1,272 research outputs found
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 -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.
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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 -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 be the fraction field of a 2-dimensional, henselian, excellent local
domain with finite residue field . When the characteristic of is not 2,
we prove that every quadratic form of rank is isotropic over 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
. The latter result is proved by making a careful study of ramification and
cyclicity of division algebras over the field , 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 of prime
order different from the characteristic of , if it is cyclic of degree
over the completed field for every discrete valuation of ,
then the same holds over . This local-global principle for cyclicity is also
established over function fields of -adic curves with the same method.Comment: Final version, 31 pages, may be slightly different from the published
versio
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