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A diophantine equation over a function field

Part of: Surfaces and higher-dimensional varieties

Published online by Cambridge University Press:  09 April 2009

J. W. S. Cassels
J. W. S. Cassels
Affiliation:
Department of Pure Mathematics and Mathematical Statistics 16 Mill Lane Cambridge CB2 1SB, United Kingdom
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Abstract

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Let x0, x1, x2, x3 be polynomials in a variable t and with coefficients in a field k of character of characteristic 0. If and , then x0 = x1 = x2 = x3 = 0. This partially answers a question of Pjatetskii-Š;apiro and Šafarevič about the K3-surface . The proof uses a technique of M. R. Christie.

MSC classification

Secondary: 14J25: Special surfaces
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 25 , Issue 4 , June 1978 , pp. 489 - 496
Copyright
Copyright © Australian Mathematical Society 1978

References

Christie, M. R. (1976), “Positive definite functions of two variables which are not the sum of three squares”, J. Number Theory 8, 224–232.CrossRefGoogle Scholar
Dem'janenko, V. A. (1977), “indeterminate equation” (Russian), Zap. Naucn, Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI), 67, 163–166.Google Scholar
Hellegouarch, Y. (1970), “Étude des points d'ordre fini des variétés abeliennes de dimension un définies sur un anneau principal”, J. reine. angew. Math. 244, 20–36.Google Scholar
Pjatetskii-Sapiro, I. I. and Safarevic, I. R. (1971), “Torelli's theorem for K3 algebraic surfaces”, (Russian), Izv. Akad. Nauk SSSR (ser. mat.) 35, 530–572, especially the last section.Google Scholar