Electronic Resource
Springer
Compositio mathematica
110 (1998), S. 335-367
ISSN:
1570-5846
Keywords:
elliptic curve
;
Mordell-Weil group
;
rank
;
torsion
;
integral points
;
L-series
;
Birch and Swinnerton-Dyer conjecture
;
height
;
regulator
;
elliptic logarithm
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract In an earlier paper we developed an algorithm for computing all integral points on elliptic curves over the rationals Q. Here we illustrate our method by applying it to Mordell's Equation y2=x3+k for 0 ≠ k ∈ Z and draw some conclusions from our numerical findings. In fact we solve Mordell's Equation in Z for all integers k within the range 0 〈 | k | ≤ 10 000 and partially extend the computations to 0 〈 | k | ≤ 100 000. For these values of k, the constant in Hall's conjecture turns out to be C=5. Some other interesting observations are made concerning large integer points, large generators of the Mordell–Weil group and large Tate–Shafarevič groups. Three graphs illustrate the distribution of integer points in dependence on the parameter k. One interesting feature is the occurrence of lines in the graphs.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1023/A:1000281602647
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