The solubility of diagonal cubic surfaces

Peter Swinnerton-Dyer

Annales scientifiques de l'École Normale Supérieure (2001)

  • Volume: 34, Issue: 6, page 891-912
  • ISSN: 0012-9593

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Swinnerton-Dyer, Peter. "The solubility of diagonal cubic surfaces." Annales scientifiques de l'École Normale Supérieure 34.6 (2001): 891-912. <http://eudml.org/doc/82561>.

@article{Swinnerton2001,
author = {Swinnerton-Dyer, Peter},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {elliptic curve; projective diagonal cubic surface; Hasse principle; Tate-Shafarevich group},
language = {eng},
number = {6},
pages = {891-912},
publisher = {Elsevier},
title = {The solubility of diagonal cubic surfaces},
url = {http://eudml.org/doc/82561},
volume = {34},
year = {2001},
}

TY - JOUR
AU - Swinnerton-Dyer, Peter
TI - The solubility of diagonal cubic surfaces
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2001
PB - Elsevier
VL - 34
IS - 6
SP - 891
EP - 912
LA - eng
KW - elliptic curve; projective diagonal cubic surface; Hasse principle; Tate-Shafarevich group
UR - http://eudml.org/doc/82561
ER -

References

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  1. [1] Bender A.O., Sir Swinnerton-Dyer P., Solubility of certain pencils of curves of genus 1, and of the intersection of two quadrics in P4, Proc. London Math. Soc.83 (3) (2001) 299-329. Zbl1018.11031MR1839456
  2. [2] Cassels J.W.S., Arithmetic on curves of genus 1, I. On a conjecture of Selmer, J. Reine Angew. Math.202 (1959) 52-99. Zbl0090.03005MR109136
  3. [3] Cassels J.W.S., Guy M.J.T., On the Hasse principle for cubic surfaces, Mathematika13 (1966) 111-120. Zbl0151.03405MR211966
  4. [4] Colliot-Thélène J.-L., Surfaces cubiques diagonales, in: Séminaire de théorie des nombres, Paris 1984–85, Progress in Math., 63, Birkhäuser, 1986, pp. 51-66. Zbl0595.14029MR897341
  5. [5] Colliot-Thélène J.-L., Kanevsky D., Sansuc J.-J., Arithmétique des surfaces cubiques diagonales, in: Diophantine Approximation and Transcendence Theory, Springer Lecture Notes, 1290, 1987, pp. 1-108. Zbl0639.14018MR927558
  6. [6] Colliot-Thélène J.-L., Skorobogatov A.N., Sir Swinnerton-Dyer P., Hasse principle for pencils of curves of genus one whose Jacobians have rational 2-division points, Invent. Math.134 (1998) 579-650. Zbl0924.14011MR1660925
  7. [7] Hasse H., Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper, Teil I, Jahresbericht der D.M.V.35 (1926) 1-55. Zbl52.0150.19JFM52.0150.19
  8. [8] Heath-Brown D.R., The solubility of diagonal cubic diophantine equations, Proc. London Math. Soc. (3)79 (1999) 241-259. Zbl1029.11010MR1702242
  9. [9] Kanevsky D., Application of the conjecture on the Manin obstruction to various Diophantine problems, Astérisque147–148 (1987) 307-314. Zbl0625.14010MR891437
  10. [10] Milne J.S., Arithmetic Duality Theorems, Academic Press, 1986. Zbl0613.14019MR881804
  11. [11] Selmer E.S., Sufficient congruence conditions for the existence of rational points on certain cubic surfaces, Math. Scand.1 (1953) 113-119. Zbl0051.03202MR57908
  12. [12] Sir Swinnerton-Dyer P., Some applications of Schinzel's Hypothesis to Diophantine Equations, in: Györy, Iwaniec, Urbanowicz (Eds.), Number Theory in Progress, 1999. Zbl0937.11024

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