Cohomology of Congruence Subgroups of SL_4(Z). III

Details Cohomology of Congruence Subgroups of SL_4(Z). III Cohomology of Congruence Subgroups of SL_4(Z). III

2009-03-18

arxiv.org/abs/0903.3201v2

Mathematics

Number Theory

incorporates referees' comments

Scientific paper

In two previous papers [AGM1, AGM2] we computed cohomology groups H^5(\Gamma_0 (N); \C) for a range of levels N, where \Gamma_0 (N) is the congruence subgroup of SL_4 (\Z) consisting of all matrices with bottom row congruent to (0,0,0,*) mod N. In this note we update this earlier work by carrying it out for prime levels up to N = 211. 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 non-zero entries. We also make two conjectures concerning the contributions to H^5(\Gamma_0 (N); \C) for N prime coming from Eisenstein series and Siegel modular forms.

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