mark McConnell, the
latter with Lizhen Ji.Comment: Dedicated to Robert MacPherson on the occasion of his 60th birthda
Torsion in the cohomology of congruence subgroups of SL(4,Z) and Galois representations
We report on the computation of torsion in certain homology theories of
congruence subgroups of SL(4,Z). Among these are the usual group cohomology,
the Tate-Farrell cohomology, and the homology of the sharbly complex. All of
these theories yield Hecke modules. We conjecture that the Hecke eigenclasses
in these theories have attached Galois representations. The interpretation of
our computations at the torsion primes 2,3,5 is explained. We provide evidence
for our conjecture in the 15 cases of odd torsion that we found in levels up to
31
Modular forms and elliptic curves over the cubic field of discriminant -23
Let F be the cubic field of discriminant -23 and let O be its ring of
integers. By explicitly computing cohomology of congruence subgroups of
GL(2,O), we computationally investigate modularity of elliptic curves over F.Comment: Incorporated referee's comment
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