David Harari

Arithmetic Duality Theorems for 1-Motives

Abstract The argument for proving Corollary 3.5 is insufficient; we fill in the gap here. Also, the first two statements of Proposition 4.1 may not be true in general in the case i = 1, but for the main results it suffices to use them over a sufficiently small U, where they hold. These results are used in the proof of Theorem 0.2, but its statement remains unchanged in the crucial case i = 1; in the (uninteresting) case i = 0 it needs to be modified slightly. We also correct a few other minor inaccuracies.

Non-abelian descent and the arithmetic of Enriques surfaces

The Brauer-Manin obstruction to the Hasse principle and weak approximation provides a fruitful general approach to rational points on varieties over number fields. A fundamental problem here can be stated as follows: is it possible to describe in purely geometric terms the class of smooth projective varieties for which the Brauer-Manin obstruction is the only obstruction to the Hasse principle and weak approximation? In recent examples where the Brauer-Manin obstruction is not the only one (see [1, 7, 14]), the key role is played by etale Galois coverings with a non-abelian Galois group. This has left open the question whether similar examples exist for varieties with an abelian geometric fundamental group. The case of principal homogeneous spaces of abelian varieties and that of rational surfaces (which are geometrically simply connected), where the Brauer-Manin obstruction is expected to be the only one, might seem to suggest that as long as the geometric fundamental group is abelian, the Brauer-Manin obstruction should still be the only one. The Manin obstruction was linked to the classical abelian descent by ColliotThelene and Sansuc [2]. In [8], the authors introduced the non-abelian descent as a new tool for studying rational points. The present paper enriches the non-abelian theory with a general method for constructing non-abelian torsors, and then applies it to an example which answers the above question in the negative.

Non-abelian Cohomology and Rational Points

Using non-abelian cohomology we introduce new obstructions to the Hasse principle. In particular, we generalize the classical descent formalism to principal homogeneous spaces under noncommutative algebraic groups and give explicit examples of application.

Local-global principles for

Building upon our arithmetic duality theorems for 1motives, we prove that the Manin obstruction related to a finite subquotient B(X) of the Brauer group is the only obstruction to the Hasse principle for rational points on torsors under semiabelian varieties over a number field, assuming the finiteness of the TateShaferevich group of the abelian quotient. This theorem answers a question by Skorobogatov in the semiabelian case, and is a key ingredient of recent work on the elementary obstruction for homogeneous spaces over number fields. We also establish a Cassels–Tate type dual exact sequence for 1-motives, and give an application to weak approximation. A Jean-Louis Colliot-Thelene, pour ses 60 ans

1

Abstract We introduce a new obstruction to weak approximation which is related to etale non-abelian coverings of a proper and smooth algebraic variety X defined over a number field k. This enables us to give some counterexamples to weak approximation which are not accounted for by the Brauer–Manin obstruction, for example bielliptic surfaces.

-motives

We discuss the question of whether the Brauer–Manin obstruction is the only obstruction to the Hasse principle for integral points on affine hyperbolic curves. In the case of rational curves we conjecture a positive answer, we prove that this conjecture can be given several equivalent formulations and we relate it to an old conjecture of Skolem. Finally, we show that for elliptic curves minus one point a strong version of the question (describing the set of integral points by local conditions) has a negative answer.

Weak approximation and non-abelian fundamental groups

Given a smooth projective curve X of genus at least 2 over a number field k, Grothendieck’s Section Conjecture predicts that the canonical projection from the étale fundamental group of X onto the absolute Galois group of k has a section if and only if the curve has a rational point. We show that there exist curves where the above map has a section over each completion of k but not over k. In the appendix Victor Flynn gives explicit examples in genus 2. Our result is a consequence of a more general investigation of the existence of sections for the projection of the étale fundamental group ‘with abelianized geometric part’ onto the Galois group. We also point out the relation to the elementary obstruction of Colliot-Thélène and Sansuc.

The Brauer–Manin obstruction for integral points on curves

We establish a link between the descent obstruction against rational points and sections of the fundamental group extension that has applications to the Brauer–Manin obstruction and to the birational case of the section conjecture in anabelian geometry.

Galois sections for abelianized fundamental groups

— Over a global fieldK (number field, or function field of a curve over a finite fieldF ), arithmetic duality theorems for the Galois cohomology of tori and finite Galois modules have long been known. More re cent work investigates the case where K is the function fields of of a curve over a p-adic field. ForK the function field of a curve over the formal series field F = C((t)), we establish analogous duality theorems. We thus control th e obstruction to the local-global principle and to weak approximation for homog eneous spaces of tori. There are differences with the afore described cases. For ex ample the Hasse principle need not hold for principal homogeneous spaces of aK-rational torus. 2 JEAN-LOUIS COLLIOT-THÉLÈNE & DAVID HARARI

Descent Obstruction and Fundamental Exact Sequence

We prove that the Brauer–Manin obstruction is the only obstruction to the existence of integral points on affine varieties over global fields of positive characteristic