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Diagonal Cubic Equations in four Variables with Prime Coefficients

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Rational Points on Algebraic Varieties

Part of the book series: Progress in Mathematics ((PM,volume 199))

Abstract

The aim of this paper is to give an alternative proof of a theorem of R. Heath-Brown regarding the existence of non-zero integral solutions of the equation

$$ p_1 X_1^3 + p_2 X_2^3 + p_3 X_3^3 + p4X_4^3 = 0 $$

where the pj are prime integers congruent to 2 modulo 3.

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References

  1. J.W.S. CasselsArithmetic on curves of genus 1. I. On a conjecture of SelmerJ. reine angew. Math. 202 (1959), 52–99.

    Google Scholar 

  2. J.W.S. CasselsLectures on Elliptic CurvesCambridge University Press, 1991.

    Google Scholar 

  3. R. Heath-BrownThe solubility of diagonal cubic Diophantine equationsProc. London Math. Soc. (3) 79 (1999), 241–259.

    Article  MathSciNet  MATH  Google Scholar 

  4. J.-P. SerreA course in ArithmeticNew York: Springer—Verlag, 1973.

    Book  MATH  Google Scholar 

  5. J.-P. SerreGalois cohomologySpringer—Verlag Berlin Heidelberg, 1997.

    Google Scholar 

  6. J.H. SilvermanThe Arithmetic of Elliptic CurvesNew York: Springer—Verlag, 1986.

    Google Scholar 

  7. [7] H.P.F. Swinnerton—DyerThe solubility of diagonal cubic surfacesto appear in Ann. Sci. École Norm. Sup.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Imperial College of Science, Technology and Medicine, London

    Carmen Laura Basile

  2. Department of Pure Mathematics and Mathematical Statistics, Sidney Sussex College, Cambridge

    Thomas Anthony Fisher

Authors
  1. Carmen Laura Basile
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  2. Thomas Anthony Fisher
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Editor information

Editors and Affiliations

  1. Institut Fourier, UFR de Mathématiques, UMR 5582, Université de Grenoble I et CNRS, B.P. 74, 38402, Saint-Martin d’Hères, France

    Emmanuel Peyre

  2. Department of Mathematics, Princeton University, Washington Road, Princeton, 08544-1000, NJ, USA

    Yuri Tschinkel

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© 2001 Springer Basel AG

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Basile, C.L., Fisher, T.A. (2001). Diagonal Cubic Equations in four Variables with Prime Coefficients. In: Peyre, E., Tschinkel, Y. (eds) Rational Points on Algebraic Varieties. Progress in Mathematics, vol 199. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8368-9_1

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  • DOI: https://doi.org/10.1007/978-3-0348-8368-9_1

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9536-1

  • Online ISBN: 978-3-0348-8368-9

  • eBook Packages: Springer Book Archive

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