相关论文: Admissibility of groups over function fields of p-…

In this paper we generalize an argument of Neukirch from birational anabelian geometry to the case of arithmetic curves. In contrast to the function field case, it seems to be more complicate to describe the position of decomposition groups…

Let $C$ be an affine plane curve. We consider additive functions $f: K\rightarrow K$ for which $f(x)f(y)=0$, whenever $(x,y)\in C$. We show that if $K=\mathbb{R}$ and $C$ is the hyperbola with defining equation $xy=1$, then there exist…

Let $k$ be a field of characteristic $p>0$, which has infinitely many discrete valuations. We show that every finite embedding problem for $\Gal(k)$ with finitely many prescribed local conditions, whose kernel is a $p$-group, is properly…

Let $K$ be a $p$-adically closed field and $G$ a group interpretable in $K$. We show that if $G$ is definably semisimple (i.e. $G$ has no definable infinite normal abelian subgroups) then there exists a finite normal subgroup $H$ such that…

For any number field K, it is unknown which finite groups appear as Galois groups of extensions L/K such that L is a maximal subfield of a division algebra with center K (a K-division algebra). For K=Q, the answer is described by the long…

Let $R$ be a discrete valuation ring of field of fractions $K$ and of residue field $k$ of characteristic $p > 0$. In an earlier work, we studied the question of extending torsors on $K$-curves into torsors over $R$-regular models of the…

Let $K$ be the function field of a $p$-adic curve, $G$ a semisimple simply connected group over $K$ and $X$ a $G$-torsor over $K$. A conjecture of Colliot-Th\'el\`ene, Parimala and Suresh predicts that if for every discrete valuation $v$ of…

Given a field F, one may ask which finite groups are Galois groups of field extensions E/F such that E is a maximal subfield of a division algebra with center F. This question was originally posed by Schacher, who gave partial results over…

Suppose that $\mathbf{G}$ is a connected reductive group over a finite extension $F/\mathbb{Q}_p$, and that $C$ is a field of characteristic $p$. We prove that the group $\mathbf{G}(F)$ admits an irreducible admissible supercuspidal, or…

We use the Aubry-Perret bound for singular curves, a generalization of the Hasse-Weil bound, to prove the following curious result about rational functions over finite fields: Let $f(X),g(X)\in\Bbb F_q(X)\setminus\{0\}$ be such that $q$ is…

We prove the following result which was conjectured by Stichtenoth and Xing: let $g$ be the genus of a projective, irreducible non-singular curve over the finite field $\Bbb F_{q^2}$ and whose number of $\Bbb F_{q^2}$-rational points…

Fix a non-negative integer g and a positive integer I dividing 2g-2. For any Henselian, discretely valued field K whose residue field is perfect and admits a degree I cyclic extension, we construct a curve C over K of genus g and index I.…

Let $A$ be an abelian variety over the function field $K$ of a curve over a finite field. We describe several mild geometric conditions ensuring that the group $A(K^{\rm perf})$ is finitely generated and that the $p$-primary torsion…

Working on Berkovich analytic curves, we propose a geometric approach to the study of the Hasse principle over function fields of curves defined over a complete discretely valued field. Using it, we show the Hasse principle to be verified…

Let k be a global field of characteristic not 2. The classical Hasse-Minkowski theorem states that if two quadratic forms become isomorphic over all the completions of k, then they are isomorphic over k as well. It is natural to ask whether…

We consider the problem of defining polynomials over function fields of positive characteristic. Among other results, we show that the following assertions are true. 1. Let $\G_p$ be an algebraic extension of a field of $p$ elements and…

For any finite field k of characteristic exceeding 3, the Hasse principle and weak approximation is established for non-singular cubic hypersurfaces X over the function field k(t), provided that X has dimension at least 6.

Croke-Kleiner admissible groups firstly introduced by Croke-Kleiner belong to a particular class of graph of groups which generalize fundamental groups of $3$--dimensional graph manifolds. In this paper, we show that if $G$ is a…

We define the notion of accessibility for a pro-$p$ group. We prove that finitely generated pro-$p$ groups are accessible given a bound on the size of their finite subgroups. We then construct a finitely generated inaccessible pro-$p$…

We study the collection of group structures that can be realized as a group of rational points on an elliptic curve over a finite field (such groups are well known to be of rank at most two). We also study various subsets of this collection…