In 1993 Swinnerton-Dyer obtained the first non-trivial results on the existence of rational
points for surfaces endowed with a pencil of curves of genus 1 (over a number field). Since then, the
methods used in his proof have been extended and refined by several authors, thereby leading to significant
progress towards several classical arithmetic conjectures. In this talk I will attempt to describe the ideas
that constitute the core of the technique.

This speaker has declined to have their material posted on the web.

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