Page 289 -

266 13. Elliptic Curves over Finite Fields

h

Proof. We have the inclusion of fields k(E) ⊃ k(E ) ⊃ k(E) p h = k(E (p ) ).By
h
h
p
p

Lemma (5.3) we have [k(E) : k(E )] = [k(E) : k(E) ] and thus k(E ) = k(E) .
h

This defines the isomorphism E → E (p ) with the desired factorization property.
Now we wish to apply this discussion to multiplication by p, denoted p E : E →
2
E,onacurve E over a field k of positive characteristic p. The degree of p E is p ,
and it factors as
2
p = deg(p E ) = deg(p E ) s · deg(p E ) i .
Since p E is not separable, it follows that the height h of p E is 1or2,and # ker(p E )
i 2−h
is p or 1, respectively, i.e., # i E(k) = (p ) .
¯
p
(5.5) Proposition. For an elliptic curve E over a perfect field k of positive charac-
teristic with formal group E , the height h of p E equals the height of the formal
group E .
p
Proof. Multiplication by p on E , denoted by [p] E (t) = c 1 t + c 2 t p 2 +· · · is
induced by p E on E. The inseparable degree can be calculated in terms of the em-
bedding k[[t]] → k[[t]] given by sustitution f (t) → f ([p] E (t)). This is of degree
2
p for c 1 = 0 and of degree p otherwise.
(5.6) Theorem. Let E be an elliptic curve over a perfect field k of characteristic
p > 0 with formal group E . The following are equivalent:
(1) p E(k s ) = 0, i.e., the curve has no points of order p.
(2) p E is a purely inseparable isogeny.
(3) The formal group E has height 2.
(4) The invariant differential ω is exact, i.e., E is supersingular.
2
Further, for a supersingular E we have an isomorphism E → E (p ) and E is defined
over F 2.
p
Proof. The equivalence of (1) and (2) follows from the formula for the order of the
kernel of an isogeny in terms of its separable degree. The equivalence of (2) and (3)
is (5.5) and of (3) and (4) is 6(6.7). The last assertion follows from (5.4) and the
remarks preceding (5.3). This proves the theorem.
Since every supersingular elliptic curve is defined over F 2. This gives proof that
p
there are only finitely many up to isomorphism over F p .
¯

§6. The Endomorphism Algebra and Supersingular Curves

It might happen that the Frobenius endomorphism π E of an elliptic curve E over F q
might have some power in the subring Z of End(E). Now we study the implications
0
of this possibility. Recall π E is in End k (E) (resp. End (E)) Which is a subring (resp.
k
0
(E) (resp. End (E)).
division subring) of End ¯ k
¯ k