Intersections de deux quadriques et pinceaux de courbes de genre 1

TL;DR

This monograph studies rational points on genus 1 fibrations and intersections of two quadrics over number fields, developing techniques and establishing the Hasse principle under certain conjectures.

Contribution

It advances the understanding of rational points on genus 1 fibrations and intersections of quadrics, extending techniques and proving the Hasse principle conditionally.

Findings

Develops techniques for studying rational points on genus 1 fibrations.

Establishes the Hasse principle for smooth intersections of two quadrics in higher dimensions.

Proves the Hasse principle for certain del Pezzo surfaces of degree 4 under conjectural assumptions.

Abstract

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 takes up and develops further the technique initiated by Swinnerton-Dyer in 1993, and later generalised by Colliot-Th\'el\`ene, Skorobogatov and Swinnerton-Dyer, for studying rational points on pencils of curves of genus 1. The second half, which builds upon the first, is devoted to quartic del Pezzo surfaces and to higher-dimensional intersections of two quadrics. Conditionally on two well-known conjectures (Schinzel's hypothesis and the finiteness of Tate-Shafarevich groups of elliptic curves), it establishes the Hasse principle for all smooth n-dimensional intersections of two quadrics with n>2 as well as for a large class of del Pezzo surfaces of degree 4.