81 problems

Let $L$ be a lattice of signature $(n,2)$, let $X$ be the associated orthogonal Shimura variety, and for $1\leq g\leq n$ let $\mathcal Z(M)$ be the codimension-$g$ special cycle as…

Let $E/\mathbb{Q}$ be an elliptic curve. An elliptic curve over $\mathbb{Q}$ is modular if there exists a primitive Hilbert modular newform over $\mathbb{Q}$ of parallel weight $2$…

Modularity conjecture. Any rigid Calabi–Yau threefold $X$ defined over $\mathbb{Q}$ is modular: its $L$-series $L(X,s)$ coincides up to finitely many Euler factors with the Mellin…

Let $X$ be a Shimura variety of Type I or Type IV, and let $X\hookrightarrow X^\Sigma$ be a smooth toroidal compactification. Let $\overline{\mathcal Z}(m)$ denote the Zariski clos…

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 $\mathcal{A}/\mathbb{Q}$ be an abelian surface of conductor $N$ with $\operatorname{End}_{\mathbb{Q}}\mathcal{A}=\mathbb{Z}$. Let $F$ be a weight $2$ cuspidal paramodular newfo…

Modularity conjecture. The counting matrix $a_{ij}$ of a generic Fano threefold in the Iskovskikh family with parameters $(N,d)$ is in the same class as the corresponding matrix in…

Let a family of Calabi–Yau threefolds be given, and consider the action of the Galois group on the relevant third cohomology of its members. A splitting of this Galois action into…

Conjecture 3. The variety $\tilde{W}_2$ is modular. One can isolate a two-dimensional piece $U$ in its third cohomology group whose good Euler factors coincide with those of the ne…

The paper studies deformation rings of two-dimensional Galois representations. In particular, these rings parametrize potentially Barsotti–Tate Galois representations and are used…

A Calabi–Yau threefold $X$ is a smooth projective threefold over $\mathbb{Q}$ with trivial canonical bundle and … It is rigid if … For such an $X$ defined over $\mathbb{Q}$, let…

Let $A$ be an abelian variety of ${\mathrm{GL}}_2$-type over ${\mathbb Q}$, meaning that its rational endomorphism algebra contains a number field of degree equal to ${\rm dim} A$.…

Generalized Shimura–Taniyama–Weil conjecture. Under these circumstances, $J(C)$ should be modular. The conjecture predicts modularity from the isomorphism between the abelian varie…

Let $E$ denote the Frey elliptic curve associated with a putative nontrivial solution of Fermat's equation. Serre's epsilon-conjecture. The conjecture that $E$ is not modular was n…

Let $E$ be an elliptic curve over $\mathbb Q$, and let $f$ be a suitable cusp form with completed $L$-function $\Lambda(f,s)$. Weil–Shimura–Taniyama conjecture. The zeta function o…

The modularity conjecture for abelian varieties of GL2-type. Every abelian variety of GL2-type is a simple factor of $J_0(N)$ for a suitable $N$.

Strict Nash equilibrium conjecture. An amodular graph cannot have an equilibrium partition associated with a strict Nash equilibrium.

Let $\mathcal M$ be a PEL-type Shimura variety with special cycles $\mathcal Z(m)$ and closures $\overline{\mathcal Z}(m)$ in a compactification. Generalized Kudla compactification…

Let $1<r\leq n/2$, let $\mathcal Z_r(m)$ be the special cycles on $\mathcal A_n$ associated with principally polarized abelian varieties of dimension $r$ as isogeny factors, and de…

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

Modularity conjecture at $\phi=-1/7$. The semisimplification of the representation is isomorphic to

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…