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Full-Text Articles in Physical Sciences and Mathematics
Cohomology Of Congruence Subgroups Of Sl4(Z). Iii, A Ash, Pe Gunnells, M Mcconnell
Cohomology Of Congruence Subgroups Of Sl4(Z). Iii, A Ash, Pe Gunnells, M Mcconnell
Mathematics and Statistics Department Faculty Publication Series
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.
Cohomology Of Congruence Subgroups Of Sl(4, Z) Ii, A Ash, Pe Gunnells, M Mcconnell
Cohomology Of Congruence Subgroups Of Sl(4, Z) Ii, A Ash, Pe Gunnells, M Mcconnell
Mathematics and Statistics Department Faculty Publication Series
In a previous paper [3] we computed cohomology groups H5(..0(N),C), where ..0(N) is a certain congruence subgroup of SL(4,Z), for a range of levels N. In this note we update this earlier work by extending the range of levels and describe cuspidal cohomology classes and additional boundary phenomena found since the publication of [3]. The cuspidal cohomology classes in this paper are the first cuspforms for GL(4) concretely constructed in terms of Betti cohomology.
Lattice Polytopes, Hecke Operators, And The Ehrhart Polynomial, Pe Gunnells, Fr Villegas
Lattice Polytopes, Hecke Operators, And The Ehrhart Polynomial, Pe Gunnells, Fr Villegas
Mathematics and Statistics Department Faculty Publication Series
Let P be a simple lattice polytope. We define an action of the Hecke operators on E(P), the Ehrhart polynomial of P, and describe their effect on the coefficients of E(P). We also describe how the Brion–Vergne formula for E(P) transforms under the Hecke operators for nonsingular lattice polytopes P.
Cohomology Of Congruence Subgroups Of Sl4(Z), A Ash, Pe Gunnells, M Mcconnell
Cohomology Of Congruence Subgroups Of Sl4(Z), A Ash, Pe Gunnells, M Mcconnell
Mathematics and Statistics Department Faculty Publication Series
Let N>1 be an integer, and let Γ=Γ0(N)SL4( ) be the subgroup of matrices with bottom row congruent to (0, 0, 0, *) modN. We compute H5(Γ; ) for a range of N and compute the action of some Hecke operators on many of these groups. We relate the classes we find to classes coming from the boundary of the Borel–Serre compactification, to Eisenstein series, and to classical holomorphic modular forms of weights 2 and 4.