358 problems

Let $k>4$ and let … be the Fermat–Euler surface in $\mathbf{P}^3$. Consider the rational points on $\mathcal E_k$ outside the union of its curves of genus $0$ and $1$. Bombieri–Lan…

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 $n$.

Cassels–Swinnerton-Dyer conjecture. If $X$ has an $L$-point for some field extension $L\supset K$ of degree $[L:K]$ coprime to $3$, then $X$ has a $K$-point.

Let $p$ be a prime with $p\equiv 4,7,8\pmod 9$. A prime is a rational cube sum if there exist $a,b\in\mathbb Q$ such that $p=a^3+b^3$. Sylvester's rational cube-sum conjecture. It…

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,

Mordell–Lang conjecture. The Zariski closure of $S_A(K)$ and $S_B(K)$ consists of a finite union of elliptic curves and points.

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

Abramovich–Harris conjecture. All potentially dense sets of points of degree $d$ on $C$ should be geometrically explained by a morphism of degree $d$ from $C$ to a curve of genus…

Batyrev–Manin conjecture. It is conjectured that this number grows as

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 $\mathbb{Q}$.…

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 $k$-…

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…

Corvaja–Zannier's conjecture. Any smooth projective variety with a potentially dense set of rational points has the weak Hilbert property potentially.

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

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

Let $E$ be an elliptic curve over $\mathbb{Q}$ with a specified Weierstrass equation. Consider arithmetic progressions contained in the set of $x$-coordinates of points in…

Let $n,Re\mathbb{N}$ and $d=n+R$, and let $\mathcal{M}\subseteq\mathbb{R}^d$ be a bounded, immersed, $n$-dimensional smooth submanifold with boundary. For $Q\in\mathbb{N}$ and…

Loughran–Smeets conjecture. Under some mild assumptions, the upper bound of Loughran and Smeets for $N_{\mathrm{loc}}(\pi,B)$ is sharp.

Caporaso–Harris–Mazur uniformity conjecture. There is a constant $C_g$ such that every curve of genus $g$ defined over $\mathbb{Q}$ has at most $C_g$ rational points:

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 $C$ be a hyperelliptic curve over $\mathbb{Q}$ of genus $g \geq 2$, defined by a model … for $f\in\mathbb{Z}[x]$ without repeated roots. For each squarefree integer $d$, let…

Let $E/\mathbb{Q}$ be an elliptic curve, and let its isogeny class be the set of elliptic curves over $\mathbb{Q}$ that are $\mathbb{Q}$-isogenous to $E$. Mazur's conjecture. The s…

Let $k$ be a number field and let $X$ be a smooth projective variety over $k$. The cotangent bundle $\Omega^1_X$ is nef when it is a nef vector bundle. Nef-cotangent conjecture. If…

Let $\underline{d}=(d_1,\ldots,d_e)$ be a tuple of positive integers, and let $X_{n,\underline{d}}$ denote the universal complete intersection of type $\underline{d}$ in…

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…