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§8. Tate’s Description of Homomorphisms 271

u 0
Proof. In an extension of Q diagonalize the action of π E to on V (E) and
0 v

u 0 2
the action of π E to , where u,v are the conjugate roots of T f E (1/T ) and
0 v

ab
2

u ,v of T f E (1/T ).For to be in Hom (Fr) (V (E), V (E )) it is necessary

cd
and sufficient that

au bv a b u 0 u 0 a b au bu
= = = .
cu cu c d 0 v 0 u c d cv dv

ab
If one entry of is nonzero, then one eigenvalue of π E is equal to one eigen-
cd

value of π E and since they are conjugate pairs, the sets {u,v} and {u ,v } are equal.

When u = v, the supersingular case, we see that Hom (Fr) consists of all homo-
morphisms and has rank 4. When the eigenvalues are distinct, we cannot have both

a = 0and c = 0 otherwise u = u and u = v contradicting u = v and, sim-
ilarly, we cannot have both b = 0and d = 0 or it would contradict u = v. Thus

a 0 0 b
Hom (Fr) (V (E), V (E )) consists of all or all and hence has rank 2.

0 d c 0
This proves the proposition.
(8.3) Remark. For two isogenies u : E → E and v : E → E the composite

vu : E → E is an isogeny, in particular, nonzero. This means that the pairings

End k (E) × Hom k (E, E ) → Hom k (E, E )
are nondegenerate. We know that rank End k (E) = 2 for E ordinary (not su-
persingular) and rank End k (E) = 4 for E supersingular. Hence we see that for

Hom k (E, E ) = 0, it follows that rank Hom k (E, E ) = rank Hom (Fr) (T (E), T (E )),

and this proves Theorem (8.1) in the case where Hom k (E, E ) = 0. In other words,

T :Hom k (E, E ) ⊗ Z → Hom (Fr) (T (E), T (E ))
is a monomorphism between two modules of the same rank with torsion free coker-
nel, and, therefore, T is an isomorphism.
(8.4) Theorem. Let E and E be two elliptic curves over a finite field k=F q . Then

the following are equivalent:

(1) E and E are isogenous over k.
(2) V (E) and V (E ) are Gal(k/k)-isomorphic modules.

(3) #E(k) = #E (k), i.e., the two curves have the same number of elements over k.

(4) ζ E (s) = ζ E (s), the two curves have the same zeta functions.

Proof. The equivalence of (1) and (2) follows by Tate’s theorem, (8.1), since

Hom(E, E ) ⊗ Q and Hom (V (E), V (E ))
Gal( ¯ k/k)