28 problems

Let $k$ be an infinite base field of characteristic $p$, where $p$ is prime, and let $n\geqslant 1$ be an integer. Write $\mathbb Z/p^n\mathbb Z$ for the cyclic group of order…

Let $k$ be a field, let $d\geq 1$, and let $D$ be a central division algebra of degree $d$ over $k$. For a positive rational number $r$, write $\operatorname{Mod}_{D,r}$ for the fu…

A central simple algebra is a finite-dimensional associative algebra whose center is the base field and which becomes a matrix algebra after extending scalars. Its degree is the sq…

Let $E$ be an elliptic curve over a number field. Its essential dimension is denoted by $\operatorname{ed}E$. Infinite essential-dimension conjecture for elliptic curves. Then … Th…

Let $k$ be a field, let $G$ be a finite group, and let $X$ be a $k$-variety with a faithful $G$-action. For every prime $p$, let $G_p$ be a Sylow $p$-subgroup of $G$. Duncan's conj…

Let $X$ be a projective variety and let $f:X\to X$ be a polarized endomorphism, meaning that there is an ample divisor $H$ and a positive integer $q>1$ such that … A generically fi…

Let $k$ be a field of characteristic zero, and let $\mathfrak{S}_n$ be the symmetric group on $n$ letters. The essential dimension conjecture for symmetric groups. … The source pre…

Let $G$ be a finite group over the field of complex numbers. Its essential dimension is the minimal dimension of a faithful $G$-variety admitting a compression from a faithful line…

For a finite group $G$, let $\operatorname{ed}(G)$ denote its essential dimension and let $S_n$ be the symmetric group on $n$ letters. The essential-dimension conjecture for symmet…

Let $k$ be a field of positive characteristic and let $G$ be a finite commutative unipotent $k$-group scheme. Write $\mathrm{ed}_k(G)$ for the essential dimension of $G$, and let…

Merkurjev's nonexistence conjecture. No rational parameterisation of quadratic forms in $I_n^3(k)$ exists for $n>14$.

Essential dimension conjecture. When ${\rm{gd}}_k G$ is finite, one should have

Let $A$ be an abelian variety over a field $L$ of characteristic zero, and let $[p]:A\to A$ denote multiplication by a prime $p$. Brosnan's conjecture. For every prime $p$, the map…

Let $p$ be a prime, $D$ a Hermitian symmetric domain, $G=\operatorname{Aut}^0(D)$, and let $\Gamma$ be an arithmetic subgroup of $G$ as in the source. For a subgroup…

Let $\mathbb{H}$ be a torsion-free integral variation of Hodge structure on a smooth irreducible complex variety $B$. Let $\varphi:B^{\mathrm{an}}\to\Gamma\backslash D$ be its peri…

Let $p$ be a prime integer and let $k$ be a $p$-closed field of characteristic different from $p$. Let $G$ be an affine algebraic group defined over $k$ whose connected component…

Essential dimension conjecture. One has

Let $k$ be an algebraically closed field, let $G$ be a connected reductive linear algebraic group over $k$, let $R$ be a regular local ring containing $k$, and let … be a $G$-torso…

Tossici's conjecture.

Verschiebung lower-bound conjecture. The essential dimension of $G$ over $k$ satisfies

Let $k$ be a field of positive characteristic, and let $G$ be a finite commutative unipotent group scheme over $k$. Let…

Stable-cohomology coefficient conjecture. The top degree in which stable cohomology with arbitrary coefficients is nonzero is the same as the top degree in which stable cohomology…

Generalized fixed-point conjecture. If $\operatorname{rdim}_k(H_i)\leq d$ for all $i=1,\dots,r$, then there exists a $d$-dimensional $k$-variety $X$ with a faithful $G$-action and…

Dolgachev's inequality.

Sylow-versality conjecture. If $X$ is $G_p$-versal for every prime $p$, then $X$ is $G$-versal.