10 problems

Let $K$ be a $p$-adic field, let $\mathcal{X}_K$ be smooth and proper over $K$ of pure dimension $d-1$, let $n\in\mathbb Z$, and let $A$ be a locally compact abelian group in…

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 $\Gamma$ be a congruence subgroup of $\operatorname{SL}_2(\mathbb{Z})$, let $E$ be a real quadratic field, and let $H(\Gamma,E)$ denote the image of the connecting homomorphism…

Let $\Gamma$ be a congruence subgroup of $\operatorname{SL}_3(\mathbb{Z})$, let $E$ be a real quadratic field, and let $H(\Gamma,E)$ be the Hecke-stable subspace of…

Let $\pi$ be as in the paper, let $\mathrm T(\mathbb Z_p)_p$ be the relevant torus quotient, let $Y_{1,1}$ be the arithmetic manifold, and let $\rho_\pi$ be its conjectural Galois…

Let $\Gamma$ be the congruence subgroup under consideration and let $\eta$ be the associated coefficient character. A Galois representation attached to a Hecke eigenclass in…

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…