73 problems

Conjecture. The only values of $n\in\mathbb{N}_{\geq 3}$ such that $M(n)=2$ are

Potential-density conjecture. Integral points are potentially Zariski-dense in the $K$-scheme $X_{G,\underline{C}}(Y)_K$: there exists a finite extension $L/K$ such that the Zarisk…

Campana's conjecture. $X$ is special if and only if the set of integral points on $X$ is potentially dense.

Orbifold Mordell conjecture. Assume that

The Lang–Vojta conjecture. Under these hypotheses, every set of $(D,S)$-integral points in $X$ is Zariski degenerate. These problems are central in Diophantine geometry and, for th…

Let $U_m$ be the affine Markoff surface … where $m$ is an integer, and let $U_m(\mathbb{A}_{\mathbb{Z}})$ denote its integral adelic points. Ghosh–Sarnak conjecture. The number of…

Let $k$ be an algebraically closed field, let $B$ be a nonsingular irreducible algebraic curve over $k$ with function field $K=k(B)$, and let $(X,D)$ be a pair of log general type…

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

Let $A$ be an abelian variety defined over $\mathbb{C}$, let $U$ be an open affine subset of $A$, and let $F\subset \mathbb{C}^n$ be a finitely generated subgroup. Let…

Let $K$ be a number field, let $S$ be a finite set of places of $K$ containing all archimedean places, and let $f:\mathbb{P}^1\to\mathbb{P}^1$ be a rational map of degree at least…

Let $X$ be a nice curve, meaning a smooth, projective, geometrically integral curve, of genus at least $2$ over $\overline{\mathbb{Q}}$. For a point $x\in X(\overline{\mathbb{Q}})$…

Correlation conjecture. The intersection ${\mathcal P}\cap H$ is correlated with respect to the morphism $H\to B$.

Let $C\subset\mathbb{C}^2$ be a curve defined over $\mathbb{Z}$ and irreducible over $\mathbb{C}$. Let $C_{\mathbb{R}}$ be an irreducible component of $C\cap\mathbb{R}^2$ whose int…

Let $X$ be a smooth proper variety over a number field $k$, let $D$ be a normal crossings divisor, and let $A$ be an ample line bundle on $X$. Let $r$ be a positive integer and let…

Let $(Y/\boldsymbol{\Delta})$ be a smooth projective Campana constellation supported by a firmament $\boldsymbol{\Gamma}$. Campana's conjecture. Points on $Y$ integral with respect…

Let $K$ be a global function field, let $M_K$ be its set of places, let $S\subseteq M_K$ be finite, and let $\phi$ be a Drinfeld module over $K$. A point $\gamma$ is $S$-integral f…

Dynamical Siegel conjecture. For any nonpreperiodic point $\beta\in\mathbb{P}^1_{\mathfrak{o}_K}(\overline{K})$, there are at most finitely many preperiodic points of $\varphi$ in…

Let $k$ be a number field, let $S$ be a finite set of places of $k$ containing the archimedean places, and let ${\mathcal O}_{k,S}$ be the ring of $S$-integers. Let $R(x)\in k(x)$…

Let $k$) be a number field, let $S$ be a finite set of places of $k$ containing the archimedean places, and let ${\mathcal O}_{k,S}$ be the ring of $S$-integers. Let $A/k$ be an ab…

Relative undecidability conjecture. The finiteness problem for integral points is undecidable relative to the existence problem.

Let $X$ be a variety over a number field $F$, let $D$ be a divisor on $X$, and let $L$ be a big divisor. For a finite set of places $S$ and a model $(\mathcal U,\mathcal L)$ of a d…

Let $(\underline{X},\Delta)$ be a log pair and set $\underline{U}=\underline{X}\setminus\operatorname{Supp}(\Delta)$. Assume that $\underline{U}$ is geometrically $\mathbb A^1$-con…

Campana's generalized Lang conjecture. If $\dim X \geq 1$, then $(\mathcal{X},\Delta)(\mathcal{O}_{K,S})$ is not dense in $X$.

Weakly Special Conjecture. If $X$ is a weakly special variety over a number field $K$, then $X$ has a potentially dense set of integral points.

Lang–Vojta–Chevalley–Weil conjecture. If $X$ satisfies potential density of integral points, then $X$ is weakly special.