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§5. Rationality Properties of Frobenius Elements in -Adic Representations 301

§5. RationalityProperties of Frobenius Elements in -Adic
Representations: Variation of

In this section we continue with the notations of (4.1) and for an -adic representation
¯
ρ :Gal(K/K) → GL(V ) the notation of (4.6) for the characteristic polynomials
P v,ρ (T ) = det(1 − ρ w (Fr w )T ).
(5.1) Definition. An -adic representation ρ :Gal(K/K) → GL(V ) of a number
¯
field K is integral (resp. rational) provided there is a finite set of places S of K such
that ρ is unramified at all v outside of S and such that P v,ρ (T ) ∈ Z[T ] (resp. Q[T ])
for those v.
(5.2) Examples. For a number field K, the one-dimensional -adic representation
¯
Q (1)(K) is unramified at all v not dividing . The characteristic polynomial for v
unramified is
P v,ρ = 1 − (Nv)T
since Fr w (z) = z Nv on th power roots of unity, in particular on the th power roots
of unity for w|v. Observe that P v,ρ (T ) is in 1 + T Z[T ], so that the representation is
integral and is independent of in this case.
For an elliptic curve E over a number field K, the two-dimensional -adic rep-
resentation V (E) is unramified at all v not dividing where E has good reduction.
The characteristic polynomial for v unramified is

P v,ρ (T ) = 1 − a v T + (Nv)T 2
√
where a v is a rational integer with |a v |≤ 2 Nv by the Riemann hypothesis. Here
1 − a v + Nv = P v,ρ (1) is the number of points on the reduced curve E v with
coordinates in k(v). Observe that P v,ρ (T ) is in 1 + T Z[T ], is independent of ,and

for the factorization P v,ρ (T ) = (1 − α v T )(1 − α T ),wehave |α v |, |α |≤ (Nv) 1/2 .
v v
The representation V (E) is in particular integral.
The importance of the notion of rational representations lies in the fact that it is
possible to compare -adic representations for different rational primes using the
fact that the P v,ρ (T ) are independent of , see Examples (5.2).

¯
(5.3) Definition. A rational -adic representation ρ :Gal(K/K) → GL(V ) is com-

patible with a rational -adic representation ρ :Gal(K/K) → GL(V ) provided
¯
there exists a finite set of places S such that ρ and ρ are unramified outside S and

P v,ρ (T ) = P v,ρ (T ) for all v outside S.

Examples of compatible pairs of representations are contained in (5.2) by using
different rational primes. Compatibility is clearly an equivalence relation.
(5.4) Proposition. Let ρ :Gal(K/K) → GL(V ) be a rational -adic representa-
¯

tion and a rational prime. There exists at most one semisimple rational -adic

representation compatible with ρ up to isomorphism.