rank of an elliptic curve
Theorem 2 (Weak Mordell-Weil).
is finite for all .
The Mordell-Weil theorem implies that for any elliptic curve the group of points has the following structure:
where denotes the set of points of finite order (or torsion group), and is a non-negative integer which is called the of the elliptic curve. It is not known how big this number can get for elliptic curves over . The largest rank known for an elliptic curve over is 28 http://www.math.hr/ duje/tors/tors.htmlElkies (2006).
Note: see Mazur’s theorem for an account of the possible torsion subgroups over .
The elliptic curve has rank 0 and .
Let , then . The torsion group is generated by the point .
Let , then . See http://math.bu.edu/people/alozano/Examples.htmlgenerators here.
- 1 James Milne, Elliptic Curves, online course notes. http://www.jmilne.org/math/CourseNotes/math679.htmlhttp://www.jmilne.org/math/CourseNotes/math679.html
- 2 Joseph H. Silverman, The Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1986.
- 3 Joseph H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1994.
- 4 Goro Shimura, Introduction to the Arithmetic Theory of Automorphic Functions. Princeton University Press, Princeton, New Jersey, 1971.
|Title||rank of an elliptic curve|
|Date of creation||2013-03-22 13:49:12|
|Last modified on||2013-03-22 13:49:12|
|Last modified by||alozano (2414)|
|Defines||weak Mordell-Weil theorem|
|Defines||rank of an elliptic curve|