A. O. Bender and Sir Peter Swinnerton-Dyer

The main part of the paper finds necessary conditions for solubility of a pencil of curves of genus 1, each of which is a 2-covering of an elliptic curve with at least one 2-division point. As in previous work, these are proved subject to Schinzel's Hypothesis and to the finiteness of the Tate--\u{S}afarevi\u{c} group of elliptic curves defined over a number field. It thus generalizes earlier work of Colliot-Thélène, Skorobogatov and the second author.

The final section gives necessary conditions (though of a rather ugly nature) for the solubility of a Del Pezzo surface of degree 4.

2000 Mathematical Subject Classification: 11D25.