Frobenius morphism
Let be a field of characteristic and let . Let
be a curve defined over contained in , the
projective space
of dimension . Define the homogeneous ideal
of
to be (the ideal generated by):
For , of the form we define . We define a new curve as
the zero set of the ideal (generated by):
Definition 1.
In order to check that the Frobenius morphism is well defined we need to prove that
This is equivalent to
proving that for any we have .
Without loss of generality we can assume that is a generator
of , i.e. is of the form for some
. Then:
as desired.
Example: Suppose is an elliptic curve defined over
, the field of elements. In this case the
Frobenius map is an automorphism
of , therefore
Hence the Frobenius morphism is an endomorphism (or isogeny) of the elliptic curve.
References
- 1 Joseph H. Silverman, The Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1986.
Title | Frobenius morphism |
---|---|
Canonical name | FrobeniusMorphism |
Date of creation | 2013-03-22 13:51:45 |
Last modified on | 2013-03-22 13:51:45 |
Owner | alozano (2414) |
Last modified by | alozano (2414) |
Numerical id | 4 |
Author | alozano (2414) |
Entry type | Definition |
Classification | msc 14H37 |
Related topic | FrobeniusAutomorphism |
Related topic | FrobeniusMap |
Related topic | ArithmeticOfEllipticCurves |
Defines | Frobenius morphism |