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Algebraic points on quartic curves over function fields. (English) Zbl 0581.14015
Let \(\Gamma\) be an irreducible algebraic projective variety of degree \(d\) defined over a field \(k\). Let \(K\) be a finite extension of \(k\) with \([K:k]\) prime to \(d\). One is interested in knowing whether the nontrivialiy of \(\Gamma (K)\) implies that of \(\Gamma (k)\). When the ground field \(k\) is the function field \(\mathbb Q(\lambda)\) of transcendence one, the author, D. J. Lewis and P. Morton [Arch. Math. 43, 344–350 (1984; Zbl 0532.14011)] have given two examples of a quartic curve over \(k\) which has no \(k\)-points but does have points in extension fields \(K\) of all odd degrees. The present paper is concerned with pointing the above examples into an infinite family of examples.
MSC:
14G25 Global ground fields in algebraic geometry
14H25 Arithmetic ground fields for curves
11D25 Cubic and quartic Diophantine equations
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References:
[1] Pfister, Systems of quadratic forms, Colloque sur les formes quadratiques 59 pp 115– (1979)
[2] DOI: 10.1007/BF01224978 · Zbl 0426.14008
[3] DOI: 10.1007/BF01196658 · Zbl 0532.14011
[4] DOI: 10.1007/BF01222721 · Zbl 0405.10016
[5] Coray, Ada Arith. 30 pp 267– (1976)
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