Related papers: Noether's problem for p-groups of order p^{5}

We use Heegner points to prove the existence of nontorsion rational points on the elliptic curve $y^2 = x^3 + D$ for any rational number $D=a/b$ such that $a$ and $b$ are squarefree integers for which $6$, $a$, and $b$ are pairwise…

Let $G$ be a semiabelian variety defined over an algebraically closed field $K$ of characteristic $0$. Let $\Phi\colon G\dashrightarrow G$ be a dominant rational self-map. Assume that an iterate $\Phi^m \colon G \to G$ is regular for some…

Let $G$ be a finite group and let $\pi(G)=\{p_1, p_2, \ldots, p_k\}$ be the set of prime divisors of $|G|$ for which $p_1<p_2<\cdots<p_k$. The Gruenberg-Kegel graph of $G$, denoted ${\rm GK}(G)$, is defined as follows: its vertex set is…

Assume $G$ is a solvable group whose elementary abelian sections are all finite. Suppose, further, that $p$ is a prime such that $G$ fails to contain any subgroups isomorphic to $C_{p^\infty}$. We show that if $G$ is nilpotent, then the…

Let $k$ be a field with characteristic different from $2$. In this paper, we describe the $k$-rational orbit spaces in some irreducible prehomogeneous vector spaces $(G,V)$ over $k$, where $G$ is a connected reductive algebraic group…

In this survey article we review Kac-Moody and Heisenberg algebra actions on the categories $\mathcal{O}$ of the rational Cherednik algebras associated to groups $G(\ell,1,n)$. Using these actions we solve basic representation theoretic…

Let $G$ be a finite graph of genus $g$. Let $d$ and $r$ be non-negative integers such that the Brill-Noether number is non-negative. It is known that for some $k$ sufficiently large, the $k$-th homothetic refinement $G^{(k)}$ of $G$ admits…

Let $N$ be a non-squarefree positive integer and let $\ell$ be an odd prime such that $\ell^2$ does not divide $N$. Consider the Hecke ring $\mathbb{T}(N)$ of weight $2$ for $\Gamma_0(N)$, and its rational Eisenstein primes of…

In previous work, Eli Aljadeff and the first-named author attached an algebra B_H of rational fractions to each Hopf algebra H. The generalized Noether problem is the following: for which finite-dimensional Hopf algebra H is B_H the…

A theorem of Grothendieck asserts that over a perfect field k of cohomological dimension one, all non-abelian H^2-cohomology sets of algebraic groups are trivial. The purpose of this paper is to establish a formally real generalization of…

A finite group $G$ is called monomial if every irreducible character of $G$ is induced from a linear character of some subgroup of $G$. One of the main questions regarding monomial groups is whether or not a normal subgroup $N$ of a…

For various nonsolvable groups $G$, we prove the existence of extensions of the rationals $\mathbb{Q}$ with Galois group $G$ and inertia groups of order dividing $ge(G)$, where $ge(G)$ is the smallest exponent of a generating set for $G$.…

Let $E/k$ be a non-isotrivial elliptic curve over a global function field $k$ of characteristic $p>3$, and $G\subset \mathrm{Gal}(k^{\mathrm{sep}}/k)$ be a topologically finitely generated subgroup. We prove that if $E/k$ has analytic rank…

A first-order theory is Noetherian with respect to the collection of formulae $\mathcal{F}$ if every definable set is a Boolean combination of instances of formulae in $\mathcal{F}$ and the topology whose subbasis of closed sets is the…

We present two criteria for a group $G$ to satisfy the following statements: any $G$-graded gr-prime (gr-semiprime) right gr-Goldie ring admits a gr-semisimple graded right classical quotient ring. The criterion for gr-semiprime rings is…

We obtained some sufficient and necessary conditions of existence of faithful irreducible representations of a soluble group $G$ of finite rank over a field $k$. It was shown that the existence of such representations strongly depends on…

The inverse Galois problem asks whether any finite group can be realised as the Galois group of a Galois extension of the rationals. This problem and its refinements have stimulated a large amount of research in number theory and algebraic…

In this paper, the spectrum and the decomposability of a multivariate rational function are studied by means of the effective Noether's irreducibility theorem given by Ruppert. With this approach, some new effective results are obtained. In…

Let G be one of the groups SL_n C, Sp_2n C, SO_m C, O_m C, or G_2. For a generically free G-representation V, we say that N is a level of stable rationality for V/G if V/G x P^N is rational. In this paper we improve known bounds for the…

For a class of groups $G$ over a field $\mathbb{F}$, including certain Lie groups, Algebraic groups and finite groups, we develop a general method to determine rational and real elements, thereby unifying earlier group-specific results into…