36 problems

Virtual modularity conjecture. The series $\Theta^{2,\mathrm{vir}}_{\mathcal A_n}$ is a holomorphic Siegel modular form of weight $2n$ for $\operatorname{Sp}_4(\mathbb Z)$, valued…

Greer–Lian's modularity conjecture. The series $\Theta_{\mathcal A_n}(q)$ is a holomorphic modular form of weight $2n$ for $\operatorname{SL}_2(\mathbb Z)$, valued in…

Modularity conjecture at $\phi_{\pm}$. The semisimplification of the representation of $G_K$ is isomorphic to the restriction to $G_K$ of

Let $K$ be a number field, let $p$ be a rational prime unramified in $K$, and let $G_K$ denote the absolute Galois group of $K$. Let…

Prokhorenkova et al.'s modularity conjecture. With high probability,

Let $\sigma_2(q/p)$ denote the genus-two signature at a rational number $q/p$. For coprime odd integers $0<q<p$, consider the transformed fraction $q/(2q+p)$. Modular transformatio…

Let $K$ be a totally real field and let $p\geq3$ be prime. A continuous representation … is geometric if it satisfies the geometricity conditions of Fontaine and Mazur. The Fontain…

Equivalent modularity criteria. The following statements are equivalent:

Let $\mathcal{C}_p$ be the vertex operator algebra considered in the paper. A vertex operator algebra is quasi-lisse when its associated variety has finitely many symplectic leaves…

Let $G_n^h=(V,E)$ be a preferential attachment graph, where $n$ is the number of vertices and $h$ is the number of edges added per step. Write $\operatorname{mod}(G_n^h)$ for its m…

Let $m\le n$ be fixed. Let $X'/X$ be the given double cover, let $\operatorname{U}_{2m}$ be the associated quasi-split unitary group, and let…

Let $K$ be a totally real field and let $\{\rho_{\pi, \lambda}: \mathrm{Gal}_{K} \to \mathrm{GL}_n(\overline E_{\lambda})\}_{\lambda}$ be the compatible system of $K$ defined over…

Integral Galois representation conjecture. There exists a unique continuous representation

Let $\mathrm{M}_2^6$ and $\mathrm{M}_3^{(2,2)}$ be the motives introduced in the paper, and let $(-3)$ denote the Tate twist. Motive-isomorphism conjecture. The two motives … are i…

Let $K/\mathbf{F}_q$ be a function field and let $E/K$ be a nonisotrivial elliptic curve. For $r\geq 1$, $E$ is called $r$-modular when it is $(r,\Sigma_\infty)$-modular for every…

Modularity conjecture. Under the stated assumption, the arithmetic generating series $z(g,\phi)_{{\mathfrak e},{\mathfrak a}}^{\overline{\mathcal L}}$ is holomorphic and modular, w…

For a constant $c>0$, let $G_{n,c/n}$ be the binomial Erdős–Rényi random graph with edge-probability $c/n$, and define … For $0<c\leq1$, the source states that $\bar{q}(n,c)\to1$ a…

Modularity conjecture. Every elliptic curve over a totally real field $F$ is modular.

Let $\mathcal A$ be the set of cuspidal automorphic representations defined above. For $\pi\in\mathcal A$, let … where $S$ ranges over the relevant finite sets and…

Let $E$ be an elliptic curve over the rational numbers. The modularity conjecture. Every elliptic curve is modular. This is a central theorem in the arithmetic of elliptic curves a…

Let $X$ be a smooth, projective, geometrically irreducible curve with function field $K$, and let $E$ be an elliptic curve over $K$ with conductor $N$ and irreducible associated Ga…

Consider each $K3$ hypersurface listed above, after desingularization when necessary. $K3$ modularity conjecture. Each such $K3$ hypersurface is modular with the weight-$3$ newform…

Let $S$ be the surface under consideration, let $c_1(S)$ denote its first Chern class, and let $v_0(S)$ be the null vector specified in the source. Write…

Let $K$ be a quadratic imaginary field of class number one, let $p>3$ be a prime, and let $E$ be an elliptic curve over $K$. Write $G_K$ for the absolute Galois group of $K$, and l…

Let $S$ be a K3 surface with polarization $H$, first Chern class $c_1$, and prime rank $r$. Let $\mathsf Z_{K3,H,r,c_1}^{\mathrm{inst}}(q,y)$ and…