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SOLUBILITY OF CERTAIN PENCILS OF CURVES OF GENUS 1, AND OF THE INTERSECTION OF TWO QUADRICS IN ℙ4

Published online by Cambridge University Press:  23 August 2001

A. O. BENDER  and
PETER SWINNERTON-DYER
A. O. BENDER
Affiliation:
Robinson College, Cambridge CB3 9ET, a.o.bender@dpmms.cam.ac.uk
PETER SWINNERTON-DYER
Affiliation:
Trinity College, Cambridge CB2 1TQ, hpfs100@newton.cam.ac.uk
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Abstract

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.

Type
Research Article
Information
Proceedings of the London Mathematical Society , Volume 83 , Issue 2 , September 2001 , pp. 299 - 329
Copyright
2001 London Mathematical Society

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