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ˇ
§7. Tate’s Conjecture, Safareviˇ c’s Theorem, and Faltings’ Proof 305
(6.6) with n = 2g and weight w = 1 with the case of elliptic curves being g = 1. The
Tate conjecture allows one to reconstruct an abelian variety up to isogeny from its V
for suitable among the representations satisfying the hypotheses of the finiteness
theorem.
The other application of weight properties arises with the convergence assertion
of the L-function L ρ (s) associated with (ρ ) in (5.7).
(6.8) Remark. Let (ρ ) be a compatible family of -adic representations of a num-
ber field K with exceptional set S.Ifall ρ have weight w, then the L-function L ρ (s)
of (5.7) converges for Re(s)> 1 + w/2 by 11(6.6).

ˇ
§7. Tate’s Conjecture, Safareviˇ c’s Theorem, and Faltings’ Proof

In Corollary (3.4) we saw how the Galois modules T (A) and V (A) could give
information about isogenies, but actually this is only the beginning. In fact, the ideas
are related with the Mordell conjecture, solved by Faltings [1983], and they are part
of a broader story which was considered in part in Chapter 12, §6 and Chapter 13,
§8. Recall the following commutative diagram for any field k and prime to char(k):

T

Hom k (A, A ) ⊗ Z −−−−→ Hom Gal(k s /k) (T (A), T (A ))
∩ ∩
V
Hom k (A, A ) ⊗ Q −−−−→ Hom Gal(k s /k) (V (A), V (A ))

where A and A are elliptic curves, or more generally abelian varieties, over the field

k and the endomorphisms are defined over k. We know that for any field k the maps
T and V are injective and that T has a torsion-free cokernel by 12(6.1) and 12(6.4).
For the case of abelian varieties, see also Mumford [1970, Theorem 3, p. 176].
Tate’s theorem, 13(8.1) says that T and hence V are isomorphisms for k a finite
field, and, moreover, that the action of Gal(k/k) is semisimple on V (A). For the
¯
case of abelian varieties, see also Tate [1966].
In Tate [1974], the following conjecture appears as Conjecture 5, p. 200, for
elliptic curves and it is a natural extension of Tate’s work over finite fields.
(7.1) Conjecture of Tate. For an algebraic number field K, the homomorphisms T

and V are isomorphisms for all abelian varieties A and A over K. The action of
¯
Gal(K/K) on V (A) is semisimple.
As a special case, Serre [1968] showed, for elliptic curves over a number field,

that if V (E) and V (E ) are isomorphic Galois modules and if j(E) is not an integer
in K, then E and E are isogenous over K. This was proved using the Tate model for

the curve.
Faltings [1983] has shown that the Tate conjecture is true not just for elliptic
curves but, in fact, also for all abelian varieties over a number field. This work of