Skip to Main Content

Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2024 MCQ for Mathematics of Computation is 1.78.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

The maximal rank of elliptic Delsarte surfaces
HTML articles powered by AMS MathViewer

by Bas Heijne;
Math. Comp. 81 (2012), 1111-1130
DOI: https://doi.org/10.1090/S0025-5718-2011-02529-9
Published electronically: August 4, 2011
PDF | Request permission

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
  • Peter Beelen and Ruud Pellikaan, The Newton polygon of plane curves with many rational points, Des. Codes Cryptogr. 21 (2000), no. 1-3, 41–67. Special issue dedicated to Dr. Jaap Seidel on the occasion of his 80th birthday (Oisterwijk, 1999). MR 1801161, DOI 10.1023/A:1008323208670
  • J. W. S. Cassels, Lectures on elliptic curves, London Mathematical Society Student Texts, vol. 24, Cambridge University Press, Cambridge, 1991. MR 1144763, DOI 10.1017/CBO9781139172530
  • Jasbir Chahal, Matthijs Meijer, and Jaap Top, Sections on certain $j=0$ elliptic surfaces, Comment. Math. Univ. St. Paul. 49 (2000), no. 1, 79–89. MR 1777155
  • Stanley Rabinowitz, A census of convex lattice polygons with at most one interior lattice point, Ars Combin. 28 (1989), 83–96. MR 1039134
  • Matthias SchĂĽtt and Tetsuji Shioda, “Ellipitic Surfaces,” Advanced Studies in Pure Mathematics, 60 (2010), 51-160.
  • Tetsuji Shioda, An explicit algorithm for computing the Picard number of certain algebraic surfaces, Amer. J. Math. 108 (1986), no. 2, 415–432. MR 833362, DOI 10.2307/2374678
  • Tetsuji Shioda, Some remarks on elliptic curves over function fields, AstĂ©risque 209 (1992), 12, 99–114. JournĂ©es ArithmĂ©tiques, 1991 (Geneva). MR 1211006
  • Joseph H. Silverman “The Arithmetic of Ellipitc Curves,” GTM 106, Springer-Verlag, New York 1986.
  • Joseph H. Silverman, Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 151, Springer-Verlag, New York, 1994. MR 1312368, DOI 10.1007/978-1-4612-0851-8
  • Hisashi Usui, On the Mordell-Weil lattice of the elliptic curve $y^2=x^3+t^m+1$. I, Comment. Math. Univ. St. Paul. 49 (2000), no. 1, 71–78. MR 1777154
Similar Articles
  • Retrieve articles in Mathematics of Computation with MSC (2010): 11G05, 14J27
  • Retrieve articles in all journals with MSC (2010): 11G05, 14J27
Bibliographic 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
  • 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)
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 81 (2012), 1111-1130
  • MSC (2010): Primary 11G05, 14J27
  • DOI: https://doi.org/10.1090/S0025-5718-2011-02529-9
  • MathSciNet review: 2869052