Related papers: Essential dimension in mixed characteristic

We give a simple formula for the essential dimension of a finite pseudo-reflection group at a prime p and determine the absolute essential dimension for most irreducible pseudo-reflection groups. We also study the "poor man's essential…

We fix a field $\kk$ of characteristic $p$. For a finite group $G$ denote by $\delta(G)$ and $\sigma(G)$ respectively the minimal number $d$, such that for any finite dimensional representation $V$ of $G$ over $\kk$ and any $v\in…

In this article the $p$-essential dimension of generic symbols over fields of characteristic $p$ is studied. In particular, the $p$-essential dimension of the length $\ell$ generic $p$-symbol of degree $n+1$ is bounded below by $n+\ell$…

Let $k$ be a field and let $G$ be an affine algebraic group over $k$. Call a $G$-torsor weakly versal for a class of $k$-schemes $\cal C$ if it specializes to every $G$-torsor over a scheme in $\cal C$. A recent result of the first author,…

We define and study the essential dimension of an algebraic stack. We compute the essential dimension of the stacks Mgn and MgnBar of smooth, or stable, n-pointed curves of genus g. We also prove a general lower bound for the essential…

Let $p$ be an odd prime and $F$ be a number field whose $p$-class group is cyclic. Let $F_{\{\mathfrak{q}\}}$ be the maximal pro-$p$ extension of $F$ which is unramified outside a single non-$p$-adic prime ideal $\mathfrak{q}$ of $F$. In…

Let k be a non-archimedean local field with residual characteristic p. Let G be a connected reductive group over k that splits over a tamely ramified field extension of k. Suppose p does not divide the order of the Weyl group of G. Then we…

Let $G$ be a finite group and $K$ a number field. We construct a $G$-extension $E/F$, with $F$ of transcendence degree $2$ over $K$, that specializes to all $G$-extensions of $K_\mathfrak{p}$, where $\mathfrak{p}$ runs over all but finitely…

If $G$ be a finite $p$-group and $\chi$ is a non-linear irreducible character of $G$, then $\chi(1)\leq |G/Z(G)|^{\frac{1}{2}}$. In \cite{fernandez2001groups}, Fern\'{a}ndez-Alcober and Moret\'{o} obtained the relation between the character…

The representation dimension of a finite group G is the smallest positive integer m for which there exists an embedding of G in GL_m(C). In this paper we find the largest value of representation dimensions, as Granges over all groups of…

Let G be a connected linear algebraic group defined over an algebraically closed field k and H be a finite abelian subgroup of G whose order is prime to char(k). We show that the essential dimension of G is bounded from below by rank(H) -…

For every given $p\in [1,+\infty)$ and $n\in\mathbb{N}$ with $n\ge 1$, the authors identify the strong $L^p$-closure $L_{\mathbb{Z}}^p(D)$ of the class of vector fields having finitely many integer topological singularities on a domain $D$…

We give an upper bound for the essential dimension of a smooth unipotent algebraic group over an arbitrary field. We also show that over a field $k$ which is finitely generated over a perfect field, a smooth unipotent algebraic $k$-group is…

In 1878, Jordan proved that if a finite group $G$ has a faithful representation of dimension $n$ over $\mathbb{C}$, then $G$ has a normal abelian subgroup with index bounded above by a function of $n$. The same result fails if one replaces…

The faithful dimension of a finite group $\mathrm G$ over $\mathbb C$, denoted by $m_\mathrm{faithful}(\mathrm G)$, is the smallest integer $n$ such that $\mathrm G$ can be embedded in $\mathrm{GL}_n(\mathbb C)$. Continuing our previous…

Let $G$ be a group and $H \le K \le G$. We say that $H$ is $c$-embedded in $G$ with respect to $K$ if there is a subgroup $B$ of $G$ such that $G = HB$ and $H \cap B \le Z(K)$. Given a finite group $G$, a prime number $p$ and a Sylow…

Let $p$ be a prime and $\mathbb{F}_p$ be a finite field of $p$ elements. Let $\mathbb{F}_pG$ denote the group algebra of the finite $p$-group $G$ over the field $\mathbb{F}_p$ and $V(\mathbb{F}_pG)$ denote the group of normalized units in…

We give a short proof of the fact that if all characteristic p simple modules of the finite group G have dimension less than p, then G has a normal Sylow p-subgroup.

We prove for various finite groups $G$ and integers $n\geq 1$ that there are families of equations with Galois group $G$ that cannot be simplified to a one-parameter family even after adjoining a root of a polynomial of degree at most $n$.…

Let R be an associative ring with possible extra structure. R is said to be weakly small if there are countably many 1-types over any finite subset of R. It is locally P if the algebraic closure of any finite subset of R has property P. It…