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 . In this article we find all elliptic curves over for which his method is applicable. For these curves we also compute the maximal Mordell-Weil rank.
2.J.
W. S. Cassels, Lectures on elliptic curves, London
Mathematical Society Student Texts, vol. 24, Cambridge University
Press, Cambridge, 1991. MR 1144763
(92k:11058)
7.Tetsuji
Shioda, Some remarks on elliptic curves over function fields,
Astérisque 209 (1992), 12, 99–114.
Journées Arithmétiques, 1991 (Geneva). 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 elliptic
curves, Graduate Texts in Mathematics, vol. 151, Springer-Verlag,
New York, 1994. MR 1312368
(96b:11074)
10.Hisashi
Usui, On the Mordell-Weil lattice of the elliptic curve
𝑦²=𝑥³+𝑡^{𝑚}+1. I, Comment.
Math. Univ. St. Paul. 49 (2000), no. 1, 71–78.
MR
1777154 (2001g:11087)
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
. In this article we find all elliptic curves over 
for which his method is applicable. For these curves we also compute the maximal Mordell-Weil rank. 