1,121 problems

Let … be the unique normalized cusp form of weight $12$ for $SL(2,\mathbb{Z})$, with Fourier coefficients $\tau(n)$. Lehmer's conjecture. … The conjecture asserts that the Ramanuja…

Let $A_0$ be the optimal elliptic curve in the $\mathbb{Q}$-isogeny class of an elliptic curve $A$, and let $\pi_{A_0}:X_0(N)\to A_0$ be its modular parametrization. The Manin cons…

Let $f\in S_k(\Gamma_1(N))$ be a newform, let $\mathfrak{p}$ be a prime such that the associated mod $\mathfrak{p}$ representation $\rho_{f,\mathfrak{p}}$ is irreducible, and let…

Let $S_k(\mathrm{SL}_2(\mathbb{Z}))$ be the space of cusp forms of weight $k$ and level $1$, and let the Hecke polynomial denote the characteristic polynomial of a Hecke operator o…

Nonvanishing conjecture. The first Fourier coefficient satisfies

Let $A$ be a rational symmetric positive-definite $r\times r$ matrix. For $B\in\mathbb{Q}^r$ and $C\in\mathbb{Q}$, define … A triple $(A,B,C)$ is called modular when $f_{A,B,C}(q)$…

Zariski-density conjecture. If $C$ contains an irreducible one-dimensional point, then the set of modular points contained in $C$ is Zariski dense.

Let $f\in S_k(N)$ be a non-CM newform of weight $k\ge 4$, and let $a(p)$ denote its normalized Fourier coefficient at a prime $p$. Atkin–Serre conjecture. For every $\varepsilon>0$…

Fix a positive integer $N$ and a prime $p mid N$. For each integer $k$, let $M{k,N}=Mk(Gamma1(N)capGamma0(p))$, let $Up$ be the Hecke operator on this space, and let $P{k,N}(X)$ be…

Thompson's conjecture. There is an infinite-dimensional graded $\mathbb{M}$-module $V$ whose graded dimension is $J(\tau)$ and whose McKay–Thompson series $T_g(\tau)$ are distingui…

Kac–Wakimoto's conjecture. These are all irreducible highest weight $\mathfrak{g}$-modules $L(\Lambda)$ with this property.

Let $(X,H)$ be a smooth polarized surface with $H_1(X,\mathbb{Z})=0$ and $h^{2,0}(X)>0$. Let $r$ be prime and let $c_1\in H^2(X,\mathbb{Z})$ be algebraic. Let…

Modularity conjecture. The function

Norton's generalized moonshine conjecture. For every commuting pair $(g,h)$ in $\mathbb M$, there exists such a genus-zero modular function $Z(g,h,\tau)$; explicitly, if…

For a positive integer $n$, let $\overline{p}(n)$ denote the number of overpartitions of $n$. A Ramanujan-type congruence is a congruence of the form…

Let $E$ be an elliptic curve over ${\mathbb Q}$. A curve is modular if it is parametrized by a modular form. Taniyama–Shimura–Weil conjecture. Every elliptic curve over…

Let $K_\infty$ be the relevant infinite extension and let the signed Selmer groups constructed using Wach modules be the signed Selmer groups attached to a modular form that is non…

Let $T_r=A_{2r}/\mathbb{Z}_2$ be the tadpole Dynkin diagram, and let $\mathbf{T}_r$ be its Cartan matrix. For $\mathbf{x}_r=(x_1,\ldots,x_r)$, define the tadpole Nahm sum … The sum…

For four-dimensional supersymmetric Yang–Mills theories with $\mathcal{N}=4$ supersymmetry and gauge groups $U(N)$ of arbitrary rank, the associated Vafa–Witten invariants are defi…

Let $d_k(n)$ be defined by the generating function … In particular, $d_2(n)$ counts the relevant 2-elongated plane partitions. Andrews–Paule conjecture. For all integers $n\geq 0$…

Let $t_o(n)$ be defined by … where $q$ is a complex number with $|q|<1$ and $(a;q)_n$ denotes the standard $q$-Pochhammer symbol. Andrews–El Bachraoui's conjecture. If $8n+9$ has a…

Let $N$ be a fixed level, and let $\mathbf{NCM}(N,k)$ denote the number of non-CM Galois orbits of newforms of weight $k$ and level $N$. Tsaknias's generalized Maeda conjecture. Th…

Let $F$ be a totally real field, let $p$ be a prime, and let $\rho$ be a mod $p$ Galois representation. For a regular weight $(k,l)$, geometric modularity means that $\rho$ is modu…

Let a prime-detecting quasimodular form be a quasimodular form whose Fourier coefficients detect primes. A quasimodular Eisenstein series is a quasimodular form in the Eisenstein s…

Let $Q\colon\mathbb{Z}^d\to\mathbb{Z}$ be positive definite and unimodular, let $b$ be the corresponding even bilinear form, and let $\Theta_Q=\Theta_b$ be its theta function. Let…