37 problems

Let $N\geq 1$. Write $M(\Gamma_0(N),\mathbb{Z}[\frac{1}{6N}])$ for the graded algebra of modular forms for $\Gamma_0(N)$ with coefficients in $\mathbb{Z}[\frac{1}{6N}]$. Weight-six…

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

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…

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 $\Gamma$ 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…

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 $p$ 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…

Let $p$ be prime, let $r\in\mathbb{Z}$ satisfy $\gcd(r,p)=1$, and suppose that $r/p\in\mathbb{Q}\cap(0,4)$. Let … Let $\overline{\Gamma}_1^{(p)}(r)$ denote the subgroup generated b…

Non-square congruence subgroup conjecture. Both $\Gamma_1(\alpha T+\beta)$ and $\Gamma_0(\alpha T+\beta)$ may be non-square congruence subgroups for any choice of $\alpha\neq 0$ an…

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…

Stronger congruence-subgroup conjecture. Then

Uniform bounded generation conjecture. There is a minimal constant $K:=K(\Phi)$ proportional to ${\rm rank}(\Phi)$ and independent of $R$ and $I$ such that

Let $\vec v(q)$ be a vector-valued modular function with components $v_i(q)$, and write the Fourier expansion of $v_i(q)$ with leading exponent $m_0^{(i)}$ and Fourier coefficients…

Let $N$ be an integer greater than $2$, and let the size of an irreducible solution of $(E_N)$ mean the size notion defined earlier in the paper. Uniform size-bound conjecture. The…

Let $N\in\mathbb{N}^{*}$ with $N\geq 2$. A solution of $(E_N)$ is called irreducible when it cannot be decomposed as a sum of two nontrivial solutions, as in the preceding discussi…

Steinberg-module stability conjecture. For each $i,j\geq 0$, the $\mathrm{VB}_{\mathbb{Z}}/\Gamma(p)$-module

Let $\mathcal{B}$ be a super-modular category of rank $2k$, and let $\hat{S}$ and $\hat{T}^2$ be the corresponding matrices. Let…

Let $\Gamma$ be an arithmetic group, let $\Gamma_k$ range over congruence subgroups of $\Gamma$ of increasing level, and let $\delta$ denote the deficiency of the associated real s…

Let $\mathcal{R}_q$ be the resonance set associated with the congruence subgroup $\Gamma(q)$, and let $\delta$ denote the Hausdorff dimension of the limit set. Uniform essential sp…

Let $M(\Gamma_0(p),\mathbb{Z})$ be the graded algebra of integral modular forms on $\Gamma_0(p)$, and let $T\in M_{p-1}(\Gamma_0(p),\mathbb{Z})$ be the $T$-form. Generator-weight c…