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

Proof. This is an immediate corollary of Theorem (4.11).

The next definition describes Examples (5.2) where there is a representation for
each rational prime. For a number field K, let [ ] denote the set of places of K
dividing a rational prime , i.e., the set of v with p v = .
(5.5) Definition. Let K be an algebraic number field, and for each rational prime
¯
let ρ :Gal(K/K) → GL(V ) be a rational -adic representation of K.The
system (ρ ) is called compatible provided ρ and ρ are compatible for any pair of

primes. The system (ρ ) is called strictly compatible provided there exists a finite
(T ) has rational
set S of places of K such that ρ is unramified outside S ∪ [ ], P v,ρ
coefficients, and

(T ) = P v,ρ (T ) for all v outside S ∪ [ ] ∪ [ ]
P v,ρ

When a system (ρ ) is strictly compatible, there is a smallest finite set S hav-
ing the strict compatibility properties of the previous definition. This S is called the
exceptional set of the system.

(5.6) Remark. By (4.6), the characteristic polynomials P v,ρ (T ) canbedefinedeven
for ramified places. In (5.1), (5.3), and (5.5), it is interesting to consider systems
where all P v,ρ (T ) have rational coefficients or integral coefficients as is the case in
Examples (5.2).

Observe that for a characteristic polynomial P v,ρ (T ) over Q, the expression
P v,ρ(Nv) −s equals P(p −s ) for some polynomial P(T ) ∈ Q[T ].
v
Now we come to the notion of an Euler factor, which is an expression of the form
for the representation ρ at the place v

1 1
= .
−s
−s
−s
P v,ρ ((Nv) ) (1 − c 1 p v )...(1 − c m p v )
a
Here m is dim ρ times a where Nv = p .
v
(5.7) Definition. Let (ρ ) be a strictly compatible family of -adic representations
of a number field K with exceptional set S. The formal Dirichlet series below is the
L-function of (ρ ).
1
L ρ (s) = where choose , v| .
−s
P v,ρ ((Nv) )
v/∈s
This definition can be used for a single -adic representation as well without ref-
erence to compatibility. Also, the polynomials P v,ρ (T ) are independent of which
representation from the compatible family is used to determine them.
The question of the convergence of these Euler products was discussed in Chap-
ter 11, §6, and the conditions considered in the next section lead to convergence
properties.