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§6. Endomorphisms and the Tate Module 247

T :Hom(E, E ) → Hom Z (T (E), T (E ))
for each prime . Tensoring with Z we obtain homomorphism of Z -modules also
denoted with the same letter

T :Hom(E, E ) ⊗ Z → Hom Z (T (E), T (E )).
The basic result is the following which holds also for abelian varieties and where the
proof follows the one in Mumford [1974, pp. 176–178].

(6.1) Theorem. For a prime unequal to the ground field characteristic the natural
map

T :Hom(E, E ) ⊗ Z → Hom Z (T (E), T (E ))

is injective. Moreover, the group Hom(E, E ) is finitely generated and free abelian.

Proof. Since every nonzero E → E is surjective, the group Hom(E, E ) is torsion
0

free, and we can think of Hom(E, E ) as a subgroup of in Hom (E, E ).

Assertion. For any finitely generated subgroup M of Hom(E, E ) the subgroup

QM ∩ Hom(E, E ) ={λ : E → E with nλ in M for some n = 0}
is again finitely generated.

To prove the assertion, we note that Hom(E, E ) = 0 when E and E are not
isogenous, and using the injection Hom(E, E ) → End(E) induced by an isogeny

E → E, we are reduced to the case E = E . The norm deg : End(E) → N where

0
deg(λ) > 0 if and only if λ = 0 extends to deg : End (E) → Q.Now QM is a finite
0
dimensional space, and the set if λ in End (E) with N(λ) < 1 is a neighborhood V
of zero in QM. Since U ∩ End(E) = (0), it follows that QM ∩ End(E) is discrete
in QM, and thus it is finitely generated.
By the above assertion it suffices to proove that for ant finitely generated M in

Hom(E, E ) satisfying M = QM ∩ Hom(E, E ), the restricted homomorphism is a
monomorphism

(T (E), T (E )).
T : M ⊗ Z → Hom Z
Let λ 1 ,... ,λ m be a basis of the abelian group M. Since the right-hand side is Z -
free, we would have T (c 1 λ 1 +· · · + c m λ m ) = 0 where we may assume that one
c j a unit in Z if T were not injective. This would mean that there are integers
a 1 ,... , a m not all divisible by such that for λ = a 1 λ 1 +· · · + a m λ m we have

T (λ)(T (E)) ⊂ T (E ) and hence, f ( E) = 0in E . Then λ = µ, where µ is in

Hom(E, E ), and, since
µ is in QM ∩ Hom(E, E ) = M,