ﻻ يوجد ملخص باللغة العربية
We introduce descent methods to the study of strong approximation on algebraic varieties. We apply them to two classes of varieties defined by P(t)=N_{K/k}(z): firstly for quartic extensions of number fields K/k and quadratic polynomials P(t) in one variable, and secondly for k=Q, an arbitrary number field K and P(t) a product of linear polynomials over Q in at least two variables. Finally, we illustrate that a certain unboundedness condition at archimedean places is necessary for strong approximation.
We prove new results on splitting Brauer classes by genus 1 curves, settling in particular the case of degree 7 classes over global fields. Though our method is cohomological in nature, and proceeds by considering the more difficult problem of splitt
Let us consider an algebraic function field defined over a finite Galois extension $K$ of a perfect field $k$. We give some conditions allowing the descent of the definition field of the algebraic function field from $K$ to $k$. We apply these result
Fix $d ge 2$ and a field $k$ such that $mathrm{char}~k mid d$. Assume that $k$ contains the $d$th roots of $1$. Then the irreducible components of the curves over $k$ parameterizing preperiodic points of polynomials of the form $z^d+c$ are geometric
Let $X$ be a hyperbolic curve over a field $k$ finitely generated over $mathbb{Q}$. A Galois section $s$ of $pi_{1}(X)tomathrm{Gal}(bar{k}/k)$ is birational if it lifts to a section of $mathrm{Gal}(overline{k(X)}/k(X))tomathrm{Gal}(bar{k}/k)$. Grothe
We give conditions for the Mayer-Vietoris property to hold for the algebraic K-theory of blow-up squares of toric varieties in any characteristic, using the theory of monoid schemes. These conditions are used to relate algebraic K-theory to topologic