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

Proof. Let E be the subalgebra of E generated by X. Then E = E + E so that the

finitely generated E/E satisfies (E/E ) = E/E . Thus from the structure theorem

for modules over Z we have E/E = 0or E = E . This proves the proposition.

(6.6) Theorem (Faltings). Let be a prime. Let S be a finite set of places in an
algebraic number field K, and let n and w be natural numbers. Then the set of
isomorphism classes of n-dimensional, semisimple, integral -adic representations
of weight w unramified outside S is finite.
Proof. Extend S to S( ) by adding the finite number of places dividing . Now apply
(6.4) to S( ) and m = 2n 2 to obtain a finite Galois extension L of K with G =
Gal(L/K). Let S be a finite set of places v outside of S( ) such that G is the union
∗
of all Fr w with w|v and v ∈ S . This is possible by (4.9).
∗

Assertion. Let ρ :Gal(K/K) → GL(V ) and ρ :Gal(K/K) → GL(V )

¯
¯
be two n-dimensional, semisimple, integral -adic representations of weight w un-

ramified outside S.IfTr(ρ(Fr v )) = Tr(ρ (Fr v )) for all v ∈ S , then ρ and ρ are
∗
isomorphic.
Proof of the assertion. By (2.4) we can choose Galois stable lattices of V and
V and further a basis of these lattices so that we can view ρ, ρ as morphisms

¯
Gal(K/K) → GL n (Z ).For s ∈ G,let ρ (S) = (ρ(s), ρ (s)), and let E be the
∗

2
∗
subalgebra over Z of M n (Z ) generated by the elements ρ (s), s ∈ G. By (4.11),
it suffices to show that for each x = (x 1 , x 2 ) ∈ E,
Tr(x 1 ) = Tr(x 2 )
since the representations are semisimple. By hypothesis, we know that Tr(ρ(Fr w )) =
∗
Tr(ρ (Fr w )) for all w|v and v ∈ S , and thus it suffices to show that these ρ generate
∗

E over Z .
∗
∗
Let ¯ρ denote ρ followed by reduction GL n (Z ) → GL n (F ).Ifwecan show
∗
∗
that the set of ¯ρ (Fr w ) for all w|v and v ∈ S generates E/ E, then by Nakayama’s
∗
lemma (6.5) we have established the assertion. For this, observe that ¯ρ factors
through Gal(K/K) = Gal(L/K) = G and all elements of Gal(L/K) are of the
¯
∗
∗
form Fr w with w|v and v ∈ S . Hence im( ¯ρ ) generates E/ E and E is generated
by ρ (Fr w ) where w|v and v ∈ S . This proves the assertion.
∗
∗
Finally, returning to the proof of the theorem, we see by (6.3) that there are
only finitely many choices for P v,ρ (T ) at each v ∈ S . Since P v,ρ (T ) determines
∗
Tr(ρ(Fr v )), there are by the assertion only finitely many representations ρ up to
isomorphism of the type in the statement of the theorem. This proves the finiteness
theorem.
ˇ
(6.7) Remark. It is this finiteness theorem which Faltings uses to deduce the Safare-
viˇ c conjecture, see (7.1) or Satz 5 in Faltings [1983], from the assertion of the
Tate conjecture, see (7.2) or Satz 3 and Satz 4. The Galois modules V (A) for a g-
dimensional abelian variety over K satisfy the hypotheses of the finiteness theorem