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The arithmetic of certain quartic curves

Published online by Cambridge University Press:  14 November 2011

J. W. S. Cassels
J. W. S. Cassels
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, 16 Mill Lane, University of Cambridge, Cambridge CB2 1SB, U.K.
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Synopsis

Let F(X, Y, Z) be a non-singular quadratic form with rational coefficients. The curve EF(x2, y2, z2) = 0 is of genus 3. A procedure is described for deciding whether there is an effective divisor on E of degree 3 defined over the rationals. There is such a divisor if and only if there is a point on E defined over some algebraic number field of odd degree. An example is constructed for which there is no such divisor although (i) there are points on E defined over all p-adic fields and over the reals and (ii) there are infinitely many rational points on each of the three curves F(X, y2, z2) = 0, F(x2, Y, z2) = 0 and F(x2, y2, Z) = 0.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 100 , Issue 3-4 , 1985 , pp. 201 - 218
Copyright
Copyright © Royal Society of Edinburgh 1985

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