Related papers: Retract Rational Fields

Let $\Gamma$ be the fundamental group of a manifold modeled on three dimensional Sol geometry. We prove that $\Gamma$ has a finite index subgroup $G$ which has a rational growth series with respect to a natural generating set. We do this by…

We generalize a recent result of Pavic--Schreieder regarding the surjectivity of the obstruction morphism defined in [PS23]. As a consequence of this result, we show that geometrically (retract) rational varieties over a Laurent field of…

Seeking for a converse to a well-known theorem by Borel-Tits, we address the question whether the group of rational points G(k) of an anisotropic reductive k-group may admit a split spherical BN-pair. We show that if k is a perfect field or…

Consider a finite l-group acting on the affine space of dimension n over a field k, whose characteristic differs from l. We prove the existence of a fixed point, rational over k, in the following cases: --- The field k is p-special for some…

In the first part of this paper, we develop a general framework that permits a comparison between explicit class field theories for a family of rational function fields $\mathbb{F}_s(t)$ over arbitrary constant fields $\mathbb{F}_s$ and…

Yanchevski\u{i} had asked whether conic bundle surfaces over $\mathbf{P}^1_k$ are unirational when $k$ is a finite field. We give a partial answer to his question by showing that for quasi-finite fields $k$ (e.g. finite fields) a regular…

The projective orthogonal and symplectic groups $PO_n(F)$ and $PSp_n(F)$ have a natural action on the $F$ vector space $V' = M_n(F) \oplus ... \oplus M_n(F)$. Here we assume $F$ is an infinite field of characteristic not 2. If we assume…

Let $k$ be a perfect field such that for every $n$ there are only finitely many field extensions, up to isomorphism, of $k$ of degree $n$. If $G$ is a reductive algebraic group defined over $k$, whose characteristic is very good for $G$,…

Let $k$ be a field of characteristic zero, let $G$ be a connected reductive algebraic group over $k$ and let $\mathfrak{g}$ be its Lie algebra. Let $k(G)$, respectively, $k(\mathfrak{g})$, be the field of $k$-rational functions on $G$,…

We define $k$-rationalized $G$-equivariant elliptic cohomology, for a field of characteristic zero $k$ and a compact Lie group $G$, via adelic descent. We also give adelic descriptions of rationalized $G$-equivariant singular cohomology and…

We prove the following extension of Tits' simplicity theorem. Let $k$ be an infinite field, $G$ an algebraic group defined and quasi-simple over $k,$ and $G(k)$ the group of $k$-rational points of $G.$ Let $G(k)^+$ be the subgroup of $G(k)$…

We study the 0-th stable A^1-homotopy sheaf of a smooth proper variety over a field k assumed to be infinite, perfect and to have characteristic unequal to 2. We provide an explicit description of this sheaf in terms of the theory of…

We report on progress in the qualitative study of rational points on rationally connected varieties over number fields, also examining integral points, zero-cycles, and non-rationally connected varieties. One of the main objectives is to…

In this paper, we focus on a question of M. Newman on isomorphic subgroups of solvable groups. We get a reduction theorem of this question: for each prime q, assume that this question holds for every characteristic q-groups, then this…

Given a number field $k$, we show that, for many finite groups $G$, all the Galois extensions of $k$ with Galois group $G$ cannot be obtained by specializing any given finitely many Galois extensions $E/k(T)$ with Galois group $G$ and $E/k$…

We classify stably/retract rational norm one tori in dimension $n-1$ for $n=2^e$ $(e\geq 1)$ is a power of $2$ and $n=12, 14, 15$. Retract non-rationality of norm one tori for primitive $G\leq S_{2p}$ where $p$ is a prime number and for the…

We extend (scheme-theoretic) Bruhat-Tits theory to quasi-reductive groups i.e. with trivial split unipotent radical over discretely valued henselian non-archimedean fields $K$, whose ring of integers is excellent and residue field is…

We give an 'arithmetic regularity lemma' for groups definable in finite fields, analogous to Tao's 'algebraic regularity lemma' for graphs definable in finite fields. More specifically, we show that, for any $M>0$, any finite field…

We prove the conjecture of Kac-Wakimoto on the rationality of exceptional W-algebras for the first non-trivial series, namely, for the Bershadsky-Polyakov vertex algebras $W_3^{(2)}$ at level $k=p/2-3$ with $p=3,5,7,...$. This gives new…

We show that the number of rational points on the fibres of a proper morphism of smooth varieties over a finite field k whose generic fibre has a ``trival'' Chow group of zero cycles is congruent to 1 mod |k|. As a consequence we prove that…