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J. -L. Colliot-Thélène

Publications and source records attributed to J. -L. Colliot-Thélène.

7 recordsLinked to original sources
arXiv2019

Brauer-Manin obstruction for Markoff surfaces

Ghosh and Sarnak have studied integral points on surfaces defined by an equation x^2+y^2+z^2-xyz= m over the integers. For these affine surfaces, we systematically study the Brauer group and the Brauer-Manin obstruction to the integral Hasse principle. We prove that strong approximation for integral points on any such surface, away from any finite set of places, fails, and that, for m\neq 0, 4, the Brauer group does not control strong approximation.

math.NT↗
arXiv2011

Potential density for some families of homogeneous spaces

For a smooth, projective family of homogeneous varieties defined over a number field, we show that if potential density holds for the rational points of the base, then it also holds for the total space. A conjecture of Campana and Peternell, known in dimension at most 4 and for certain higher dimensional cases, would then imply potential density for the rational points of smooth projective varieties over number fields whose tangent bundle is nef.

math.AG↗
arXiv2010

Good reduction of the Brauer-Manin obstruction

For a smooth and projective variety over a number field with torsion free geometric Picard group and finite transcendental Brauer group we show that only the archimedean places, the primes of bad reduction and the primes dividing the order of the transcendental Brauer group can turn up in the description of the Brauer-Manin set.

math.AG↗
arXiv2010

Groupe de Chow des zéro-cycles sur les variétés p-adiques

Ceci est un rapport sur l'article "A finiteness theorem for zero-cycles over p-adic fields" (arXiv:math/0605165) de Shuji Saito et Kanetomo Sato. ----- This is a survey on the paper "A finiteness theorem for zero-cycles over p-adic fields" (arXiv:math/0605165) by Shuji Saito and Kanetomo Sato.

math.AG↗
arXiv2009

Une version du théorème d'Amer et Brumer pour les zéro-cycles

M. Amer and A. Brumer have shown that, for two homogeneous quadratic polynomials f and g in at least 3 variables over a field k of characteristic different from 2, the locus f=g=0 has non-trivial solution over k if and only if, for a variable t, the equation f+tg=0 has a non-trivial solution over k(t). We consider a modified version of this result, and show that the projective variety over k defined by f_0=...=f_r=0, where the f_i are homogeneous polynomials over k of the same degree d\ge2 in n+1 variables (with n+1\ge r+2) , has a 0-cycle of degree 1 over k if and only if the generic hypersurface f_0+t_1f_1+...+t_rf_r=0 has a 0-cycle of degree 1 over k(t_1,...,t_r).

math.AG↗
arXiv2004

The density of rational points in curves and surfaces

Let $X$ be an algebraic variety, defined over the rationals. This paper gives upper bounds for the number of rational points on $X$, with height at most $B$, for the case in which $X$ is a curve or a surface. In the latter case one excludes from the counting function those points that lie on lines in the surface. The bounds are uniform for all $X$ of a given degree. They are best possible in the case of curves. As an application it is shown that if $F$ is an irreducible binary form of degree 3 or more then almost all integers represented by $F$ have essentially one such representation.

math.NT↗