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294 15. Elliptic Curves over Global Fields and -Adic Representations

(2.1) Remark. If V is an n-dimensional complex vector space, then a continuous
homomorphism Gal(k s /k) → GL(V ) = GL(n, C) factors through a finite quotient

of the form Gal(k s /k) → Gal(k /k), where k ⊂ k s and k /k is a finite Galois
extension. This follows from the fact that the only compact, totally disconnected
subgroups of GL(n, C) are finite subgroups.
If we replace the complex numbers C by the -adic numbers Q ,thenasusual
finite dimensional vector spaces V and their endomorphism algebras End(V ) have a
natural product topology, and GL(V ) has the subspace topology of End(V ) making
it into a locally compact, totally disconnected topological group. It is this fact which
makes -adic representations more useful in studying the Galois group Gal(k s /k)
than complex representations.

(2.2) Definition. An n-dimensional -adic representation of the Galois group
Gal(k s /k),orofthe field k, is a continuous homomorphism ρ :Gal(k s /k) →
GL(V ) = GL(n, Q ),where V is an n-dimensional Q vector space.

In this case, the image of ρ can be infinite as can be seen by the two basic exam-
ples of (2.3) and Tate modules.

(2.3) Example. For prime to the characteristic of k, Q (1)(k s ) is a one-dimensional
-adic representation, and for any elliptic curve E, the rational Tate vector space
V (E) is a two-dimensional -adic representation (of Gal(k s /k) or of k). For the
details, see (5.1), (5.2), and (5.3).

The Z -submodules of the -adic representation spaces in the examples Z (1)(k s )
⊂ Q (1)(k s ) and T (E) ⊂ V (E) are stable under the action of Gal(k s /k),and
introducing a basis, we can factor the -adic representation through the subgroup of
integral matrices

Gal(k s /k) → GL(n, Z ) ⊂ GL(n, Q ),

where n = 1inthe first caseand n = 2inthe case of T (E) ⊂ V (E). More
generally, a module T over Z of a vector space V over Q is called a lattice provided
Z → V is an isomorphism. A basis of T is also
the natural homomorphism T ⊗ Z
a basis of V yielding compatible isomorphisms
∼
GL(T ) −−−−→ GL(n, Z )
∩ ∩
∼
GL(V ) −−−−→ GL(n, Q )
(2.4) Proposition. If ρ is an -adic representation of Gal(k s /k) on the vector space
V over Q , then there exists a lattice T in V with ρ(s)T = T for all s in Gal(k s /k) =
G.