168 problems

Let $D$ be a diophantine subset of $\mathbb{C}\left(z\right)$, and for each $n\in\mathbb{Z}_{\geq 1}$ let $\mathbb{C}\left[z\right]_n$ be the $(n+1)$-dimensional $\mathbb{C}$-vecto…

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 $x\in X(\overline{\mathbb{Q}})$…

Bilu–Luca conjecture. There exists a real number $c>0$ such that, for every sufficiently large integer $N$, at least $cN$ of the number fields…

Let $C$ be an elliptic curve defined over $\mathbb Q$, embedded in projective space, and let $x_n$ be a fixed coordinate in a coprime integral representative $[x_0,\ldots,x_N]$ of…

Let $A$ be a complex abelian variety, let $\Gamma$ be a subgroup of finite rank in $A(\mathbb{C})$, and let $X$ be a subvariety of $A$. Here $\overline{X(\mathbb{C}) \cap \Gamma}$…

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

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 $X\subset \mathbf{G}_m^N(\overline{\mathbf{Q}})$ be a variety, and let $U_X^{\mathbf{Q}}$ be the smallest union of torsion varieties containing $X$ that is defined over…

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…

An affine cube of dimension $d$ in $\mathbb Z$ is a set … where $b_0,\ldots,b_d$ are nonzero integers. Solymosi's conjecture. There exists an integer $d>0$ such that no affine cube…

Let $F=\mathbb{Q}$ and let $X$ be a hyperbolic curve. For the étale unipotent fundamental group $U^{\mathrm{et}}$ and its level-$n$ quotient $U_n^{\mathrm{et}}$, let…

Browning–Heath-Brown–Salberger's conjecture. For every $\varepsilon>0$,

Generalized Lang conjecture. The set of pre-periodic points contained in $Y$ is not Zariski-dense in $Y$.

The fractal intersection conjecture. The closure $\overline{Z(\bar K)\cap F}^{\,\mathrm{Zar}}$ is a union of finitely many points and finitely many components $B_i$ such that…

David–Philippon's conjecture. There is a constant $c(A,L)$ depending only on $A$ and $L$ such that

Mazur's conjecture for abelian varieties. The preceding assertion should hold for every such $A$ and $G$.

Prime divisibility embedding conjecture. There is a Euclidean motion $T$ such that

Divisibility embedding conjecture. There is a set $N\subset\mathbb{Z}^n$ such that $k\cdot N$ is congruent to $M$.

Embedding conjecture. One can find a Euclidean motion $T$ such that

Integer-coordinate conjecture. Every Heron simplex can be represented with integer coordinates.

Let $\Lambda$ be a finitely generated subgroup of $\overline{\mathbb{Q}}^\times$ containing $\zeta_{m-1}$, with $m\ge 2$. Say that algebraic numbers are multiplicatively dependent…

Let $K$ be a number field, let $M_K$ be its set of places, and define the height $\operatorname{h}$ and conductor $\operatorname{cond}$ as in the preceding number-field abc formula…

The abc-conjecture for a number field. For every $\kappa>1$ and every $x\in K$ distinct from $0$ and $1$,

Let $K=\mathbb C(B)$, where $B$ is a smooth projective curve, and let $X$ be a projective variety over $K$ containing no rational curve. Let $h_L$ be a Weil height attached to an a…

Let $K=k(B)$, where $B$ is a smooth projective curve, and let $X$ be a projective variety of general type over $K$. A sequence of $K$-rational points in $X$ is generic if every inf…