×

zbMATH — the first resource for mathematics

Some varieties with points only in a field extension. (English) Zbl 0532.14011
Several authors have investigated the following problem. Let \(\Gamma\) be an irreducible algebraic variety of degree \(d\), in projective \(n\)-space \(\mathbb P^ n\), defined over a field \(k\); and suppose \(K\) is a finite extension of \(k\) with \([K:k]\) prime to \(d\). If \(\Gamma\) has a point defined over \(K\), then does it necessarily have a point defined over \(k\)? For instance, the result is known to be true for quadrics in \(\mathbb P^ n\), and for cubic plane curves; it is also true for the intersection of two quadrics, whereas it fails to hold for three quadrics. The current paper comprises several further counterexamples, which are the following.
(I) Two cubics in \(\mathbb P^ 2\) having a common zero over a quadratic extension of \(\mathbb Q\), but having no common zero in \(\mathbb Q\).
(II) Two examples of quartic curves defined over \(\mathbb Q\), everywhere locally solvable, having no rational point, yet possessing points defined over a cubic extension of \(\mathbb Q\).
(III) Two examples akin to those of (II), but now over the function field \(\mathbb Q(t)\).
(IV) A quartic form in 16 variables with a point over a cubic extension of \(\mathbb Q\), but with no point in \(\mathbb Q\) itself.
Reviewer: Andrew Bremner

MSC:
14G25 Global ground fields in algebraic geometry
14H25 Arithmetic ground fields for curves
11D25 Cubic and quartic Diophantine equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] M.Amer, Quadratische Formen ?ber Funktionenk?rpern. Dissertation, Mainz 1976.
[2] A. Bremner andP. Morton, A new characterization of the integer 5906. Manuscripta Math.44, 187-229 (1983). · Zbl 0516.10012
[3] A. Brumer, Remarques sur les couples de formes quadratiques. C.R. Acad. Sci. Paris286 A, 679-681 (1978). · Zbl 0392.10021
[4] J. W. S. Cassels, On a problem of Pfister about systems of quadratic forms. Arch. Math.33, 29-32 (1979). · Zbl 0405.10016
[5] D. F. Coray, On a problem of Pfister about intersections of three quadrics. Arch. Math.34, 403-411 (1980). · Zbl 0435.14004
[6] D. F. Coray, Algebraic points on cubic hypersurfaces. Acta Arith.30, 267-296 (1976). · Zbl 0294.14012
[7] H. Davenport andD. J. Lewis, Homogeneous additive equations. Proc. Royal Soc. A.274, 443-460 (1963). · Zbl 0118.28002
[8] B. N.Delone and D. K.Faddeev, Theory of irrationalities of the third degree. Amer. Math. Soc. Transl. 1964. · Zbl 0133.30202
[9] A. M. Legendre, M?m. Acad. Roy. Sc. de l’Inst. de France6, 41, Sec. 49 (1823).
[10] A. Pfister, Systems of Quadratic Forms. Colloque sur les Formes Quadratiques, 2 Bull. Soc. Math. France Mem.59, 115-123 (1979). · Zbl 0407.10017
[11] H. Poincar?, Sur les propri?t?s arithm?tiques des courbes alg?briques. J. Math. Pures Appl.7, 161-233 (1901). · JFM 32.0564.06
[12] T. A. Springer, Sur les formes quadratiques d’indice z?ro. C.R. Acad. Sci. Paris234, 1517-1519 (1952). · Zbl 0046.24303
[13] A.Weil, Sur les courbes alg?briques et les vari?t?s qui s’en d?duisent. Paris 1948. · Zbl 0036.16001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.