5 problems

Let $E$ be a real quadratic field and let $\Gamma\subset\operatorname{SL}_3(\mathbb{Z})$ be a finite-index subgroup. Let $H(\Gamma,E)$ be the Steinberg-generated subspace of…

Let $E$ be a real quadratic field and let $\Gamma\subset\operatorname{SL}_3(\mathbb{Z})$ be a finite-index subgroup. Let $H(\Gamma,E)$ be the Steinberg-generated subspace, and let…

Let $p$ be a prime and let $\bar{\mathbb{F}}_p$ be an algebraic closure of $\mathbb{F}_p$. Let $\rho:G_{\mathbb{Q}}\to\operatorname{GL}(3,\bar{\mathbb{F}}_p)$ be a Galois represent…

Modularity conjecture. There is a Hecke eigenclass in $H^*(\Gamma_0(N),M_{\mathbf{x}})$ that matches $\rho$.

Let $K\in\mathcal{F}^+$ be unramified outside a finite set of places $\Sigma$, and let $C_K$ be the associated compactly supported arithmetic complex. Generalized Tate-motive main…