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§5. Points of Order p and Supersingular Curves 265

§5. Points of Order p and Supersingular Curves

Recall that the degree of an endomorphism λ : E → E is deg(λ) = [k(E) : k(E )]

coming from the induced embedding k(E ) → k(E) factors into separable and
purely inseparable parts
deg(λ) = deg(λ) s deg(λ) i ,

where deg(λ) s = [k(E) : k(E )] s and deg(λ) i = [k(E) : k(E )] i . The purely

h
inseperable degree is always a power of p, deg(λ) i = p , where h is called the
height of λ. The kernel ker(λ)of λ is a finite subgroup of E(k s ).
(5.1) Remark. The number of elements in the kernel is the separable degree. Let
k = k s . Given a finite subgroup G in E(k s ) there exists a unique separable isogeny
G
λ G : E → E/G where k(E/G) = k(E) , the subfield invariant by the action of G
on D(E) induced by translations of G on E. The isogeny and the quotient elliptic
curve are defined over an intermediate field k 1 between k and its separable algebraic
G
closure k s . Then ker(λ G ) = G and deg(λ G ) = deg(λ g ) s = #G = [k(E) : k(E) ]
by Galois theory. This is an indication of how to construct all separable isogenies.
We do not supply the details of the construction of E/G except to give k(E/G) and
the interested reader can consult Mumford, Abelian Varieties, Section 7 for further
details.
h
(5.2) Definition. The iterated absolute Frobenius Fr h : E → E (p ) is defined by
E
h
h
h
p
Fr (x, y) = (x p h , y ) where E (p ) has Weierstrass equation with coefficients the
E
h
p th power of the coefficients of E, i.e.,
h
p
2
3
2
y + a p h xy + a p h y = x + a p h x + a p h x + a .
1 3 2 4 6
h
h (p ) p h
For k perfect Fr is purely inseparable of height h since k(E ) = k(E) in
E
h
k(E). Observe that E is defined over the finite field F h if and only if E and E (p )
p
h
are isomorphic and in this case Fr = π E .
E
(5.3) Lemma. Let K = k(x, y) be a separable algebraic extension of a purely tran-
p
scendental extension k(x) of the perfect field k. Then [K : K ] = p.
p
Proof. Since K = k(x, y) = k(x, y ), the element x generates K over K p =
p
p
p
p
p
k (x , y ) = k(x , y ). Hence the degree is either p or 1. In the latter case x is in
p
p
K and x = t where t is both separable and purely inseparable over k(x), which is
p
impossible. Hence [K : K ] = p.
(5.4) Proposition. Let k be a perfect ground field, and let λ : E → E be a purely

h

inseparable isogeny of height h. Then there exists an isomorphism u : E → E (p )
h
with uλ = Fr .
E