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Cet ouvrage est consacré à l'arithmétique des surfaces fibrées en courbes de genre 1 au-dessus de la droite projective, et à l'arithmétique des intersections de deux quadriques dans l'espace projectif. Swinnerton-Dyer introduisit en 1993 une technique permettant d'étudier les points rationnels des pinceaux de courbes de genre 1. La première moitié de l'ouvrage reprend et développe cette technique ainsi que ses généralisations ultérieures. La seconde moitié, qui repose sur la première, porte sur les surfaces de del Pezzo de degré 4 et sur les intersections de deux quadriques de dimension supérieure; les résultats annoncés dans [C. R. Math. Acad. Sci. Paris 342 (2006), no. 4, 223--227] y sont démontrés.
This research monograph focuses on the arithmetic, over number fields, of surfaces fibred into curves of genus 1 over the projective line, and of intersections of two quadrics in projective space. The first half contains a complete account of the technique initiated by Swinnerton-Dyer in 1993 for studying rational points on pencils of curves of genus 1, while incorporating and generalising most of its subsequent refinements. The second half, which builds upon the first, is devoted to quartic del Pezzo surfaces and higher-dimensional intersections of two quadrics. It culminates in the proof of the results announced in [C. R. Math. Acad. Sci. Paris 342 (2006), no. 4, 223--227].
This research monograph focuses on the arithmetic, over number fields, of surfaces fibred into curves of genus 1 over the projective line, and of intersections of two quadrics in projective space. The first half contains a complete account of the technique initiated by Swinnerton-Dyer in 1993 for studying rational points on pencils of curves of genus 1, while incorporating and generalising most of its subsequent refinements. The second half, which builds upon the first, is devoted to quartic del Pezzo surfaces and higher-dimensional intersections of two quadrics. It culminates in the proof of the results announced in [C. R. Math. Acad. Sci. Paris 342 (2006), no. 4, 223--227].
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From the reviews:
"The monograph under review grew out of the author's Ph.D thesis. ... The monograph is written in extremely careful manner. ... There are two introductions: the English one makes emphasis on the general context, whereas the French one contains more technical descriptions and references. To sum up, the monograph can be recommended to everyone interested in the state of the art of methods and results concerning arithmetic of rational varieties." (Boris Kunyavskii, Zentralblatt MATH, Vol. 1122 (24), 2007)
"The monograph under review grew out of the author's Ph.D thesis. ... The monograph is written in extremely careful manner. ... There are two introductions: the English one makes emphasis on the general context, whereas the French one contains more technical descriptions and references. To sum up, the monograph can be recommended to everyone interested in the state of the art of methods and results concerning arithmetic of rational varieties." (Boris Kunyavskii, Zentralblatt MATH, Vol. 1122 (24), 2007)