1,077 problems

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…

Vojta's stronger conjecture. For every $\epsilon>0$, there is a proper Zariski-closed subset $Z$ of $X$ such that, for all $c\in\mathbb{R}$,

Let $\phi_\lambda$ be an $L^2$-normalized Laplace eigenfunction on an arithmetic surface, with eigenvalue parameter $\lambda$. Iwaniec's conjecture. For every $\varepsilon>0$, one…

Let $X$ be a smooth projective curve of genus at least two over a number field $K$. Mordell's conjecture. The set $X(K)$ is finite. This foundational theorem is known as Faltings'…

Grothendieck conjecture. The geometry of anabelian schemes over fields finitely generated over $\mathbb{Q}$ is completely determined by their étale fundamental groups; in particula…

Shafarevich's conjecture. For every positive integer $e$, only finitely many lattices occur as $\operatorname{Pic}(\overline{X})$ for K3 surfaces $X$ over number fields $K$ with…

Lang–Vojta conjecture. No set of $S$-integral $k$-rational points on $X\setminus D$ is Zariski dense.

Hasse–Weil conjecture. The function $L(s,C/K)$ has an analytic continuation, and the completed function satisfies

Kato's residue conjecture. The map $\Delta_j$ is an isomorphism for all $r\geq 1$ and $j\geq 0$. This conjecture concerns comparison of Kato homology on a regular proper flat model…

Suppose that … is a tower of covers of congruence arithmetic hyperbolic 3-manifolds whose injectivity radius approaches infinity. Write $H_1(M_i,\mathbb{Z})_{tors}$ for the torsion…

Cassels–Swinnerton-Dyer conjecture. If $X$ has an $L$-point for some field extension $L\supset K$ of degree $[L:K]$ coprime to $3$, then $X$ has a $K$-point.

Let $k$ be a finite field of characteristic $\ell\ne p$, let $X$ be a geometrically connected smooth curve over $k$, and let … be the exact sequence for the arithmetic fundamental…

Colmez's conjecture. The identity

Let $R\subset\mathbb{C}$ be a finitely-generated $\mathbb{Z}$-algebra, and let $S$ be a smooth $R$-scheme. Let $(\mathscr{E},\nabla)$ be a flat vector bundle on $S/R$. Vanishing…

Szpiro's conjecture. For every $\varepsilon>0$, there exists some constant $c>0$ depending only on $\varepsilon$, such that all elliptic curves $E$ over $\boldsymbol{\mathbb{Q}}$ s…

Weil's conjecture. The rational function $Z(X_p,T)$ has the form

Deligne's weight-monodromy conjecture. The action of Frobenius on $\operatorname{gr}^M_j H^i(Y_{\overline{K}},\mathbb{Q}_\ell)$ should be pure of weight $i+j$.

Let $F/F_0$ be a quadratic extension, let $(V,\phi)$ be a symmetric space of dimension $2n+2$, and let $I\subset [0,n]$ be non-empty. Let $\RM^\pm_I$ be the spin local model over…

Manin–Mumford conjecture. A smooth algebraic curve of genus $g\geq 2$ has finitely many torsion points.

Let $X$ be an arithmetic scheme, meaning a separated scheme of finite type over $\operatorname{Spec}\mathbb{Z}$, and let $n<0$. Write $X_{\text{ét}}$ for its étale site and…

Iwaniec–Sarnak conjecture. For every $\varepsilon>0$,

Bombieri–Dwork conjecture. The minimal differential equation of a $G$-function should come from geometry.

Let $E/K$ be an elliptic curve over a number field $K$, and let $G_K:=\operatorname{Gal}(\overline{K}/K)$. Let $G$ be a topologically finitely generated subgroup of $G_K$, and writ…

Grothendieck's section conjecture. Every section $s_X\in\mathcal{S}(\pi_1(X)/\pi_1(k))$ arises from a point $x\in X(k)$, and

Modularity conjecture. The function