On the number of Mordell-Weil generators for cubic surfaces

Let S be a smooth cubic surface over a field K. It is well-known that new K-rational points may be obtained from old ones by secant and tangent constructions. A Mordell-Weil generating set is a subset B of S(K) of minimal cardinality which generates S(K) via successive secant and tangent constructions. Let r(S,K) be the cardinality of such a Mordell-Weil generating set. Manin posed what is known as the Mordell-Weil problem for cubic surfaces: if K is finitely generated over its prime subfield then r(S,K) is finite. In this paper, we prove a special case of this conjecture. Namely, if S contains two skew lines both defined over K then r(S,K) = 1. One of the difficulties in studying the secant and tangent process on cubic surfaces is that it does not lead to an associative binary operation as in the case of elliptic curves. As a partial remedy we introduce an abelian group H_S(K) associated to a cubic surface S/K, naturally generated by the K-rational points, which retains much information about the secant and tangent process. In particular, r(S, K) is large as soon as the minimal number of generators of H_S(K) is large. In situations where weak approximation holds, H_S has nice local-to-global properties. We use these to construct a family of smooth cubic surfaces over the rationals such that r(S,K) is unbounded in this family. This is the cubic surface analogue of the unboundedness of ranks conjecture for elliptic curves

Brauer-Manin pairing, class field theory and motivic homology

For a smooth proper variety over a $p$-adic field, the Brauer group and abelian fundamental group are related to the higher Chow groups by the Brauer-Manin pairing and the class field theory. We generalize this relation to smooth (possibly non-proper) varieties, using the motivic homology and the tame version of Wiesend's ideal class group. Several examples are discussed.Comment: 25 page

Rationality problems and conjectures of Milnor and Bloch-Kato

We show how the techniques of Voevodsky's proof of the Milnor conjecture and the Voevodsky- Rost proof of its generalization the Bloch-Kato conjecture can be used to study counterexamples to the classical L\"uroth problem. By generalizing a method due to Peyre, we produce for any prime number l and any integer n >= 2, a rationally connected, non-rational variety for which non-rationality is detected by a non-trivial degree n unramified \'etale cohomology class with l-torsion coefficients. When l = 2, the varieties that are constructed are furthermore unirational and non-rationality cannot be detected by a torsion unramified \'etale cohomology class of lower degree.Comment: 15 pages; Revised and extended version of http://arxiv.org/abs/1001.4574 v2; Comments welcome

Brauer-Manin obstruction for integral points of homogeneous spaces and representation by integral quadratic forms

An integer may be represented by a quadratic form over each ring of p-adic integers and over the reals without being represented by this quadratic form over the integers. More generally, such failure of a local-global principle may occur for the representation of one integral quadratic form by another integral quadratic form. We show that many such examples may be accounted for by a Brauer-Manin obstruction for the existence of integral points on schemes defined over the integers. For several types of homogeneous spaces of linear algebraic groups, this obstruction is shown to be the only obstruction to the existence of integral points. ----- Une forme quadratique enti\`ere peut \^etre repr\'esent\'ee par une autre forme quadratique enti\`ere sur tous les anneaux d'entiers p-adiques et sur les r\'eels, sans l'\^etre sur les entiers. On en trouve de nombreux exemples dans la litt\'erature. Nous montrons qu'une partie de ces exemples s'explique au moyen d'une obstruction de type Brauer-Manin pour les points entiers. Pour plusieurs types d'espaces homog\`enes de groupes alg\'ebriques lin\'eaires, cette obstruction est la seule obstruction \`a l'existence d'un point entier.Comment: 53 pages, in Englis

\'Etale homotopy equivalence of rational points on algebraic varieties

It is possible to talk about the \'etale homotopy equivalence of rational points on algebraic varieties by using a relative version of the \'etale homotopy type. We show that over $p$-adic fields rational points are homotopy equivalent in this sense if and only if they are \'etale-Brauer equivalent. We also show that over the real field rational points on projective varieties are \'etale homotopy equivalent if and only if they are in the same connected component. We also study this equivalence relation over number fields and prove that in this case it is finer than the other two equivalence relations for certain generalised Ch\^atelet surfaces.Comment: New title, rewritten introduction, 48 pages. To appear in Algebra & Number Theor

Division Algebras and Quadratic Forms over Fraction Fields of Two-dimensional Henselian Domains

Let $K$ be the fraction field of a 2-dimensional, henselian, excellent local domain with finite residue field $k$. When the characteristic of $k$ is not 2, we prove that every quadratic form of rank $\ge 9$ is isotropic over $K$ using methods of Parimala and Suresh, and we obtain the local-global principle for isotropy of quadratic forms of rank 5 with respect to discrete valuations of $K$. The latter result is proved by making a careful study of ramification and cyclicity of division algebras over the field $K$, following Saltman's methods. A key step is the proof of the following result, which answers a question of Colliot-Th\'el\`ene--Ojanguren--Parimala: For a Brauer class over $K$ of prime order $q$ different from the characteristic of $k$, if it is cyclic of degree $q$ over the completed field $K_v$ for every discrete valuation $v$ of $K$, then the same holds over $K$. This local-global principle for cyclicity is also established over function fields of $p$-adic curves with the same method.Comment: Final version, 31 pages, may be slightly different from the published versio