Cohomology of congruence subgroups of 𝑆𝐿₄(ℤ). III

Ash, Avner; Gunnells, Paul; McConnell, Mark

doi:10.1090/S0025-5718-10-02331-8

Cohomology of congruence subgroups of $\mathrm {SL}_4(\mathbb {Z})$. III
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by Avner Ash, Paul E. Gunnells and Mark McConnell PDF

Math. Comp. 79 (2010), 1811-1831 Request permission

Abstract:

In two previous papers we computed cohomology groups $H^{5}(\Gamma _{0} (N); \mathbb {C})$ for a range of levels $N$, where $\Gamma _{0} (N)$ is the congruence subgroup of $\mathrm {SL}_{4} (\mathbb {Z})$ 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 nonzero entries. We also make two conjectures concerning the contributions to $H^{5}(\Gamma _{0} (N); \mathbb {C})$ for $N$ prime coming from Eisenstein series and Siegel modular forms.

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