66 problems

Let $\Gamma$ be a congruence subgroup in the setting of the Weyl law for the discrete spectrum of a semisimple Lie group. Sarnak's conjecture. The Weyl law holds in complete genera…

Serre's congruence subgroup property conjecture. Every such arithmetic lattice $\Gamma$ has the congruence subgroup property.

Let $G=SO(n,1)$ and let $\Gamma$ be an arithmetic lattice in $G$. Serre's conjecture. The group $\Gamma$ fails the congruence subgroup property. The notes say this conjecture was a…

Let $k$ be a global field, let $S$ be a finite set of places of $k$, and let $\mathfrak{G}$ be a simple simply connected algebraic group over $k$. The $S$-arithmetic group…

Let $f_i(\tau)$ be a component of a vector-valued modular form with Fourier expansion … where $n^{(i)}_0=p_i/N_i$ and $(p_i,N_i)=1$. Suppose that all coefficients $a_n^{(i)}$ are a…

Let $n\geq 3$ and $m$ be a positive integer, and let $\Gamma_{n,m}$ denote the principal congruence subgroup of $\mathrm{SL}(n,\mathbb{Z})$ of level $m$. Lubotzky's conjecture. The…

Let $K$ be an algebraic number field, let $S$ be a finite set of maximal ideals of its ring of integers, and let $\Lambda_S^\times$ be the group of units of an $S$-localized maxima…

Let $\mathcal C$ be a modular tensor category, and let $\rho_{\mathcal C}^{\lambda,1}$ be its modular representation; equivalently, consider the corresponding representation for a…

Let $F$ be a finite field, let $n\ge 3$, and let … be the subgroup of matrices congruent to the identity modulo $t$. Write $H_1(K)$ for its first homology group and…

Grigorchuk–Lubotzky–Pyber conjecture. The two asymptotic constants coincide and equal

Let $\Gamma(1)=SL(2,\mathbb{Z})$ be the modular group, and let the kernel be the normal subgroup of $\Gamma(1)$ whose elements are represented by the identity matrix in the modular…

Fix a nonzero rational number $q=\tfrac{s}{t}\in\mathbb{Q}\cap(-4,4)$ in lowest terms, with $\gcd(s,t)=1$. Let $\Gamma_q$ be the group generated by the root unipotents associated t…

Serre's conjecture. The group $\operatorname{SL}_1(\mathcal{O})$ satisfies the subgroup congruence property.

For $\alpha\in\mathbb{C}$, let … Let $\frac{q}{p}\in\mathbb{Q}\cap(-4,4)$ with $\gcd(p,q)=1$. Write $\pi$ for the relevant reduction homomorphism. Nyberg Brodda's congruence subgro…

Let $\Gamma$ be a non-spherical small Coxeter graph, with associated Artin group $A[\Gamma]$, Coxeter group $W[\Gamma]$, and level-two congruence subgroup $A[\Gamma][2]$. Write…

Let $n\geqslant 3$ and let $\Gamma_n=\mathrm{SL}_n(\mathbf Z)$. For a natural number $m\geqslant 0$, let $\Gamma_n(m)$ be the principal congruence subgroup consisting of matrices c…

Let $\Gamma$ be a finite-index subgroup of $\operatorname{Mcg}(\Sigma)$, and let $\alpha$ be a simple closed curve on $\Sigma$. A subgroup of $\operatorname{Mcg}(\Sigma)$ is congru…

Let $\Sigma$ be a surface of genus at least $3$. A finite-index subgroup of $\operatorname{Mcg}(\Sigma)$ is a subgroup of finite index containing the kernel of the action on a fini…

Let $\Sigma$ be a connected, orientable, hyperbolic surface of finite type, meaning that it is obtained by deleting finitely many points from a closed surface. Write…

Almost-all non-congruence presentations conjecture. As the order of $S$ tends to infinity, the proportion of non-congruence presentations among all presentations tends to $1$:

Congruence kernel conjecture. The projective representation of $\operatorname{SL}(2,\mathbb Z)$ given by $\rho$ has kernel that is a congruence subgroup of level…

Atkin–Lehner invariance conjecture. The action of Atkin–Lehner involutions on $\mathcal{M}_k(\Gamma_1(N))$ restricts to the magnetic subspaces

SL2-action conjecture. For every $\gamma\in\mathrm{SL}_2(\mathbb{Z})$, the action

Let $\Gamma$ be a fixed arithmetic group, let $L$ be a coefficient lattice, and let $\{\Gamma_j\}_{j\in\mathbb{N}}$ be a family of congruence subgroups with…

Let $p$ be prime and $r\in\mathbb{Z}$ satisfy $\gcd(r,p)=1$, with $r/p$ admissible as defined in the source. Let $\Delta_{r/p}$ be generated by the two parabolic matrices $A$ and…