35 problems

Colliot-Thélène–Sansuc conjecture. The Brauer–Manin obstruction is the only obstruction to the local-to-global principle for $Y$.

A K3 surface is a smooth, simply connected projective surface with trivial canonical bundle. For a variety $X$ over a number field $k$, write $X(\mathbb{A}_k)$ for its adelic point…

Let $\mathcal{A}\subseteq\mathbb{N}$ be finite, let $v$ be the primitive vector from the associated orbit, and let $L$ be a linear functional. Define … and let…

Let $k$ be a number field and let $X\subset \mathbb{P}^n_k$, with $n\geq 6$, be a non-conical geometrically integral intersection of two quadrics. Suppose that $X$ has a smooth…

Let $F$ be the function field of a $p$-adic curve, let $G$ be a connected linear algebraic group over $F$, and let $Z$ be a projective homogeneous space under $G$. Colliot-Thélène–…

Let $K$ be a number field. A proper variety $X$ over $K$ violates the local-global principle if $X(K_v)\neq\emptyset$ for every nontrivial place $v$ of $K$, while $X(K)=\emptyset$.…

Apollonian packing local-global conjecture. In any integral Euclidean Apollonian packing, all sufficiently large integers not ruled out by congruence conditions modulo $24$ are rep…

Let $E$ be a moiety of a primitive Eisenstein circle packing. A positive integer is called sporadic if it is admissible in $E$, does not appear in $E$, and does not belong to one o…

Local-global norm conjecture. The local-global principle for norms holds in non-cyclic extensions of number fields.

Let $\mathcal{P}$ be a primitive integral Kleinian $(d-1)$-sphere packing in $\mathbb{R}^d\cup\{\infty\}$ with an orientation-preserving automorphism group $\Gamma$ of Möbius trans…

Let ${\mathcal{P}^{\operatorname{full}}}$ be a primitive integral packing, and let $\Sigma$ be the union of the residue classes modulo $24$ that have representatives in…

Let $\mathbb H^3$ be the upper half-space model of hyperbolic $3$-space, and let $\Gamma$ be a geometrically finite subgroup of…

Folklore local-to-global conjecture. If $X(\mathbf{Q}_p)\neq\emptyset$ for every prime $p$, then

Define … The table of congruence and reciprocity obstructions specifies the stated restrictions on numerators and denominators in these orbits. Local-global conjecture for $\Psi_1$…

Let ${\mathcal{A}}$ be a primitive Apollonian circle packing. Let $S_{\mathcal{A}}$ be the set of missing curvatures that do not lie in any of the quadratic or quartic obstruction…

Let ${\mathcal{A}}$ be a primitive integral Apollonian circle packing. A positive curvature $c$ is missing in ${\mathcal{A}}$ if curvatures congruent to $c$ modulo $24$ occur in…

Let $X$ be a smooth projective variety over a global field $K$, and let $\ell$ be a prime invertible in $K$. Write $\widehat{\otimes}\,\mathbb{Z}_\ell$ for the inverse-limit $\ell$…

Let $K$ be a number field, let $\ell$ be a prime, and let $(A,\ell)$ be a strong counterexample to the local-global principle for isogenies of prime degree between abelian surfaces…

Let $p\neq 3$ be a prime number, let $P=p$ or $2p$, and let … be the fundamental unit of $\mathbb{Q}(P^{1/3})$, where $\alpha,\beta,\gamma\in(1/3)\mathbb{Z}$. Shimizu–Hirakawa conj…

Let $n\geq 3$ and let $h$ be a fixed integer. Consider integral binary forms of degree $n$ that locally represent $h$, meaning that the equation $F(x,y)=h$ has a solution over…

Let $p\neq 3$ be a prime number, let $P=p$ or $2p$, and let … be the fundamental unit of $\mathbb{Q}(P^{1/3})$, where $\alpha,\beta,\gamma\in\mathbb{Z}$. The cubic analogue of the…

Let $F$ be a global field, let $T$ be a nonempty finite set of places of $F$ containing the archimedean places, and let $R=R_{(T)}\subseteq F$ be the ring of $T$-integers. Let $M$…

Let $K$ be a global function field over a finite field $k$ of positive characteristic, let $S$ be a finite set of places of $K$, and let $\Gamma\subset O_S^*$ be a subgroup. Fix a…

The Local-Global Conjecture. Assume that $A\ge2$, so that $\Gamma_A$ is Zariski dense in $\operatorname{SL}_2$, and that $F(\Gamma_A)$ is infinite; equivalently, the Zariski closur…

Let $f\in\mathbf Q[x]$ be a quadratic polynomial, and let $n>3$ be an integer. Let $\Phi_{n,f}$ denote the period-$n$ dynatomic polynomial of $f$, and let $\mathcal S_{\Phi_{n,f}}$…