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The maximal rank of elliptic Delsarte surfaces


Author: Bas Heijne
Journal: Math. Comp. 81 (2012), 1111-1130
MSC (2010): Primary 11G05, 14J27
DOI: https://doi.org/10.1090/S0025-5718-2011-02529-9
Published electronically: August 4, 2011
MathSciNet review: 2869052
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Abstract: Shioda described a method to compute the Lefschetz number of a Delsarte surface. In one of his examples he uses this method to compute the rank of an elliptic curve over $ k(t)$. In this article we find all elliptic curves over $ k(t)$ for which his method is applicable. For these curves we also compute the maximal Mordell-Weil rank.


References [Enhancements On Off] (What's this?)

  • 1. Peter Beelen and Ruud Pellikaan, ``The Newton Polygon of Plane Curves with Many Rational Points,'' Designs, Codes and Cryptography, 21, 41-67, 2000. MR 1801161 (2003c:14024)
  • 2. J.W.S. Cassels ``Lectures on Elliptic Curves,'' LMSST 24, Cambridge University Press, Cambridge, 1991. MR 1144763 (92k:11058)
  • 3. J. Chahal, M. Meijer and J. Top, ``Sections on Certain $ j=0$ Elliptic Surfaces,'' Comm. Math. Univ St. Pauli, 49 (2000), 79-89. MR 1777155 (2001j:14053)
  • 4. Stanley Rabinowitz, ``A Census of Convex Lattice Polygons with at most one Interior Point,'' Ars Combinatoria, 28 (1989), 83-96. MR 1039134 (91f:52019)
  • 5. Matthias Schütt and Tetsuji Shioda, ``Ellipitic Surfaces,'' Advanced Studies in Pure Mathematics, 60 (2010), 51-160.
  • 6. Tetsuji Shioda ``An Explicit Algorithm for Computing the Picard Number of Certain Algebraic Surfaces,'' American Journal of Mathematics, vol. 108, No. 2 (April 1986), pp 415-432. MR 833362 (87g:14033)
  • 7. Tetsuji Shioda ``Some remarks on elliptic curves over function fields,'' Astérique, 209 (1992), 99-114. MR 1211006 (94d:11046)
  • 8. Joseph H. Silverman ``The Arithmetic of Ellipitc Curves,'' GTM 106, Springer-Verlag, New York 1986.
  • 9. Joseph H. Silverman ``Advanced Topics in the Arithmetic of Ellipitc Curves,'' GTM 151, Springer-Verlag, New York 1994. MR 1312368 (96b:11074)
  • 10. Hisashi Usui ``On the Mordell-Weil Lattice of the Elliptic Curve $ y^2=x^3+t^m+1$,'' Comm. Math. Univ St. Pauli, 49 (2000), 71-78. MR 1777154 (2001g:11087)

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Additional Information

Bas Heijne
Affiliation: The Johann Bernoulli Institute for Mathematics and Computer Science (JBI), University of Groningen, P. O. Box 407, 9700AK Groningen, the Netherlands
Email: b.l.heijne@rug.nl

DOI: https://doi.org/10.1090/S0025-5718-2011-02529-9
Received by editor(s): July 9, 2010
Received by editor(s) in revised form: January 14, 2011
Published electronically: August 4, 2011
Additional Notes: The author would like to thank Jaap Top for several fruitful discussions
This work was supported by a grant of the Netherlands Organization for Scientific Research (NWO)
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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