11 problems

Let $G$ be a simply connected group, let $k$ be a global field, let $S$ be a set of valuations, and let $\Gamma=G(\mathcal{O}_S)$. Write $\operatorname{rk}_{k_v}(G)$ for the rank o…

Let $k$ be a number field, let $\tilde{\mathbf{G}}$ be the ambient algebraic group, and let $\Gamma\subseteq\tilde{\mathbf{G}}(k)$ be an arithmetic subgroup. Write $V_{\infty}(k)$…

Let $k$ be a global field, let $S$ be a finite set of places, and let $\Gamma=G(\mathcal{O}_S)$. Write $r_n(\Gamma)$ for the number of irreducible representations of degree $n$, an…

Let … where each $K_i$ is a local field, and let $\Gamma$ be an irreducible lattice in $H$ with $\ell\geq 2$. Serre's conjecture. The lattice $\Gamma$ has the congruence subgroup p…

Let $k$ be a number field, let $S$ be a finite set of places, and let $\mathbf{G}$ be a simply-connected, absolutely almost simple linear $k$-group. An $S$-arithmetic subgroup is a…

Let $G$ be a connected semisimple Lie group with higher rank, finite center, and no compact factors, and let $\Gamma$ be an irreducible lattice in $G$. Serre's congruence subgroup…

Let $\Gamma$ be a lattice in a rank-one group. The Serre conjecture. Such lattices should not have the congruence subgroup property. This conjecture is presented as an obstacle to…

Let $F_{4(-20)}$ be the simple real Lie group of the indicated type, and let $\Gamma$ be a lattice in $F_{4(-20)}$. CSP conjecture for $F_{4(-20)}$. Every lattice $\Gamma$ in the s…

Let $G$ be a semisimple Lie group and let $\Gamma\le G$ be an irreducible arithmetic lattice. Write $\operatorname{rank}_{\mathbb{R}}(G)$ for the real rank of $G$, and let CSP deno…

Let $k$ be a global field, ${\cal O}$ its ring of integers, $S$ a finite set of valuations of $k$ including all archimedean ones, and ${\cal O}_S$ the ring of $S$-integers. Let $G$…

An arithmetic lattice is a discrete subgroup of finite covolume in $SL_2(\mathbb C)$. Serre's conjecture. Arithmetic lattices in $SL_2(\mathbb C)$ do not have the congruence subgro…