Open Access. Powered by Scholars. Published by Universities.®
- Discipline
Articles 1 - 2 of 2
Full-Text Articles in Number Theory
Mod 2 Homology For Gl(4) And Galois Representations, Avner Ash, Paul E. Gunnells, Mark Mcconnell
Mod 2 Homology For Gl(4) And Galois Representations, Avner Ash, Paul E. Gunnells, Mark Mcconnell
Paul Gunnells
We extend the computations in [AGM11] to find the mod 2 homology in degree 1 of a congruence subgroup Γ of SL(4,Z) with coefficients in the sharbly complex, along with the action of the Hecke algebra. This homology group is related to the cohomology of Γ with F2 coefficients in the top cuspidal degree. These computations require a modification of the algorithm to compute the action of the Hecke operators, whose previous versions required division by 2. We verify experimentally that every mod 2 Hecke eigenclass found appears to have an attached Galois representation, giving evidence for a conjecture in …
Resolutions Of The Steinberg Module For Gl(N), Avner Ash, Paul E. Gunnells, Mark Mcconnell
Resolutions Of The Steinberg Module For Gl(N), Avner Ash, Paul E. Gunnells, Mark Mcconnell
Paul Gunnells
We give several resolutions of the Steinberg representation St_n for the general linear group over a principal ideal domain, in particular over Z. We compare them, and use these results to prove that the computations in [AGM4] are definitive. In particular, in [AGM4] we use two complexes to compute certain cohomology groups of congruence subgroups of SL(4,Z). One complex is based on Voronoi's polyhedral decomposition of the symmetric space for SL(n,R), whereas the other is a larger complex that has an action of the Hecke operators. We prove that both complexes allow us to compute the relevant cohomology groups, and …