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§6. The Endomorphism Algebra and Supersingular Curves 267

n
0
(6.1) Proposition. If no power π of π E is in Z contained in End(E), then End (E)
E k
0
= End (E) = Q(π E ) is a purely imaginary quadratic field.
¯ k
(E) is defined over some algebraic extension F q of the
Proof. An element ϕ in End ¯ k n
n
n
n
ground field k = F q and thus π ϕ = ϕπ . Thus ϕ is in Q(π ) from commutation
E E E
0
properties of algebras embedded in M 2 (Q), see Lang’s Algebra. In fact, End (E) is
¯ k
necessarily two-dimensional over Q and equals Q(π E ). The rest follows from (1.2).
m
(6.2) Proposition. If some power of π E is in Z, then π n = p for some m, Eis
E E
0
supersingular, and End (E) is a quaternion algebra.
¯ k
n
i
ni
Proof. Since π E is purely inseparable, deg(π E ) = p and deg(π ) = p . From the
E
n
n
m
n
2
assumption that π = c E ∈ Z and deg(π ) = c , we deduce that π = p , where
E E E E E
m = ni/2.
Next, p E is purely inseparable which by (5.5) means that E is supersingular.
So it is among a finite set of isomorphism classes of supersingular curves. For a
finite subgroup G of order prime to p, E/G is also supersingular as one can see by
looking at the points of order p. Let S be any infinite set of primes not including p.

Then there must exist a pair of distinct , is S with E/G and E/G isomorphic,

where G and G are cyclic subgroups of orders and , respectively. Consider the

following commutative diagram which defined the isogeny ϕ of degree , which is
not a square:
∼
E/G −−−−→ E/G
% 
 

ϕ
E −−−−→ E
2
Since deg(n E ) = n , we see that ϕ is not in Q which is always contained in
0
0
0
End (E) = End (E).IfEnd (E) were commutative, it would be an imaginary
¯ k
quadratic extension. In that case there would be an infinite set of primes S such
0
that they and products of distinct pairs are not norms of elements from End (E), i.e.,
0
degrees of elements from End (E). The above construction shows that no such infi-
0
nite set exists, and therefore we deduce that End (E) must be a quaternion algebra
and is, in particular, noncommutative.
(6.3) Theorem. For an elliptic curve E over a field k the following assertions are
equivalent:
(1) p E(k 2 ) = 0, i.e., the curve is supersingular.
(E) is noncommutative.
(2) End ¯ k
(3) E is defined over a finite field and there exist strictly positive m and n with π E m =
n
p .
E
Proof. That assertion (1) implies (2) was given in the last paragraph of the previous
proposition. Observe that in the argument only the condition that E is supersingular