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Principe de Hasse pour les surfaces de del Pezzo de degré 4
Principe de Hasse pour les surfaces de del Pezzo de degré 4
Cette thèse porte sur l'arithmétique des surfaces munies d'un pinceau decourbes de genre $1$, des surfaces de del Pezzo de degré $4$ et desintersections de deux quadriques dans $\mathbf{P}^n_k$. Dans les deuxpremiers chapitres, nous développons la technique introduite parSwinnerton-Dyer en 1993 et raffinée par Colliot-Thélène, Skorobogatov etSwinnerton-Dyer en 1998 permettant d'étudier les points rationnels dessurfaces munies d'un pinceau de courbes de genre $1$. Le troisièmechapitre, qui repose sur le second, est consacré aux surfaces de del Pezzode degré $4$ et aux intersections de deux quadriques dans $\mathbf{P}^n_k$.Soient $k$ un corps de nombres et $X$ une intersection lisse de deuxquadriques dans $\mathbf{P}^n_k$. On dit que $X$ satisfait au principe deHasse si l'existence d'un $k_v$-point de $X$ pour toute place $v$ de $k$suffit à assurer l'existence d'un $k$-point de $X$. Colliot-Thélène etSansuc ont conjecturé que(i) $X$ satisfait au principe de Hasse si $n\geq 5$;(ii) $X$ satisfait au principe de Hasse si $n=4$ et $\mathrm{Br}(X)/\mathrm{Br}(k)=0$.Le but du troisième chapitre est de démontrer la conjecture (i), ainsiqu'une grande partie de la conjecture (ii), en admettant l'hypothèse deSchinzel et la finitude des groupes de Tate-Shafarevich des courbeselliptiques sur les corps de nombres.
This thesis is concerned with the arithmetic of surfaces endowed with apencil of curves of genus $1$, of del Pezzo surfaces of degree $4$, and ofintersections of two quadrics in $\mathbf{P}^n_k$. In the first twochapters, we develop the technique introduced by Swinnerton-Dyer in 1993,and refined by Colliot-Thélène, Skorobogatov and Swinnerton-Dyer in 1998,for studying rational points on surfaces endowed with a pencil of curves ofgenus $1$. The third chapter, which builds upon the second one, is devotedto del Pezzo surfaces of degree $4$ and intersections of two quadrics in$\mathbf{P}^n_k$. Let $k$ be a number field and $X$ be a smoothintersection of two quadrics in $\mathbf{P}^n_k$. The Hasse principle issaid to hold for $X$ if the existence of a $k_v$-point on $X$ for everyplace $v$ of $k$ is enough to imply the existence of a $k$-point on $X$.Colliot-Thélène and Sansuc conjectured that(i) the Hasse principle holds for $X$ if $n\geq 5$;(ii) the Hasse principle holds for $X$ if $n=4$ and $\mathrm{Br}(X)/\mathrm{Br}(k)=0$.The goal of the third chapter is to establish conjecture (i) as well as alarge part of conjecture (ii), assuming Schinzel's hypothesis and thefiniteness of Tate-Shafarevich groups of elliptic curves over numberfields.
- Rice University United States
- Université Paris Diderot France
Hasse principle, surface de del Pezzo, [MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG], del Pezzo surface, Pinceau de courbes elliptiques, groupe de Brauer, principe de Hasse, obstruction de Brauer-Manin, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], Brauer group, Pencil of elliptic curves, Brauer-Manin obstruction, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], [MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT]
Hasse principle, surface de del Pezzo, [MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG], del Pezzo surface, Pinceau de courbes elliptiques, groupe de Brauer, principe de Hasse, obstruction de Brauer-Manin, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], Brauer group, Pencil of elliptic curves, Brauer-Manin obstruction, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], [MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT]
3013
[2] A. O. Bender et O. Wittenberg, A potential analogue of Schinzel's hypothesis for polynomials with coefficients in Fq[t], Int. Math. Res. Not. 2005 (2005), no. 36, 2237-2248.
[3] B. J. Birch et H. P. F. Swinnerton-Dyer, Notes on elliptic curves, I, J. reine angew. Math. 212 (1963), 7-25.
[4] , The Hasse problem for rational surfaces, J. reine angew. Math. 274/275 (1975), 164-174, Collection of articles dedicated to Helmut Hasse on his seventy-fifth birthday, III.
[5] S. Bosch, W. Lütkebohmert et M. Raynaud, Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 21, Springer-Verlag, Berlin, 1990.
[6] M. Bright, Computations on diagonal quartic surfaces, thèse, Cambridge, 2002.
[7] A. Brumer, Remarques sur les couples de formes quadratiques, C. R. Acad. Sci. Paris Sér. A-B 286 (1978), no. 16, A679-A681.
[8] J. W. S. Cassels, Second descents for elliptic curves, J. reine angew. Math. 494 (1998), 101-127.
[14] [18] J-L. Colliot-Thélène, J-J. Sansuc et Sir Peter Swinnerton-Dyer, Intersections of two quadrics and Châtelet surfaces, I, J. reine angew. Math. 373 (1987), 37-107.
[19] , Intersections of two quadrics and Châtelet surfaces, II, J. reine angew. Math.
[20] J-L. Colliot-Thélène, A. N. Skorobogatov et Sir Peter Swinnerton-Dyer, Hasse principle for pencils of curves of genus one whose Jacobians have rational 2-division points, Invent. math. 134 (1998), no. 3, 579-650. [OpenAIRE]
citations This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).0 popularity This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.Average influence This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).Average impulse This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.Average citations This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).0 popularity This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.Average influence This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).Average impulse This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.Average
Citations provided by BIP!Popularity provided by BIP!- Rice University United States
- Université Paris Diderot France
Cette thèse porte sur l'arithmétique des surfaces munies d'un pinceau decourbes de genre $1$, des surfaces de del Pezzo de degré $4$ et desintersections de deux quadriques dans $\mathbf{P}^n_k$. Dans les deuxpremiers chapitres, nous développons la technique introduite parSwinnerton-Dyer en 1993 et raffinée par Colliot-Thélène, Skorobogatov etSwinnerton-Dyer en 1998 permettant d'étudier les points rationnels dessurfaces munies d'un pinceau de courbes de genre $1$. Le troisièmechapitre, qui repose sur le second, est consacré aux surfaces de del Pezzode degré $4$ et aux intersections de deux quadriques dans $\mathbf{P}^n_k$.Soient $k$ un corps de nombres et $X$ une intersection lisse de deuxquadriques dans $\mathbf{P}^n_k$. On dit que $X$ satisfait au principe deHasse si l'existence d'un $k_v$-point de $X$ pour toute place $v$ de $k$suffit à assurer l'existence d'un $k$-point de $X$. Colliot-Thélène etSansuc ont conjecturé que(i) $X$ satisfait au principe de Hasse si $n\geq 5$;(ii) $X$ satisfait au principe de Hasse si $n=4$ et $\mathrm{Br}(X)/\mathrm{Br}(k)=0$.Le but du troisième chapitre est de démontrer la conjecture (i), ainsiqu'une grande partie de la conjecture (ii), en admettant l'hypothèse deSchinzel et la finitude des groupes de Tate-Shafarevich des courbeselliptiques sur les corps de nombres.
This thesis is concerned with the arithmetic of surfaces endowed with apencil of curves of genus $1$, of del Pezzo surfaces of degree $4$, and ofintersections of two quadrics in $\mathbf{P}^n_k$. In the first twochapters, we develop the technique introduced by Swinnerton-Dyer in 1993,and refined by Colliot-Thélène, Skorobogatov and Swinnerton-Dyer in 1998,for studying rational points on surfaces endowed with a pencil of curves ofgenus $1$. The third chapter, which builds upon the second one, is devotedto del Pezzo surfaces of degree $4$ and intersections of two quadrics in$\mathbf{P}^n_k$. Let $k$ be a number field and $X$ be a smoothintersection of two quadrics in $\mathbf{P}^n_k$. The Hasse principle issaid to hold for $X$ if the existence of a $k_v$-point on $X$ for everyplace $v$ of $k$ is enough to imply the existence of a $k$-point on $X$.Colliot-Thélène and Sansuc conjectured that(i) the Hasse principle holds for $X$ if $n\geq 5$;(ii) the Hasse principle holds for $X$ if $n=4$ and $\mathrm{Br}(X)/\mathrm{Br}(k)=0$.The goal of the third chapter is to establish conjecture (i) as well as alarge part of conjecture (ii), assuming Schinzel's hypothesis and thefiniteness of Tate-Shafarevich groups of elliptic curves over numberfields.