Emmanuel Peyre

Unramified cohomology and rationality problems

— The aim of this paper is to construct unirational function fields K over an algebraically closed field of characteristic 0 such that the unramified cohomology group H i nr(K,μ ⊗i p ) is not trivial for i = 2, 3 or 4 and p a prime number. This implies that the field K is not stably rational. For this purpose, we give a sufficient condition for an element to be unramified in H i(K,μ⊗i p ). This condition relies on computations in the exterior algebra of a vector space of finite dimension over the finite field Fp. Among the first examples of smooth projective varieties X over C which are unirational but not rational was the example constructed by Artin and Mumford using the torsion part of H3(X,Z). When X is unirational, this group may also be described as the unramified Brauer group of the function field of X . From this point of view, Saltman [Sa] and Bogomolov [Bo] gave examples related to Noether’s problem. Colliot-Thélène and Ojanguren [CTO] were the first to use the unramified cohomology groups in degree 3 to prove the non-rationality of a unirational field. The plan of this paper is the following: first we recall some basic facts about unramified cohomology. In the second section, we state the main result, Theorem 2, which enables one to characterize unramified elements by calculations in the exterior algebra. This generalizes some of the methods used in [Sa] and [Bo]. In the next section, we prove Theorem 2. In this proof, we show how one can lift the residue map in the exterior algebra of a subgroup of H(K,μp) of finite dimension. The fourth section applies the main result to the construction of several unirational non-rational fields. In this part, to prove the non-triviality of elements in H3 nr(K,μ ⊗3 p ), we use a recent result by Suslin [Su] and to have a ∗Math. Ann 296 (1993), 247–268

Unramified cohomology of degree 3 and Noether’s problem

Let G be a finite group and W be a faithful representation of G over C. The group G acts on the field of rational functions C(W). The question whether the field of invariant functions C(W)G is purely transcendental over C goes back to Emmy Noether. Using the unramified cohomology group of degree 2 of this field as an invariant, Saltman gave the first examples for which C(W)G is not rational over C. Around 1986, Bogomolov gave a formula which expresses this cohomology group in terms of the cohomology of the group G.In this paper, we prove a formula for the prime to 2 part of the unramified cohomology group of degree 3 of C(W)G. Specializing to the case where G is a central extension of an Fp-vector space by another, we get a method to construct nontrivial elements in this unramified cohomology group. In this way we get an example of a group G for which the field C(W)G is not rational although its unramified cohomology group of degree 2 is trivial.

Tamagawa numbers of diagonal cubic surfaces, numerical evidence

A refined version of Maninsconjecture aboutthe asymptotics of points of bounded height on Fano varieties has been developped by Batyrev and the authors. We test numerically this refined conjecture for some diagonal cubic surfaces.

Counting Points On Varieties Using Universal Torsors

Around 1989, Manin initiated a program to understand the asymptotic behaviour of rational points of bounded height on Fano varieties. This program led to the search of new methods to estimate the number of points of bounded height on various classes of varieties. Methods based on harmonic analysis were successful for compactifications of homogeneous spaces. However, they do not apply to other types of varieties. Universal torsors, introduced by Colliot-Thelene and Sansuc in connection with the Hasse principle and weak approximation, turned out to be a useful tool in the treatment of other varieties. The aim of this short survey is to describe the use of torsors in various representative examples.

Torseurs Universels Et Méthode Du Cercle

Ce texte decrit les premieres etapes d’une generalisation de la methode du cercle au cas d’une hypersurface lisse dans une variete presque de Fano.

TAMAGAWA NUMBERS OF DIAGONAL CUBIC SURFACES OF HIGHER RANK

We consider diagonal cubic surfaces defined by an equation of the form

Counting points of homogeneous varieties over finite fields

a{{x}^{3}} + b{{y}^{3}} + c{{z}^{3}} + d{{t}^{3}} = 0.

Points de hauteur bornée, topologie adélique et mesures de Tamagawa

Numerically, one can find all rational points of height ≤BforBin the range of up to 105thanks to a program due to D. J. Bernstein. On the other hand, there are precise conjectures concerning the constants in the asymptotics of rational points of bounded height due to Manin, Batyrev and the authors. Changing the coefficients one can obtain cubic surfaces with rank of the Picard group varying between 1 and 4. We check that numerical data are compatible with the above conjectures. In a previous paper we considered cubic surfaces with Picard groups of rank one with or without Brauer-Manin obstruction to weak approximation. In this paper, we test the conjectures for diagonal cubic surfaces with Picard groups of higher rank.