89 problems

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 $k$ 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…

Let $(X, \Delta)$ be a smooth C-pair over $\overline{\mathbb{Q}}$. Say that $(X, \Delta)$ satisfies potential density of integral points if, for some number field $K$, finite set o…

A special subvariety of an abelian variety over an algebraically closed field of characteristic zero is a translate of an abelian subvariety by a torsion point. Let $K$ be a number…

Let $K$ be a number field and let $S$ be a finite set of places of $K$ containing all archimedean ones. Let $f_1,\ldots,f_s:\mathbb{P}^1\to\mathbb{P}^1$ be rational maps of degree…

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

Harari–Voloch conjecture. The Brauer–Manin obstruction is the only obstruction to the existence of $S$-integral points on $U$.

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

Use the source's positive-integer Diophantine prefixes $\exists\forall\exists$ and $\exists\exists$, the function $\mathbf{Big}_{\mathbb{N},2}$ measuring the maximal size of positi…

For $f\in\mathbb{Z}[x_1,x_2]$, let $\mathbf{Big}_{\mathbb{N},2}(f)$ be the supremum of the maximum coordinate among the positive integral points on the plane curve $f=0$, with the…

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 $C$ be a curve of positive genus, represented using a dense encoding, and consider the height of its smallest integral point when such a point exists. Smale's conjecture. Effec…

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…

Lang's boundedness conjecture. The quantity $\rho$ is bounded.

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 $A/k$ be an abelian variety, and let $D$ be an effective ample divisor on $A$. Define $A_D({\mathcal O}_k)$ to be the set of ${\mathcal O}_k$-integra…

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.

Silverman's conjecture. The displayed bound should hold for all elliptic curves $E$ with no dependence on $\delta$.

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…

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.