The cuspidal cohomology of GL3/ℚ and cubic fields

Abstract

We investigate the subspace of the homology of a congruence subgroup $\mathrm{\Gamma }$ of ${\mathrm{\text{SL}}}_3(\mathbb{Z})$ with coefficients in the Steinberg module $\mathrm{\text{St}}({\mathbb{Q}}^3)$ which is spanned by certain modular symbols formed using the units of a totally real cubic field $E$. By Borel–Serre duality, $H_0(\mathrm{\Gamma },\mathrm{\text{St}}({\mathbb{Q}}^3))$ is isomorphic to $H^3(\mathrm{\Gamma },\mathbb{Q})$. The Borel–Serre duals of the modular symbols in question necessarily lie in the cuspidal cohomology $H_{\mathrm{\text{cusp}}}^3(\mathrm{\Gamma },\mathbb{Q})$. Their span is a naturally defined subspace $C(\mathrm{\Gamma },E)$ of $H_{\mathrm{\text{cusp}}}^3(\mathrm{\Gamma },\mathbb{Q})$. Using a computer, we study where $C(\mathrm{\Gamma },E)$ sits between $0$ and $H_{\mathrm{\text{cusp}}}^3(\mathrm{\Gamma },\mathbb{Q})$. On the basis of our computations, we conjecture that ${\sum }_{E}C(\mathrm{\Gamma },E)=H_{\mathrm{\text{cusp}}}^3(\mathrm{\Gamma },\mathbb{Q})$, and we raise the question as to whether for each $E$ individually it might always be true that $C(\mathrm{\Gamma },E)=H_{\mathrm{\text{cusp}}}^3(\mathrm{\Gamma },\mathbb{Q})$.