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

The hypothesis Tr M (a) = Tr N (a) for a( j) becomes simply the relation

m(i)Tr E( j) (a) = n(i)Tr E( j) (a).
Specialize to a = 1, and we obtain the relation m(i) = n(i) since Tr E( j) (a) =
dimE( j) = 0inthe field F ⊂ A. This shows that M and N are isomorphic and
proves the proposition.

(2.7) Corollary. Let G be a group, and let ρ and ρ be two semisimple finite dimen-
sional representations defined over a field F of characteristic zero. Then ρ and ρ
are isomorphic if and only if Tr(ρ(s)) = Tr(ρ (s)) for all s ∈ G.

The proof of the corollary of (2.6) follows from the discussion at the end of (2.5).
Later, we will also consider characteristic polynomials of ρ(s) in GL(V ) for
a representation ρ : G → GL(V ), where V is finite dimensional over F.These
polynomials are defined by

P ρ (s)(T ) = det(1 − ρ(s)T ) ∈ F[T ].

Then P ρ (s)(0) = 1 and the derivative at T = 0is P ρ (s) (0) =−Tr(ρ(s)) since
P ρ (s)(T ) = 1 − Tr(ρ(s))T + ... . In fact, from the traces of elements, one can
recover the characteristic polynomial.
The following formula comes up frequently in the context of -representations.
(2.8) Proposition. Let u : V → V be a linear transformation of a finite dimensional
vector space V over a field of characteristic zero. Then the characteristic polynomial
satisfies

∞ n n
−1 Tr(u )T
det(1 − uT ) = exp .
n
n=1
Proof. We can choose a basis of V so that u is upper triangular. Then det(1−uT ) =
(1−c 1 T )...(1−c m T ) where the c 1 ,..., c m , are the eigenvalues with multiplicities
of u and m = dim F V . Now use the classical expansion
∞
1 n n
c T
log =
1 − cT n
n=1
to prove the proposition.
ˇ
§3. Galois Representations and the N´ eron–Ogg–Safareviˇ c
Criterion in the Global Case

We keep the notations introduced in (1.4) for an algebraic number field K, in partic-
ular, for each place v of K, the valuation ring R (v) in K, the local field K v with its
valuation ring R v , the residue class field k(v) of either R (v) or R v ,and Nv equal to
the cardinality of k(v). We consider algebraic extensions L of K in a fixed algebraic
closure K = Q and extensions w to L of places v of K. The fact that w extends v is
¯
¯
denoted w|v.