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

The prime number theorem says that N(t," K ) ∼ t/ log t so that a subset X has
a density d if and only if
t t
N(t, X) = d + O
log t log t

Further, if X and X are two subsets with X − X and X − X both finite sets, and if
one has a density, then so does the other and the two densities are equal.
ˇ
(4.9) Theorem (Cebotarov DensityTheorem). Let L/K be a finite Galois exten-
sion of number fields with G = Gal(L/K). For each subset C of G stable under
conjugation, let X C denote the set of places v of K unramified in Lsuch that the
Frobenius element Fr w is in C for any w|v.Then X C has a density and it equals
#C
.
#G
In particular, for each element s of Gal(L/K), there are infinitely many unramified
places w of Lsuch that Fr w = s.

(4.10) Corollary. Let L be an algebraic Galois extension of a number field K which
is unramified outside a finite number of places of K. Then the Frobenius elements of
the unramified places of Lare dense in Gal(L/K).
Proof. By (4.9), the set of Frobenius elements maps subjectively onto every finite
quotient of Gal(L/K), and , thus, every element of Gal(L/K) is arbitrarily close to
a Frobenius element.

ˇ
For the proof of the Cebotarov density theorem, see either Serre [1968; Appendix
to Chapter 1], E. Artin, Collected Works,orS.Lang, Algebraic Number Theory,
1970, p. 169, where Lang records a simple proof due to M. Deuring.
In order to illustrate the use of these density results for the Frobenius elements,
we have the following result for semisimple -adic representations.

(4.11) Theorem. Let ρ : G → GL(V ) and ρ : G → GL(V ) be two semisimple

-adic representations of a number field K. Assume that ρ and ρ are unramified
outside a finite number of places S of K, and that P v,ρ (T ) = P v,ρ (T ) for all v

outside S. Then ρ and ρ are isomorphic.

Proof. The equality of characteristic polynomials gives Tr(ρ(Fr w )) = Tr(ρ (Fr w ))
for all w|p, where v is a place outside of S. Since the elements Fr w are dense by (4.9),
and since s → Tr(ρ(s)) is continuous on G, we deduce that Tr(ρ(s)) = Tr(ρ (s))

for all s in G. Hence by (2.7), the representations ρ and ρ are isomorphic. This
proves the theorem.