Regular Split Embedding Problems over Function Fields of One Variable over Ample Fields

Regular Split Embedding Problemsover Function Fields of One

Function Fields of One Variable over PAC Fields

Function Fields of One Variable over PAC Fields by Moshe Jarden, Tel Aviv University and Florian Pop, University of Pennsylvania In this note we give evidence for a conjecture of Serre and a conjecture of Bogomolov. Conjecture II of Serre considers a field F of characteristic p with cd(Gal(F )) ≤ 2 such that either p = 0 or p > 0 and [F : F ] ≤ p and predicts that H(Gal(F ), G) = 1 (i.e. each p...

Morphisms of Varieties over Ample Fields

We strengthen a result of Michiel Kosters by proving the following theorems: (*) Let φ: W → V be a finite surjective morphism of algebraic varieties over an ample field K. Suppose V has a simple K-rational point a such that a / ∈ φ(W (Kins)). Then, card(V (K)rφ(W (K)) = card(K). (**) Let K be an infinite field of positive characteristic and let f ∈ K[X] be a nonconstant monic polynomial. Suppos...

Equidistribution over function fields

We prove equidistribution of a generic net of small points in a projective variety X over a function field K. For an algebraic dynamical system over K, we generalize this equidistribution theorem to a small generic net of subvarieties. For number fields, these results were proved by Yuan and we transfer here his methods to function fields. If X is a closed subvariety of an abelian variety, then...

Arithmetic over Function Fields

These notes accompany lectures presented at the Clay Mathematics Institute 2006 Summer School on Arithmetic Geometry. The lectures summarize some recent progress on existence of rational points of projective varieties defined over a function field over an algebraically closed field.