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§6. Weight Properties of Frobenius Elements in -Adic Representations 303

§6. Weight Properties of Frobenius Elements in -Adic
Representations: Faltings’ Finiteness Theorem

(6.1) Definition. An n-dimensional integral -adic representation ρ : G → GL(V )
of an algebraic number field K has weight w provided there exists a finite set S of
places of K such that, for all places v outside S, the integral polynomials

P v,ρ (T ) = det(1 − ρ w (Fr w )) = (1 − α 1 T )...(1 − α n T )
have reciprocal roots α j with complex absolute value |α j |= (Nv) w/2 .

There is another convention were |α j |= (Nv) −w/2 .

(6.2) Examples. The examples in (5.2) are integral and they have weights. For
¯
Q (1)(K) we see from P v,ρ (T ) = 1 − (Nv)T that this one-dimensinal represen-
tation has weight 2. For V (E) where E is an elliptic curve over K, we see from
2

P v,ρ (T ) = 1 − a v (T ) + (Nv)T = (1 − α v T )(1 − α T )
v

with |α v |,|α |= (Nv) 1/2 that this representation has weight 1.
v
There are some finiteness properties implicit in the weight condition which can
be formulated as follows.
(6.3) Assertion. For given n, q,and w, the number of integral polynomials of the
form

n
P(T ) = 1 + a 1 T +· · · + a n T = (1 − c 1 T )...(1 − c n T )
with all c j algebraic integers having absolute value |c j |= q w/2 is a finite set.
The absolute value conditions on the roots translate into boundedness conditions
on the integral coefficients of the polynomials P(T ).
(6.4) Proposition. Let K be an algebraic number field, let S be a finite set of places
of K, and let m be an integer. There exists a finite Galois extension Lof K such that
every extension K of K unramified outside S and with [K : K] ≤ m is isomorphic

to a subextension of L.

Proof. By 8(3.5) up to isomorphism the number of K as above is finite. Let L be
the Galois closure of the composite of these extensions. This proves the proposition.
This proposition is another version of 8(3.5) used to prove the weak Mordell the-
orem. Now we have one more preliminary result before Falting’s finiteness theorem.

(6.5) Proposition (Nakayama’s Lemma). Let E be a Z -algebra of finite rank, and
let X be a subset of E whose image in E/ Egenerates E/ E asanalgebra.Then X
generates E as an algebra.