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Algebraic points on quartic curves over function fields
Published online by Cambridge University Press: 18 May 2009
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The following general problem is of interest. Let Λ be an irreducible algebraic variety of degree d, in projective n-space Pn, defined over a field k; and suppose that K is a finite extension of k with [K: k] prime to d. If Λ has a point defined over K, then does it necessarily have a point defined over k?
It has been studied in various instances by several authors: see, for example, Cassels [2], Coray [3, 4], Pfister [5], Bremner, Lewis, Morton [1]. Coray [3] shows that a quartic curve Λ over Q may possess points in extension fields of Q of every odd degree greater than one, but have no points in Q itself. Some further examples of this instance occur in the paper of Bremner, Lewis, Morton, with the additional property that the curve Λ also possesses points in every p-adic completion Qp of Q.
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- Copyright © Glasgow Mathematical Journal Trust 1985
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REFERENCES
1.Bremner, A., Lewis, D. J. and Morton, P., Some varieties with points only in a field extension, Archiv der Math. 43 (1984), 344–350.CrossRefGoogle Scholar
2.Cassels, J. W. S., On a problem of Pfister about systems of quadratic forms, Archiv der Math. 33 (1979), 29–32.CrossRefGoogle Scholar
3.Coray, D. F., Algebraic points on cubic hypersurfaces, Ada Arith. 30 (1976), 267–296.CrossRefGoogle Scholar
4.Coray, D. F., On a problem of Pfister about intersections of three quadrics, Archiv der Math. 34 (1980), 403–411.CrossRefGoogle Scholar
5.Pfister, A., Systems of quadratic forms, Colloque sur les formes quadratiques, 2 Bull. Soc.Math. France, Mem. No. 59, (1979), 115–123.Google Scholar
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