Milne. .Abelian.Varieties.(1998).[sharethefiles.com]

[ Pobierz całość w formacie PDF ]

1
2
polynomial of degree 2g in Z[t], and its roots all have absolute value q (��9,16).
Also, that P (A, t) is the characteristic polynomial of � acting on V A. Now consider
an abelian variety A over a number field K, and assume A has good reduction at v.
Let A(v) be the corresponding abelian variety over k(v), and define
Pv(A, t) =P (A(v), t).
For any prime w lying over v, the isomorphism V (A) �! V (A(v)) is compatible with
the map D(w) �! Gal(k(w)/k(v)). Since the canonical generator of Gal(k(w)/k(v))
acts on V A(v) as � (this is obvious from the definition of �), we see that Frobw
acts on V A as �, and so Pv(A, t) is the characteristic polynomial of Frobw acting on
V A. If w also lies over v, then Frobw is conjugate to Frobw, and so it has the same
characteristic polynomial.
Theorem 23.7. Let A and B be abelian varieties of dimension g over a number
field K. Let S be a finite set of primes of K containing all primes at which A or B
has bad reduction, and let be a prime different from the residue characteristics of the
primes in S. Then there exists a finite set of primes T = T (S, , g) of K, depending
only on S, , and g and disjoint from S *"{v | v| }, such that
Pv(A, t) =Pv(B, t) all v " T =�! A, B isogenous.
ABELIAN VARIETIES 91
Proof. Recall:
(a) A, B have good reduction at v " S �! V A, V B are unramified at v " S
(provided v ) (see 23.5);
(b) the action of � =df Gal(Kal/K) on V A is semisimple (see 21.5; remember we
are assuming Finiteness I);
(c) A and B are isogenous if V A and V B are isomorphic as �-modules (this is the
Tate conjecture 21.6).
Therefore, the theorem is a consequence of the following result concerning -adic
representations (take V = V A and W = V B).
Lemma 23.8. Let (V, �) and (W, �) be semisimple representations of Gal(Kal/K)
on Q -vector spaces of dimension d. Assume that there is a finite set S of primes of
K such that � and � are unramified outside S *"{v | v| }. Then there is a finite set
T = T (S, , d) of primes K, depending only on S, , and d and from disjoint from
S *"{v | v| }, such that
Pv(A, t) =Pv(B, t) all v " T =�! (V, �) H" (W, �).
Proof. According to Theorem 23.1, there are only finitely many subfields of Kal
2
containing K, of degree d" 2d over K, and unramfied outside S *"{v | v| }. Let L
be their composite it is finite and Galois over K and unramified outside S *"{v |
v| }. According to the Chebotarev Density Theorem (23.3), each conjugacy class in
Gal(L/K) is the Frobenius class (v, L/K) of some prime v of K not in S *"{v | v| }.
We shall prove the lemma with T any finite set of such v s for which
Gal(L/K) =*"v"T (v, L/K).
Let M0 be a full lattice in V , i.e., the Z -module generated by a Q -basis for
V . Then AutZ (M0) is an open subgroup of AutQ (V ), and so M0 is stabilized by
an open subgroup of Gal(Kal/K). As Gal(Kal/K) is compact, this shows that the
lattices �M0, � " Gal(Kal/K), form a finite set. Their sum is therefore a lattice M
stable under Gal(Kal/K). Similarly, W has a full lattice N stable under Gal(Kal/K).
By assumption, there exists a field &! �" Kal, Galois over K and unramified outside
the primes in S *"{v | v| }, such that both � and � factor through Gal(&!/K). Because
T is disjoint from S *"{v | v| }, for each prime w of &! dividing a prime v of T , we
have a Frobenius element Frobw " Gal(&!/K).
We are given an action of Gal(&!/K) on M and N, and hence on M � N. Let R
be the Z -submodule of End(M) � End(N) generated by the endomorphisms given
by elements of Gal(&!/K). Then R is a ring acting on each of M and N, we have
a homomorphism Gal(&!/K) �! R�, and Gal(&!/K) acts on M and N and through
this homomorphism and the action of R on M and N. Note that, by assumption, for
any w|v " T , Frobw has the same characteristic polynomial whether we regard it as
acting on M or on N; therefore it has the same trace,
Tr(Frobw|M) = Tr(Frobw|N).
If we can show that the endomorphisms of M � N given by the Frobw, w|v " T ,
generate R as a Z -module, then (by linearity) we have that
Tr(r|M) =Tr(r|N), all r " R.
92 J.S. MILNE
Then the next lemma (applied to R �" Q ) will imply that V and W are isomorphic
as R-modules, and hence as Gal(&!/K)-modules.
Lemma 23.9. Let k be a field of characteristic zero, and let R be a k-algebra of
finite dimension over k. Two semisimple R-modules of finite-dimension over k are
isomorphic if they have the same trace.
Proof. This is a standard result see Bourbaki, Alg�bre Chap 8, �12, no. 1,
Prop. 3.
It remains to show that the endomorphisms of M �N given by the Frobw, w|v " T ,
generate R (as a Z -module). By Nakayama s lemma, it suffices to show that R/ R is
generated by these Frobenius elements. Clearly R is a free Z -module of rank d" 2d2,
and so
2
#(R/ R)�
Therefore the homomorphism Gal(&!/K) �! (R/ R)� factors through Gal(K /K)
2
for some K �" &! with [K : K] d" 2d . But such a K is contained in L, and by
assumption therefore, Gal(K /K) is equal to {Frobw | w|v " T }.
Theorem 23.10. Finiteness I �! Finiteness II.
Proof. Recall the statement of Finiteness II:
given a number field K, an integer g, and a finite set of primes S of K, there are
only finitely many isomorphism classes of abelian varieties of K of dimension g having
good reduction outside S.
Since we are assuming Finiteness I, which states that each isogeny class of abelian
varieties over K contains only finitely many isomorphism classes, we can can replace
isomorphism with isogeny in the statement to be proved.
Fix a prime different from the residue characteristics of the primes in S, and
choose T = T (S, , g) as in the statement of Theorem 5.7. That theorem then says
that the isogeny class of an abelian variety A over K of dimension g and with good
reduction outside S is determined by the finite set of polynomials:
{Pv(A, t) | v " T }.
But for each v there are only finitely many possible Pv(A, t) s (they are polynomials
of degree 2g with integer coefficients which the Riemann hypothesis (141.b) shows to
be bounded), and so there are only finitely many isogeny classes of A s.
24. Finiteness II implies the Shafarevich Conjecture.
Recall the two statements:
Finiteness II For any number field K, integer g, and finite set S of primes of
K, there are only finitely many isomorphism classes of abelian varieties over K of
dimension g having good reduction at all primes not in S.
Shafarevich Conjecture For any number field K, integer g e" 2, and finite set of [ Pobierz całość w formacie PDF ]

Wątki