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268 13. Elliptic Curves over Finite Fields

was used. Conversely, if p E(k s ) is nonzero, then T p (E) is a free Z p -module of rank
1, and the representation End(E) → End(Z p ) is faithful into a commutative ring.
This proves that (2) implies (1). When (1) holds E is defined over a finite field and
now the same holds for (2) and of course (3).
By (6.2) we see that (3) implies (2) and (2) implies (3) follows from (6.1). This
proves the theorem.
Finally, we are also in a position to derive Deuring’s criterion for a curve to be
defined over a finite field in terms of complex multiplications of the curve.
(6.4) Theorem (Deuring). In characteristic p > 0 an elliptic curve E is defined
0
over a finite field if and only if dim Q End (E)> 1, i.e., E has (nontrivial) complex
multiplications.
Proof. If E is defined over a finite field, then (6.1) and (6.2) tell us exactly what
0
End (E) is, and it has the desired property.
2
3
Conversely, consider E λ given by y + y + λxy = x in characteristic 2 and by
2
y = x(x − 1)(x − λ) in characteristic different from2. Assume that λ is transcen-
2
dental over F p , and assume there exists u in End(E λ ) with u = N < 0. For a prime
not dividing pN we choose a nondivisible element x in T (E λ ), and denote by G n
m
n
the cyclic group generated by the image of x in z under T (E λ ) → E.Thenthe
separable quotient p n : E λ /G n = E λ(n) , where λ(n) is transcendental over F p .If
u n in End(E λ(n) ) corresponds to u under one of the two isomorphisms E λ → E λ(n) ,
then ±u = p n −1 u n p n and u(ker(p n )) is contained in ker(p n ) = G n for all n.Hence
u(x) = ax is an eigenvector, and, therefore, u acts as a scalar on T (E). This contra-
2
dicts u = N < 0. Hence if E is not defined over a finite field, then End(E) = Z,
and this proves the theorem.
§7. Summary of Criteria for a Curve To Be Supersingular
Before giving the table summarizing the previous results (Table 2) we add one more
criterion in terms of sheaf cohomology for a curve to be supersingular. The reader
with insufficient background can skip this result.
(7.1) Proposition. Let E be an elliptic curve over a field k in characteristic p > 0.
Then E is supersingular if and only if
1
1
∗
π : H (E, O E ) → H (E, O E )
E
is nonzero, where π is induced by the Frobenius morphism π E E → E.
∗
E
Proof. We have an commutative diagram using the fact that the ideal sheaf of E in

P 2 is isomorphic to O P (−3) and the Frobenius F on P = P 2 maps O E to O E (p)
where E (p) is the subscheme of P defined by f p = 0.
0 −−−−→ O P (−3p) −−−−→ O P −−−−→ O E (p) −−−−→ 0
  
f
 p−1  
0 −−−−→ O P (−3) −−−−→ O P −−−−→ O E −−−−→ 0.