286 problems

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 $X$ 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'…

Let $k>4$ and let … be the Fermat–Euler surface in $\mathbf{P}^3$. Consider the rational points on $\mathcal E_k$ outside the union of its curves of genus $0$ and $1$. Bombieri–Lan…

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

Kim's conjecture. For $n\gg 0$,

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…

Pink's conjecture. The intersection

Let $K$ be a number field, and call an extension $L_1/L_2$ of number fields integrally Diophantine when the ring of integers of $L_2$ is Diophantine in the ring of integers of…

Let $i$ and $q$ be non-negative integers, let $k$ be a field, and let $C_i^q$ denote the Kato–Kuzumaki property defined in terms of Milnor $K$-theory of degree $q$. Kato–Kuzumaki's…

Let $X$ be a smooth projective variety and let $D$ be a reduced effective anticanonical divisor on $X$ with at most normal crossings singularities. Anticanonical complement conject…

Let $C$ be an irreducible algebraic curve in $(\mathbb{C}^{\times})^n$. A cyclotomic point of $C$ is a point $(z_1,\ldots,z_n)\in C$ whose coordinates are all roots of unity. Lang'…

Torsion Anomalous Conjecture. An irreducible subvariety $V$ of a semi-abelian variety contains only finitely many maximal $V$-torsion anomalous varieties.

Let $K$ be a number field and let $\mathscr{X}/\mathcal{O}_K$ be a model for a hyperbolic curve. Let $\mathcal{L}$ be an allowable family of extensions $L/K$. Write…

Baragar–Bourgain–Gamburd–Sarnak conjecture. The Markoff equation $\mathbb{X}$ satisfies strong approximation at every prime $p$. Equivalently, $\operatorname{Out}(\Pi)$ acts transi…

Let $M\subset \mathbf{P}^{n-1}(\mathbf{Q})$ be a projective thin set of type II. For \text{\boldmath{x}}=[x_1:\hdots:x_n]\in \mathbf{P}^{n-1}(\mathbf{Q}), define the height … H…

Ih's conjecture. The set of torsion points $\boldsymbol\zeta\in\mathbb G_m^n$ such that $P(\boldsymbol\zeta)$ is an $S$-unit is not Zariski dense in $\mathbb G_m^n$, unless the zer…

Let $(X,\Delta)$ be a smooth proper orbifold curve over a number field $K$, where $\Delta=\sum_i(1-\frac{1}{m_i})D_i$, and call $(X,\Delta)$ Mordellic if, after every finite field…

A cubic hypersurface in $\mathbb{P}^n$ over $\mathbb{Q}$ is the vanishing locus of a cubic form in $n+1$ variables. The Hasse principle conjecture. When $n\geq 4$, every such cubic…

Let $E$ be an elliptic curve over $\mathbb{Q}$ with minimal discriminant $Delta_E$ and conductor $N_E$. Weak Szpiro's conjecture. There exist positive real numbers $\alpha$ and…

Let $A$) be an abelian variety over $\bar{Q}$ equipped with a Néron–Tate height $chat h$, let $Gamma$ be a finitely generated subgroup of $A(\bar{Q})$, and define its division grou…

Strong Lang–Vojta conjecture. The pair $(X,D)$ is of log general type if and only if there exists a proper closed subset

Let $K$ be a number field or a one-dimensional function field of characteristic zero, and let $n\ge 3$. Let $[Z_1:\cdots:Z_n]$ be the standard homogeneous coordinates on…

Modified Szpiro conjecture. For each $\epsilon>0$, there are finitely many elliptic curves, up to isomorphism, satisfying

Let $X$ be a smooth projective variety of general type with canonical line bundle $K_X$. Let $C$ be a curve in $X$ of geometric genus $g$, and let $\operatorname{deg}_C(K_X)$ denot…

Let $S$ be a nonempty open subscheme of $\operatorname{Spec}(\mathbb Z)$ and let $g\geq 1$ be an integer. For an abelian scheme $A$ over $S$ of relative dimension $g$, let $h_F(A)$…