For , of the form we define . We define a new curve as the zero set of the ideal (generated by):
The -power Frobenius morphism is defined to be:
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:
Hence the Frobenius morphism is an endomorphism (or isogeny) of the elliptic curve.
- 1 Joseph H. Silverman, The Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1986.
|Date of creation||2013-03-22 13:51:45|
|Last modified on||2013-03-22 13:51:45|
|Last modified by||alozano (2414)|