Page 269 -

246 12. Endomorphisms of Elliptic Curves

(5.2) Definition. The Tate module T (E) of an elliptic curve over k is the projective
limit where is a prime unequal to the the characteristic

··· → n+1 E(k s ) → E(k s ) →· · · → 2 E(k s ) → E(k s )
n
together with the action of Gal(k s /s) on the limit group. We define V (E) =
T (E) ⊗ Z Q together with the extended Gal(k s /k) action.
n
Since each E(k s ) is a free Z/ -module of rank 2, the limit T (E) is a free
n
Z -module of rank 2, and V (E) is a Q vector space of dimension 2. The Galois
action can be described as a representation
Gal(k s /k) → GL(T (E)) ⊂ GL(V (E))
which is referred to as the two-dimensional -adic representation of Gal(k s /k) asso-
ciated to E over k.
In the case where E = C/L over C,wehave N E(C) = (Z/ZN) ⊗ L and
n
T = lim n (Z/Z ) ⊗ L = Z ⊗ L.
←−
(5.3) Remark. There is a symplectic structure on the Tate module of E.For N = n
and passing to the inverse limit of the pairings e : E × E → µ (k s ) as n
n
n
n
n
approaches ∞, we obtain a nondegenerate symplectic pairing, also denoted e ,
e : T (E) × T (E) → Z (1)
using (3.8). Further tensoring with Q, we have a nondegenerate symplectic pairing
e : V (E) × V (E) → Q (1).
For σ in Gal(k s /k) the Galois properties of e are contained in the relation
σ σ σ
e (x , x ) = e (x, x )
which highlights the necessity of using the Tate twist Z (1) instead Z for the image
module of e .
One of the great steps forward in the theory of elliptic curves came when it was
realized that this Galois representation on the Tate module T (E) contains many
of the basic invariants of the isomorphism type of E. The isogeny invariants of E
are contained in the study of the Galois representation V (E). This is based on an
0
understanding of how faithful an action End(E) has on T (E) and End (E) has on
V (E). This is the subject of the next section.

§6. Endomorphisms and the Tate Module

Since every homomorphism λ : E → E restricts to a group homomorphism

N E → N E for every N commuting with multiplication in the inverse system defin-

ing the Tate modules, we have a canonical homomorphism of groups