Narrow Results

Starting with a numerically noncritical (at $p$) Hecke eigenclass $f$ in the homology of a congruence subgroup $\mathrm{\Gamma }$ of ${\mathrm{\text{SL}}}_3(\mathbb{Z})$ (where $p$ divides the level of $\mathrm{\Gamma }$) with classical coefficients, we first show how to compute to any desired degree of accuracy a lift of $f$ to a Hecke eigenclass $F$ with coefficients in a module of $p$-adic distributions. Then we show how to find to any desired degree of accuracy the germ of the projection $Z$ to weight space of the eigencurve around the point $z$ corresponding to the system of Hecke eigenvalues of $F$. We do this under the conjecturally mild hypothesis that $Z$ is smooth at $z$.

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 $C(\mathrm{\Gamma },E)=H_{\mathrm{\text{cusp}}}^3(\mathrm{\Gamma },\mathbb{Q})$.