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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
##### Keywords:
rationality; extension of ground field; quartic curve
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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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