20 problems

Failure of the local-global principle. The local-global principle for isotropy of quadratic forms fails to hold in dimension $2^n$ over $K$.

Let $\sG$ be the fixed primitive integral Apollonian gasket, let $\sB=\sB(\sG)$ be the set of its circle curvatures, … and call an integer represented if it lies in $\sB$. Let…

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 $F$ 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 $\mathbb H^3$ 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 $\mathcal{A}\subseteq\mathbb{N}$ be finite, let $v$ be the primitive vector from the associated orbit, and let $L$ be a linear functional. Define … and let…

Let ${\mathcal{A}}$ 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 $k$ 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 $K$ 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 $p\neq 3$ 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 $K$ 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…

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 $\mathcal S_{\Phi_{n,f}}$…

Let $f\in\mathbf Q[x]$ be a quadratic polynomial, and let $n>3$ be an integer. For a prime $p$, write $\mathbf Q_p$ for the field of $p$-adic numbers. A point has period $n$ for…

Local-global conjecture. Every sufficiently large admissible integer from $\mathcal{P}$ is a curvature. Equivalently,

Soit $K$ un corps global et soit $V$ une surface projective, lisse et géométriquement réglée sur $K$. Avec $\cyr{X}\mathcal N(V)$ défini comme le noyau de la restriction de…

Conjecture de finitude de l'obstruction $K$-théorique. Le groupe $\cyr{X}\mathcal N(V)$ est fini. Cette conjecture est un principe local-global en $K$-théorie algébrique; son statu…

Soit ${\mathbb F}$ un corps fini, soit $C/{\mathbb F}$ une courbe projective, lisse et géométriquement connexe, et posons $K={\mathbb F}(C)$. Pour chaque point fermé $v$ de $C$, so…