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§8. Image of -Adic Representations of Elliptic Curves 307

§8. Image of -Adic Representations of Elliptic Curves: Serre’s
Open Image Theorem

In this section, let E denote an elliptic curve defined over a number field K.The
¯
action of G = Gal(K/K) on the N-division points of E yields a homomorphism
n : G = Gal(K/K) → GL( N E) = GL 2 (Z/NZ).
∼
¯
and the inverse limit over all N is

¯ ∼
ρ : G = Gal(K/K) → lim GL( N E) = GL(Tors E) = GL(T (E)).
&
←−
N
This yields the following diagram of projections and quotients of ρ:

∼ GL 2 (Z )
GL(T (E)) =



¯
G = Gal(K/K) GL(T (E)) = GL 2 (Z )
∼



∼
GL( (E)) = GL 2 (F )
The main theorem of Serre is that im(ρ) is an open subgroup of the product group
when E does not have complex multiplication. Before discussing this result, we have
the following remark on the contrasting case when E has complex multiplication.
(8.1) Remark. If the elliptic curve E has complex multiplication by an imaginary
quadratic number field F, then F acts on each V (E) and embeds as a subfield of
¯
End(V (E)) which commutes with the action of Gal(K/K). Hence im(ρ ) com-
∗
mutes with every element of F ⊂ GL(V (E)) = GL 2 (Q ). Since the image of
∗
F contains nonscalar elements, im(ρ ) will not be open in GL(T (E)) = GL 2 (Z )
and, hence, im(ρ) will not be open in the product group. In the next chapter, we will
study this action of the complex multiplication and its relation to the Galois action
further.
The following theorem is the main result in the book by Serre [1968] and the
Inventiones paper by Serre [1972].
(8.2) Theorem (Serre). Let E be an elliptic curve without complex multiplication

¯
defined over a number field K. Then the image of ρ :Gal(K/K) → GL(T (E))
is an open subgroup.
The proof of this theorem would take us beyond the scope of this book, and, in
any event, it is difficult to improve on the exposition and methods of Serre. We will
be content to simply observe that the following result from the book, see p. IV–19, is
the criterion used by Serre to show the image is open.