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Jean-Louis Colliot-Thélène

  1. Rational points on some special cubic surfaces over a global function field.

    Пнд, 04/19/2010 - 15:01 — SomNambul
    Authors: Jean-Louis Colliot-Thélène, Sir Peter Swinnerton-Dyer
    Subjects: Algebraic Geometry
    Abstract

    Let F be a finite field of characteristic p. We consider smooth surfaces over
    F(t) defined by an equation f+tg=0, where f and g are forms of degree d in 4
    variables with coefficients in F, with d prime to p. We prove : For such
    surfaces over F(t), the Brauer-Manin obstruction to the existence of a
    zero-cycle of degree one is the only obstruction. For d=3 (cubic surfaces),
    this leads to the same result for rational points. Soit F un corps fini de
    caract\'eristique p.

  2. Groupe de Brauer et points entiers de deux surfaces cubiques affines.

    Чт, 11/19/2009 - 16:01 — SomNambul
    Authors: Jean-Louis Colliot-Thélène, Olivier Wittenberg
    Subjects: Number Theory
    Abstract

    Let a be a nonzero integer. If a is not congruent to 4 or 5 modulo 9 then
    there is no Brauer-Manin obstruction to the existence of integers x, y, z such
    that x^3+y^3+z^3=a. In addition, there is no Brauer-Manin obstruction to the
    existence of integers x, y, z such that x^3+y^3+2z^3=a.

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  3. Is the function field of a reductive Lie algebra purely transcendental over the field of invariants for the adjoint action?.

    Ср, 09/09/2009 - 22:08 — SomNambul
    Authors: Jean-Louis Colliot-Thélène, Boris Kunyavskiĭ, Vladimir L. Popov, Zinovy Reichstein
    Subjects: Algebraic Geometry
    Abstract

    Let $k$ be a field of characteristic zero, let $G$ be a connected reductive
    algebraic group over $k$ and let $\mathfrak{g}$ be its Lie algebra. Let $k(G)$,
    respectively, $k(\mathfrak{g})$, be the field of $k$-rational functions on $G$,
    respectively, $\mathfrak{g}$. The conjugation action of $G$ on itself induces
    the adjoint action of $G$ on $\mathfrak{g}$. We investigate the question
    whether or not the field extensions $k(G)/k(G)^G$ and
    $k(\mathfrak{g})/k(\mathfrak{g})^G$ are purely transcendental.

  4. Is the function field of a reductive Lie algebra purely transcendental over the field of invariants for the adjoint action?.

    Ср, 09/09/2009 - 22:08 — SomNambul
    Authors: Jean-Louis Colliot-Thélène, Boris Kunyavskiĭ, Vladimir L. Popov, Zinovy Reichstein
    Subjects: Algebraic Geometry
    Abstract

    Let $k$ be a field of characteristic zero, let $G$ be a connected reductive
    algebraic group over $k$ and let $\mathfrak{g}$ be its Lie algebra. Let $k(G)$,
    respectively, $k(\mathfrak{g})$, be the field of $k$-rational functions on $G$,
    respectively, $\mathfrak{g}$. The conjugation action of $G$ on itself induces
    the adjoint action of $G$ on $\mathfrak{g}$. We investigate the question
    whether or not the field extensions $k(G)/k(G)^G$ and
    $k(\mathfrak{g})/k(\mathfrak{g})^G$ are purely transcendental.

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