Hostname: page-component-6d856f89d9-gndc8 Total loading time: 0 Render date: 2024-07-16T07:05:55.392Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Alter, R. and Curtz, T. B.. A note on congruent numbers. Math. Comp. 28 (1974), 303–305.CrossRefGoogle Scholar
2Bremner, A.. Some quartic curves with no points in any cubic field. Proc. London Math. Soc. (to appear).Google Scholar
3Bremner, A., Lewis, D. J. and Morton, P.. Some varieties with points only in a field extension. Arch. Math. 43 (1984), 344–350.CrossRefGoogle Scholar
4Coray, D. F.. Algebraic points on cubic hypersurfaces. Acta Arith. 30 (1976), 267–296.CrossRefGoogle Scholar
5Faddeev, D. K.. Group of divisor classes on the curve defined by x 4+y 4= 1 (Russian). Dokl. Akad. Nauk SSSR 134 (1960), 776–777. Translation in: Soviet Math. Doklady 1 (1961), 1149–1151.Google Scholar
6Mordell, L. J.. The diophantine equation x 4 + y 4=1 in algebraic number fields. Acta Arith. 14 (1968), 347–355.CrossRefGoogle Scholar
7Schinzel, A.. Hasse's principle for systems of ternary quadratic forms and one biquadratic form. Studia Math. 77 (1983), 103–109.CrossRefGoogle Scholar