3 Finiteness I implies Finiteness II.
In this section we assume Finiteness I (up to isomorphism, there are only finitely many abelian varieties over a number field k isogenous to a fixed abelian variety). Hence we can apply Tate’s conjecture and the semisimplicity theorem.
We first need a result from algebraic number theory which is the analogue of the the-orem that a compact Riemann surface has only finitely many coverings with fixed degree unramified outside a fixed finite set.
THEOREM3.1. For any number fieldK, integerN, and finite set of primesS ofK, there are only finitely many fields L K unramified outside S and of degree N (up to K-isomorphism of course).
PROOF. First recall from ANT, 7.65, that for any primev and integer N, there are only finitely many extensions of Kv of degree dividingN (Kv Dcompletion ofK atv). This follows from Krasner’s lemma: roughly speaking, such an extension is described by a monic polynomial P .T /of degreedjN with coefficients inOv; the set of such polynomials is compact, and Krasner’s lemma implies that two such polynomials that are close define the same extension. Now, recall thatDi sc.L=K/DQ
Di sc.Lw=Kv/(in an obvious sense), and because we are assumingLis ramified only at primes inS, the product on the right is over the primeswdividing a primevinS. ThereforeDi sc.L=K/is bounded, and we can apply the the following classical result.
THEOREM3.2 (HERMITE1857). There are only finitely many number fields with a given discriminant (up to isomorphism).
PROOF. Recall (ANT 4.3) that, for an extensionK ofQof degreen, there exists a set of representatives for the ideal class group ofKconsisting of integral idealsawith
N.a/ nŠ
Heresis the number of conjugate pairs of nonreal complex embeddings ofK. SinceN.a/ >
1, this implies that boundjDiscK=Qjthen we boundn. Thus, it remains to show that, for a fixedn, there are only finitely many number fields with a given discriminantd. LetDD jdj. Let1; : : : ; r
case, defineY to be the set of n-tuples.y1; z1; : : :/such thatjy1j < 1,jz1j < p DC1, andyi2Cz2i < 1fori > 1. One checks easily that the volumes of these sets are
.X /D2rsp
1CD; .Y /D2s 1p 1CD;
and so both quotients.X /=2rp
D and.Y /=p
D are greater than 1. By Minkowski’s Theorem (ANT 4.19), there exist nonzero integers in K that are mapped into X or Y, according to the case. Let ˛be one of them. Since its conjugates are absolutely bounded by a constant depending only onD, the coefficients of the minimum polynomial of˛over Qare bounded, and so there are only finitely many possibilities for˛. We shall complete the proof by showing thatK DQŒ˛. Ifr ¤0, then1˛ is the only conjugate of˛ lying outside the unit circle (if it didn’t lie outside, thenNmK=Q.˛/ < 1). IfrD0, then1˛and
N
1˛are the only conjugates of˛ with this property, and1˛ ¤ N1˛since otherwise every conjugate of˛ would lie on the unit circle. Thus, in both cases, there exists a conjugate of
˛ that is distinct from all other conjugates, and so˛generatesK. 2 LetK be a number field, and let Lbe a Galois extension ofK with Galois groupG.
Letwbe a prime ofL. Thedecomposition group is
D.w/D f 2Gal.L=K/j wDwg:
The elements ofD.w/act continuously onLfor thew-adic topology, and therefore extend to the completion Lw ofL. In factLw is Galois over Kv with Galois groupD.w/. The groupD.w/acts on the residue fieldk.w/, and so we get a homomorphism
D.w/!Gal.k.w/=k.v//:
The kernel is called the inertia groupI.w/. WhenI.w/ D 1,Lis said to beunramified overK atw, and we define theFrobenius elementFrobw atwto be the element ofD.w/
corresponding to the canonical generator of Gal.k.w/=k.v//. Thus Frobw is the unique element ofG such that
Frobw.Pw/DPw; Frobw.a/aqv (modPw/
wherePw is the prime ideal ofLcorresponding tow,qv D#k.w/, andais any element of the ring of integers of L. BecauseLis Galois, the decomposition groups at the primes lying overv are conjugate, and so are the inertia groups. Therefore, if one primew lying overvis unramified they all are, andfFrobw jwjvgis a conjugacy class inG— we denote it by.v; L=K/.
THEOREM3.3 (CHEBOTAREV DENSITY THEOREM). Let L be a finite Galois extension of a number fieldKwith Galois groupG. LetC be a subset ofGstable under conjugation.
Then the set of primesvofLsuch that.v; L=K/DC has densityjCj=jGj:
PROOF. For a discussion of the theorem, see ANT, 8.31, and for a proof, see CFT, VIII
7. 2
REMARK3.4. The theorem is effective, i.e., given a class C, there is a known boundB such that there will be a primevwithN.v/Bfor which.v; L=K/DC.
3. FINITENESS I IMPLIES FINITENESS II. 141 Now consider aninfiniteGalois extensionLoverK with Galois groupG. Recall (FT,
7) thatG has a natural topology for which it is compact, and that the main theorem of Galois theory holds for infinite extension, except that it now provides a one-to-one corre-spondence between the intermediate fieldsM,L M K, and theclosedsubgroups of G. The above definitions of decomposition group etc. still make sense for infinite exten-sions. (One difference: the set of primes ramifying inLmay be infinite.)
LetV be a finite dimensional vector space overQ`. Arepresentationof Ddf Gal.Kal=K/
onV is a continuous homomorphism
W !GL.V /Ddf Aut.V /:
The kernel of is a closed normal subgroup of , corresponding to a (possibly infinite) Galois extensionLofK. The representationis said to beunramifiedat a primevofKif vis unramified inL.
We are especially interested in the representation of on V`A,A an abelian variety overK. Then the fieldLin the last paragraph is the smallest extension ofK such that all the`-power torsion points ofAare rational over it, i.e., such thatA.L/.`/DA.Kal/.`/.
THEOREM3.5. LetAbe an abelian variety over a number fieldK. Letvbe a finite prime of K, and let `be a prime distinct from the characteristic ofk.v/ (i.e., such that v - `).
ThenAhas good reduction atv if and only if the representation ofGal.Kal=K/onV`Ais unramified atv.
PROOF. ): For elliptic curves, this is proved in Silverman, 1986, VII 4.1. The proof for abelian varieties is not much more difficult. (: For elliptic curves, see Silverman, 1986, 7.1. As we now explain, the statement for abelian varieties is an immediate consequence of the existence of N´eron models (and hence is best called the N´eron criterion). Clearly the statement is really aboutAregarded as an abelian variety over the local fieldKv. As we noted in 20, N´eron showed that there is a canonical way to pass from an abelian variety AoverKv to a commutative algebraic groupA0 over the residue fieldk Dk.v/. For any prime`¤char.k.v//, the reduction map
A.Kv/`n !A0.k/`n
is a bijection. The algebraic groupA0doesn’t change whenKvis replaced by anunramified extension. It has a filtration whose quotients are successively a finite algebraic groupF (i.e., an algebraic group of dimension0), an abelian varietyB, a torusT, and an additive group U. We have
dimADdimBCdimT CdimU.
Moreover: #B.kal/`n D `2ndim.B/; #T .kal/`n D `ndim.T /, because Tkal Gdimm T, Gm.L/DLall fieldsLQ;#U.kal/`n D0, becauseUkal Gdima U,Ga.L/DLall fieldsLQ. Now suppose thatAhas good reduction, so thatA0 DB. For alln,
A.Kvun/`n DA0.kal/`n
has`2ndimAelements, and soA.Kvun/`n DA.Kval/`n. Therefore the action ofGal.Kval=Kv/ on V`A factors through Gal.Kvun=Kv/, which is what it means for the representation of Gal.Kval=Kv/onV`Ato be unramified. On the other hand, ifAdoes not have good reduc-tion, then
#A.Kvun/`n D#A0.kal/`n< `2ndimA
fornsufficiently large. As
A.Kvun/`n DA.Kval/Gal.Kalv=Kvun/ this shows that
A.Kval/Gal.Kval=Kvun/ ¤A.Kval/`n; n >> 0:
Therefore the representation of the Galois group onV`Ais ramified atv. 2 COROLLARY3.6. IfAandBare isogenous overK, and one has good reduction atv, then so also does the other.
PROOF. The isogeny defines an isomorphismV`A!V`B commuting with the actions of
Gal.Kal=K). 2
Recall that for an abelian varietyAover a finite fieldkwithqelements, the character-istic polynomialP .A; t /of the Frobenius endomorphism ofAis a monic polynomial of degree 2g inZŒt , and its roots all have absolute valueq12 (9,16). Also, thatP .A; t /is the characteristic polynomial of acting onV`A. Now consider an abelian varietyAover a number fieldK, and assumeAhas good reduction atv. LetA.v/be the corresponding abelian variety overk.v/, and define
Pv.A; t /DP .A.v/; t /:
For any primewlying overv, the isomorphismV`.A/!V`.A.v//is compatible with the mapD.w/!Gal.k.w/=k.v//. Since the canonical generator ofGal.k.w/=k.v//acts on V`A.v/as (this is obvious from the definition of), we see that Frobw acts onV`Aas , and soPv.A; t /is the characteristic polynomial of Frobw acting onV`A. Ifw0also lies overv, then Frobw0is conjugate to Frobw, and so it has the same characteristic polynomial.
THEOREM3.7. LetAandB be abelian varieties of dimensiong over a number fieldK.
LetSbe a finite set of primes ofKcontaining all primes at whichAorBhas bad reduction, and let`be a prime different from the residue characteristics of the primes inS. Then there exists a finite set of primesT DT .S; `; g/ofK, depending only onS,`, andgand disjoint fromS[ fv jvj`g, such that
Pv.A; t /DPv.B; t /allv 2T H)A; B isogenous.
PROOF. Recall: (a) A, B have good reduction at v 2 S ) V`A, V`B are unramified at v 2 S (provided v - `) (see 3.5); (b) the action of Ddf Gal.Kal=K/ on V`A is semisimple (see 2.5; remember we are assuming Finiteness I); (c)AandBare isogenous if V`AandV`Bare isomorphic as -modules (this is the Tate conjecture 2.6). Therefore, the theorem is a consequence of the following result concerning `-adic representations (take
V DV`AandW DV`B). 2
LEMMA3.8. Let.V; /and.W; /be semisimple representations ofGal.Kal=K/onQ` -vector spaces of dimensiond. Assume that there is a finite setS of primes ofKsuch that and are unramified outsideS[ fv jvj`g. Then there is a finite setT DT .S; `; d /of primesK, depending only onS,`, andd and from disjoint fromS[ fv jvj`g, such that
Pv.A; t /DPv.B; t /allv 2T H).V; /.W; /:
3. FINITENESS I IMPLIES FINITENESS II. 143 PROOF. According to Theorem 3.1, there are only finitely many subfields ofKalcontaining K, of degree`2d2overK, and unramfied outsideS[fv jvj`g. LetLbe their composite
— it is finite and Galois over K and unramified outsideS [ fv j vj`g. According to the Chebotarev Density Theorem (3.3), each conjugacy class in Gal.L=K/ is the Frobenius class.v; L=K/of some primevofKnot inS [ fvjvj`g. We shall prove the lemma with T any finite set of suchv’s for which
Gal.L=K/D [
v2T
.v; L=K/:
Let M0 be a full lattice in V, i.e., the Z`-module generated by aQ`-basis for V. Then AutZ`.M0/is an open subgroup ofAutQ`.V /, and soM0 is stabilized by an open sub-group of Gal.Kal=K/. As Gal.Kal=K/ is compact, this shows that the lattices M0, 2 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 assump-tion, there exists a field ˝ Kal, Galois over K and unramified outside the primes in S[ fv jvj`g, such that bothandfactor throughGal.˝=K/. BecauseT is disjoint from S[ fv jvj`g, for each primewof˝dividing a primevofT, we have a Frobenius element Frobw 2 Gal.˝=K/. We are given an action ofGal.˝=K/onM andN, and hence on M N. LetR be theZ`-submodule ofEnd.M /End.N /generated by the endomor-phisms given by elements ofGal.˝=K/. ThenRis a ring acting on each ofM andN, we have a homomorphismGal.˝=K/ !R, andGal.˝=K/acts onM andN and through this homomorphism and the action of RonM andN. Note that, by assumption, for any wjv 2 T, Frobw has the same characteristic polynomial whether we regard it as acting on M or onN; therefore it has the same trace,
Tr.FrobwjM /DTr.FrobwjN /:
If we can show that the endomorphisms ofM N given by the Frobw,wjv 2T, generate Ras aZ`-module, then (by linearity) we have that
Tr.rjM /DTr.rjN /; allr2R:
Then the next lemma (applied to R ˝Q`/ will imply that V and W are isomorphic as R-modules, and hence asGal.˝=K/-modules.
LEMMA3.9. Let k be a field of characteristic zero, and let R be a k-algebra of finite dimension overk. Two semisimpleR-modules of finite-dimension overkare isomorphic if they have the same trace.
PROOF. This is a standard result — see Bourbaki, Alg`ebre Chap 8,12, no. 1, Prop. 3. 2 It remains to show that the endomorphisms ofM N given by the Frobw,wjv 2 T, generate R (as a Z`-module). By Nakayama’s lemma, it suffices to show that R=`R is generated by these Frobenius elements. ClearlyRis a freeZ`-module of rank2d2, and so
#.R=`R/<#.R=`R/`2d2:
Therefore the homomorphism Gal.˝=K/ ! .R=`R/ factors through Gal.K0=K/for someK0˝withŒK0W K`2d2. But such aK0is contained inL, and by assumption therefore,Gal.K0=K/is equal tofFrobw jwjv 2Tg. 2
THEOREM3.10. Finiteness I)Finiteness II.
PROOF. Recall the statement of Finiteness II: given a number field K, an integerg, and a finite set of primesS ofK, there are only finitely many isomorphism classes of abelian varieties of K of dimension g having good reduction outsideS. Since we are assuming Finiteness I, which states that each isogeny class of abelian varieties overKcontains 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 inS, and chooseT DT .S; `; g/as in the statement of Theorem 3.7. That theorem then says that the isogeny class of an abelian varietyAoverK of dimensiong and with good reduction outsideSis determined by the finite set of polynomials:
fPv.A; t /jv 2Tg:
But for each v there are only finitely many possiblePv.A; t /’s (they are polynomials of degree 2g with integer coefficients which the Riemann hypothesis shows to be bounded), and so there are only finitely many isogeny classes ofA’s. 2