12 problems

Atkin–Swinnerton-Dyer basis conjecture. The Atkin–Swinnerton-Dyer basis is $\{H_1,H_2\}$ if $p\not\equiv 5 \mod 12$, and it is $\{H_1+H_2,H_1+(p^2+1)H_2\}$ when $p\equiv 5 \mod 12$…

The type II(A) character-group conjecture. For $\Gamma^0(11)$,

Let $f(\tau)=\sum_{n\geq 1}a(n)q^n$ be a holomorphic modular cusp form on a noncongruence subgroup of $\operatorname{SL}_2(\mathbb Z)$. Atkin–Swinnerton-Dyer conjecture. The Fourie…

A scalar modular form is a holomorphic function on the upper half-plane transforming with a prescribed weight under a subgroup of $\operatorname{SL}_2(\mathbb{Z})$; its Fourier coe…

Let $\Gamma=\operatorname{SL}_2(\mathbb{Z})$, and consider a scalar modular form for a subgroup of $\Gamma$. Say that it has bounded denominator when its Fourier coefficients have…

The 5-adic congruence conjecture. For $p=5$, $m\geq1$, and odd $n\geq1$, one has $c_{5^m}(f)\neq0$ and

Bounded-denominator conjecture. The form $f$ is a congruence form if and only if its Fourier coefficients have bounded denominators.

Bounded-denominator criterion. The distinction between congruence forms, whose algebraic Fourier coefficients have bounded denominators, and genuine noncongruence forms should be a…

Let $\Gamma=SL_2(\mathbb{Z})$, let $G$ be a finite-index subgroup of $\Gamma$, and let $\mathcal{M}_k(G)$ denote the space of modular forms of weight $k$ for $G$. The congruence cl…

Unbounded denominators conjecture. The Fourier coefficients of $f$ have bounded denominators if and only if $f$ is a cusp form for a congruence subgroup.

Let $p$ be an odd prime, and let … is related to a Hilbert modular form. The source gives no further specification of the relation or resolution status; the conjecture is motivated…

Let … ; suppose the cusp at … -rational with cusp width … , let $S_{}$ have dimension … , the space $S_{}$ has a $p$-adically integral basis $f_j(z)=$, depending on $p$, whose Four…