222 problems

Let $N\geq 4$. A quadratic polynomial is a polynomial $f(x)\in\mathbb{Q}[x]$ of degree $2$, and a rational point of period $N$ is an $x\in\mathbb{Q}$ such that $f^{(N)}(x)=x$ but…

Let $V(k)$) be the set of $k$-rational points of a projective variety $V$ over a number field $k$, and let $A^1_{\infty}$ be the infinite matrix associated with $V$ in the paper. D…

Matúš's conjecture. There exists a distribution $p$ in the zero set of $J_{\mathcal C}$ such that all joint probabilities of $p$ are rational.

Galbraith's conjecture. If $2\leq g^+_N\leq 5$, then $X^+_0(N)$ contains exceptional rational points if and only if

Weak Manin's conjecture. There exists a non-empty Zariski open subset $U\subset X$ such that, for every $\epsilon>0$,

Let $K$ be a number field, and let $E$ be a Legendre elliptic curve over $K$ with parameter $\lambda\in K$. For each even integer $n\geq 4$, consider the $n$ points constructed abo…

A number $t$ is represented as a ratio of two differences of fourth powers if there exist integers or rational numbers $a,b,c,d$ such that … with $c^4-d^4\neq 0$. Ratio of differen…

Let $a,b,c,d,e,fbinom{}{} inmathbb{Z}$ and consider the elliptic surface … where the coefficients $a_{i,j}$ are given by (ais). For $t\in\mathbb{Q}$, let $\mathcal{E}_t$ denote the…

The Strong Lang Conjecture. If $X$ is of general type, then there exists a proper subvariety $\Sigma\subsetneqq X$ such that for any finite extension $K'$ of $K$,

The nef anticanonical bundle converse. If the anticanonical bundle $-K_X$ of $X$ is nef, then for some finite extension $K'$ of $K$ the set $X(K')$ of $K'$-rational points of $X$ i…

The negative-canonical-bundle converse. If the canonical bundle $K_X$ of $X$ is negative, then for some finite extension $K'$ of $K$ the set $X(K')$ of $K'$-rational points of $X$…

Mazur–Merel version of the Strong Main Conjecture. Assuming the hypotheses of the Strong Main Conjecture, for all $p\notin P_{\mathbf C}$, only finitely many $V\in\mathcal V_p(J)$…

Let $C$ be a smooth projective geometrically connected curve of genus at least $2$ over a number field $k$, with $C(k)=\emptyset$. Assume that $C$ has a rational divisor class of d…

Let $k$ be a number field, and let $C$ be a smooth projective geometrically connected curve over $k$. In the paper, a curve is very good when its finite descent obstructions accoun…

Let $S$ be a Del Pezzo surface over $\mathbb{Q}$ with at most rational double points, and let $S^\circ\subset S$ be a dense open subset in the Zariski topology. Let $-K_S$ denote t…

Let $S\subseteq\mathbb{P}^d$ be a possibly singular del Pezzo surface, let $U\subseteq S$ be obtained by deleting exceptional divisors, and let $\rho_S$ be the relevant Picard rank…

Let $S$ be a finite set of points in the plane with rational coordinates. Write $\mathcal{T}(S)$ for the set of points that are intersections of two lines, each determined by a pai…

Let $n\geqslant3$, let $F\in\mathbb{Z}[X_0,\ldots,X_n]$ be a non-singular form of degree $d\geqslant2$, and define … For $B\geqslant1$,…

Weak Lang conjecture. Rational points on $V$ are not potentially dense.

Generalized potential-density conjecture. Under these assumptions, rational points on $V$ are potentially dense.

Let $\delta\geq 2$. For a polynomial $g\in\mathbb{Z}[T_1,\ldots,T_\nu]$, let $\mathfrak{h}(g)$ denote its homogeneous part of maximal degree, and let $\mathcal{J}_\delta$ be the se…

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

For positive cubefree integers $k$, consider the elliptic curves … Their rank is the rank of the finitely generated group $E_k(\mathbb{Q})$. Unbounded rank conjecture for $E_k$. Th…

Harris–Tschinkel conjecture. Rational points are potentially dense on

Lang's strong Diophantine conjecture. There is a proper, closed subset $V_0$ of $V$ such that, for any number field $K$, all but finitely many $K$-rational points of $V$ lie in…