Related papers: On the birational section conjecture over finitely…

We investigate sections of arithmetic fundamental groups of hyperbolic curves over function fields. As a consequence we prove that the anabelian section conjecture of Grothendieck holds over all finitely generated fields over $\Bbb Q$ if it…

Assume that the section conjecture holds over number fields. We prove then that it holds for a broad class of curves defined over finitely generated extensions of $\mathbb{Q}$. This class contains every projective, hyperelliptic curve,…

Let $X$ be a curve over a field $k$ finitely generated over $\mathbb{Q}$ and $t$ an indeterminate. We prove that, if $s$ is a section of $\pi_{1}(X)\to\operatorname{Gal}(k)$ such that the base change $s_{k(t)}$ is birationally liftable,…

We prove the geometric Bogomolov conjecture over a function field of characteristic zero.

We show that the semi-simplicity conjecture for finitely generated fields follows from the conjunction of the semi-simplicity conjecture for finite fields and for the maximal abelian extension of the field of rational numbers.

Assuming the finiteness of the Shafarevich-Tate group of elliptic curves over number fields we make several observations on the birational Grotendieck anabelian setion conjecture. We prove that the birational setion conjecture for curves…

We find all the possible torsion groups of $\Q$-curves over quadratic fields and determine which groups appear finitely and which appear infinitely often.

We give a new, geometric proof of the section conjecture for fixed points of finite group actions on projective curves of positive genus defined over the field of complex numbers, as well as its natural nilpotent analogue. As a part of our…

Let X be a smooth projective curve of genus >1 over a field K which is finitely generated over the rationals. The section conjecture in Grothendieck's anabelian geometry says that the sections of the canonical projection from the arithmetic…

In this note, we will show that Bogomolov conjecture holds for a non-isotrivial curve of genus 2 over a function field.

In this paper we formulate and prove a combinatorial version of the section conjecture for finite groups acting on finite graphs. We apply this result to the study of rational points and show that finite descent is the only obstruction to…

We prove a topological version of the section conjecture for the profinite completion of the fundamental group of finite CW-complexes equipped with the action of a group of prime order $p$ whose $p$-torsion cohomology can be killed by…

We reduce the classification of finite extensions of function fields (of curves over finite fields) with the same class number to a finite computation; complete this computation in all cases except when both curves have base field…

The problem of constructing curves with many points over finite fields has received considerable attention in the recent years. Using the class field theory approach, we construct new examples of curves ameliorating some of the known…

In this paper, we prove the Effective Bogomolov's Conjecture for hyperelliptic curves defined over function fields.

We prove that the finiteness of a finitely generated category of irreducible algebraic varieties over a field of characteristic zero is decidable. We also obtain a Burnside finiteness criterion for such a category, with applications to…

We give a construction of singular curves with many rational points over finite fields. This construction enables us to prove some results on the maximum number of rational points on an absolutely irreducible projective algebraic curve…

We prove a few uniform versions of the Mordell-Lang Conjecture and of the Shafarevich Conjecture for curves over function fields and their rational points. The main focus is on function fields having high transcendence degree over the…

We prove that the group of rational points of a non-isotrivial elliptic curve defined over the perfect closure of a function field in one variable over a finite field is finiteley generated.

Tate's theorem (Invent. Math. 1966)implies that the Tate conjecture holds for any abelian variety over a finite field whose Q_l-algebra of Tate classes is generated by those of degree 1. We construct families of abelian varieties over…