222 problems

Let $\mathcal{M}$ be a compact submanifold of $\mathbb{R}^n$ of dimension $m$, with codimension $k=n-m$, and let $N_{\mathcal{M}}(Q,\delta)$ denote the number of rational points of…

Asymptotic point-count conjecture. For sufficiently large genus $g$, one has

Let $\mathbb{F}_q$ be a finite field, let $3\leq k\leq q-2$ (or $4\leq k\leq q-3$ if $q$ is even), and let $h$ be a complete symmetric polynomial of positive degree $m$ in $k$ vari…

Let $\mathbb{F}_q$ be a finite field, let $3\leq k\leq q-2$ (or $4\leq k\leq q-3$ if $q$ is even), and let $h_e(x_1,\ldots,x_k)$ denote the complete symmetric polynomial of degree…

Lang's conjecture. (a) Weak form: The set $X(k)$ of $k$-rational points is not Zariski dense in $X$. (b) Strong form: There exists a proper Zariski-closed subset $Z\subset X$ such…

Colliot-Thélène's conjecture. Let $X$ be a rationally connected smooth variety over $k$. Then $X(k)$ is dense in $X(k_{\Omega})^{\operatorname{Br}_{\mathrm{nr}}}$.

Let $k$ be a number field and let $X$ be a smooth Fano variety over $k$, so that $-K_X$ is ample. Fix an adelic metrization of the anticanonical bundle $comathcal O(-K_X)$, with as…

Hutz's conjecture. There is no even degree $d>2$ and no $c\in\mathbb Q$ such that $\varphi_{d,c}$ has a rational point of exact period $n$. Moreover,

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…

Flynn–Poonen–Schaefer conjecture. There is no quadratic polynomial $\varphi_{2,c}(z)=z^2+c\in\mathbb Q[z]$ with a rational point of exact period $n$.

Let $X$ be a smooth projective variety defined over a number field $k$. Bombieri–Lang conjecture. There exists a proper Zariski closed subset $\Xi\subsetneqq X$ such that, for ever…

Let $k$ be a number field and let $X$ be a smooth projective curve of genus $g\ge 2$ defined over $k$. Mordell's conjecture. The set of rational points $X(k)$ is finite. This is th…

Let $X\subseteq\mathbb{P}_{\mathbb{Q}}^{M-1}$ be an irreducible projective variety of degree at least two defined over $\mathbb{Q}$. Let $N_X(B)$ be the number of rational points o…

Let $X$ be a unirational variety over a number field. The Colliot-Thélène–Sansuc conjecture. $X$ has the Hilbert property. This conjecture connects the abundance of rational points…

Positive-density conjecture. The set of primes $p$ of good reduction for $C$ such that $\bar{\omega}_p$ does not vanish on $C(\mathbb F_p)$ has positive density, unless $J$ splits…

Let $V$ be a smooth weak Fano variety over a number field $F$ such that $V(F)$ is Zariski dense, and let $H_F$ be a relative adelic height on the anticanonical line bundle. Manin–P…

Rational points conjecture. If $d\geq4$, then

Stoll's conjecture. The set $C(K)$ is dense in

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

Let $k$ be a number field. A non-trivial isotrivial elliptic fibration is an elliptic fibration over $\mathbb{P}^1$ that is isotrivial but not trivial. Density conjecture. The $k$-…

Linnik's conjecture. For each such $n$, there exists a representation with

Let $X$ be a smooth projective variety defined over a number field $k$, let $P\in X(k)$, and let $A$ be an ample $\mathbb{Q}$-Cartier divisor on $X$. For an algebraic point $P$, wr…

Let $k$ be a $C_{1}$ field, meaning that every hypersurface of degree at most $n$ in $\mathbb{P}^{n}_{k}$ has a $k$-rational point. A variety is separably rationally connected if t…

Let $X$ be a special smooth projective geometrically connected variety over a number field $k$. Campana–Corvaja–Zannier conjecture. There is a number field $L/k$ such that $X_L$ sa…

Let $g\geq 2$ be an integer. A smooth curve is a curve defined over $\mathbb{Q}$, and its rational points are its points over $\mathbb{Q}$. Uniformity Conjecture over $\mathbb{Q}$.…