height function
Definition 1
Let be an abelian group. A height function on is a function with the properties:
-
1.
For all there exists a constant , depending on and , such that for all :
-
2.
There exists an integer and a constant , depending on , such that for all :
-
3.
For all , the following set is finite:
Examples:
-
1.
For , a fraction in lower terms, define . Even though this is not a height function as defined above, this is the prototype of what a height function should look like.
-
2.
Let be an elliptic curve over . The function on , the points in with coordinates in , :
is a height function ( is defined as above). Notice that this depends on the chosen Weierstrass model of the curve.
- 3.
Finally we mention the fundamental theorem of “descent”, which highlights the importance of the height functions:
Theorem 1 (Descent)
Let be an abelian group and let be a height function. Suppose that for the integer , as in property (2) of height, the quotient group is finite. Then is finitely generated.
References
- 1 Joseph H. Silverman, The Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1986.
Title | height function |
Canonical name | HeightFunction |
Date of creation | 2013-03-22 13:49:09 |
Last modified on | 2013-03-22 13:49:09 |
Owner | alozano (2414) |
Last modified by | alozano (2414) |
Numerical id | 5 |
Author | alozano (2414) |
Entry type | Definition |
Classification | msc 14H52 |
Related topic | EllipticCurve |
Related topic | RankOfAnEllipticCurve |
Related topic | ArithmeticOfEllipticCurves |
Related topic | CanonicalHeightOnAnEllipticCurve |
Defines | height function |
Defines | canonical height |
Defines | descent theorem |