LEMMA14.5. Ifis a root of1such that for some algebraic integer and integern3, D1Cn, thenD1.
PROOF. If ¤1, then, after raising it to a power, we may assume that it is a primitivepth root of1for some primep. ThenNmQŒ =Q. 1/ D . 1/p 1p, because the minimum polynomial of 1is
.X C1/p 1C C1DXp 1C Cp
(see ANT, Chapter 6). Now the equation 1Dn implies that. 1/p 1pDnp 1N. /.
This is impossible becausepis prime. 2
REMARK14.6. (a) Let .A; / and .A0; 0/ be polarized abelian varieties over a perfect field k. If there exists an n 3such that both AandA0 have all their points of ordern rational overk, then any isomorphism.A; /!.A; 0/defined over some extension field ofkis automatically defined overk(apply AG 16.9).
(b) Let˝ k be fields such that the fixed field of D Aut.˝=k/is k and˝ is algebraically closed. Let .A; /be a polarized abelian variety over ˝, and let.'/2 be a descent system on .A; /, i.e., ' is an isomorphism .A; / ! .A; / and ' ı . '/D' for all; 2 . Assume that for somen3, there exists a subfieldKof˝ finitely generated overk such that'.P / D P for allP killed by nand all fixingK.
Then.A; /has a model overksplitting.'/, i.e., there exists a polarized abelian variety .A0; 0/overkand an isomorphism'WV0˝ !V such that' D' 1ı 'for all 2 (apply AG 16.33).
15 Geometric Finiteness Theorems
In this section we prove two finiteness theorems that hold for abelian varieties over any field k. The first theorem says that an abelian variety can be endowed with a polarization of a fixed degree in only a finite number of essentially different ways. The second says that, up to isomorphism, an abelian variety has only finitely many direct factors. As a corollary we find that there are only finitely many isomorphism classes of abelian varieties of a fixed dimension over a finite field. This simplifies the proof of Tate’s isogeny theorem.
THEOREM15.1. LetAbe an abelian variety over a fieldk, and letd be an integer; then there exist only finitely many isomorphism classes of polarized abelian varieties.A; /with of degreed.
Let.A; /and.A0; 0/be polarized abelian varieties. From a homomorphism˛WA! A0, we obtain a map
˛.0/Ddf ˛_ı0ı˛WA!A_:
When˛ is an isomorphism and˛.0/ D, we call˛ anisomorphism.A; /! .A0; 0/ of polarized abelian varieties.
The theorem can be restated as follows: LetPol.A/be the set of polarizations on A, and letEnd.A/act onPol.A/byu7!u_ııu; then there are only finitely many orbits under this action.
Note that End.A/ D Aut.A/. If uis an automorphism of A, and L is an ample invertible sheaf onA, theuLis also an ample invertible sheaf, anduLDu_ıLıu;
thusEnd.A/doesact onPol.A/.
Fix a polarization0of A, and letbe the Rosati involution onEnd.A/˝Qdefined by 0. The map 7! 01ıidentifiesPol.A/with a subset of the set.End.A/˝Q/ of elements ofEnd.A/˝Qfixed by. Because0 is an isogeny, there exists an isogeny
˛WA_ !Asuch that˛ı0 Dn, somen 2Z, and then01 D.nA1/ı˛. Therefore the image of
Pol.A/ ,!.End.A/˝Q/ lies inLDdf n 1End.A/.
LetEnd.A/act onEnd.A/˝Qby
˛7!uı˛ıu, u2End.A/, ˛ 2End.A/˝Q:
ThenLis stable under this action, and the mapPol.A/!End.A/˝Qis equivariant for this action, becauseu_ııu7!01ıu_ııuD0 1
ıu_ı0ı0 1
ııuD uı.0 1/ıu:
Note thatdeg.0 1
ı/Ddeg.0/ 1deg./. Also (see 10.23), for an endomorphism
˛ of A,deg.˛/is a fixed power ofNm.˛/(norm fromEnd.A/˝QtoQ/. Therefore, as runs through a subset ofPol.A/of elements with bounded degrees, then0 1
ıruns through a subset ofLof elements with bounded norms. Thus the theorem is a consequence of the following number theoretic result.
PROPOSITION15.2. Let E be a finite-dimensional semisimple algebra over Q with an involution , and letR be an order inE. LetL be a lattice inE that is stable under the action˛7!u˛uofRonE. Then for any integerN, there are only finitely many orbits for the action ofRon
S D fv2LjNm.v/Ng; i.e.,S=Ris finite.
AnorderinE is a subringRofE that is a full lattice, i.e., free of rankdim.E/overZ.
In the application,RDEnd.A/.
This proposition will be proved using a general result from the reduction theory of arithmetic subgroups — see below (15.9).
We come now to the second main result of this section. An abelian subvarietyBofA is said to be adirect factorofAif there exists an abelian subvarietyC ofAsuch that the map.b; c/7!bCcWBC !Ais an isomorphism. Two direct factorsBandB0ofAare said to be isomorphic if there exists an automorphism˛ofAsuch that˛.B/DB0. THEOREM15.3. Up to isomorphism, an abelian variety A has only finitely many direct factors.
PROOF. Let B be a direct factor of A with complement C say, and define e to be the composite
A'BC .b;c/7!.b;0/!BC 'A:
Theneis an idempotent (i.e.,e2De), andBis determined bye becauseB 'Ker.1 e/.
Conversely, for any idempotenteofEnd.A/
ADKer.1 e/Ker.e/:
15. GEOMETRIC FINITENESS THEOREMS 65 Letu 2 End.A/. Thene0 D ueu 1is also an idempotent inEnd.A/, andudefines an isomorphism
Ker.1 e/!Ker.1 e0/:
Therefore, we have a surjection
fidempotents in End.A/g=End.A/! fdirect factors of Ag=;
and so the theorem is a consequence of the following number theoretic result. 2 PROPOSITION15.4. LetE be a semisimple algebra of finite dimension overQ, and letR be an order inE. Then
fidempotents in Rg=R is finite (hereRacts on the set of idempotents by conjugation).
This proposition will again be proved using a general result from the reduction theory of arithmetic subgroups (see 15.8), which we now state.
THEOREM15.5. LetGbe a reductive group overQ, and let be an arithmetic subgroup ofG.Q/; letG !GL.V /be a representation ofG on aQ-vector spaceV, and letLbe a lattice inV that is stable under . IfX is a closed orbit ofGinV, thenL\Xis the union of a finite number of orbits of .
PROOF. Borel 1969, 9.11. (The theorem is due to Borel and Harish-Chandra, but special
cases of it were known earlier.) 2
REMARK15.6. (a) By an algebraic group we mean an affine group variety. It isreductive if it has no closed normal connected subgroup U consisting of unipotent elements (i.e., elements such thatun D1for somen). A connected algebraic groupG is reductive if and only if the identity componentZ0of its centre is a torus andG=Z0is a semisimple group.
For example,GLnis reductive. The group BD˚ a b
0 c
ˇ
ˇ ac ¤0 is not reductive, becauseU D ˚ 1 b
0 1 is a closed normal connected subgroup consisting of unipotent matrices.
(b) LetG be an algebraic group overQ. ThenG can be realized as a closed subgroup ofGLn.Q/for somen(this is often taken to be the definition of an algebraic group). Let
GLn.Z/D fA2Mn.Z/jdet.A/D ˙1g:
ThenGLn.Z/is a group, and we let 0 D GLn.Z/\G.Q/. A subgroup ofG.Q/ is said to bearithmeticif it is commensurable with 0, i.e., if \ 0is of finite index in both and 0 — this is an equivalence relation. One can show that, although 0depends on the choice of the embeddingG ,!GLn, two embeddings give commensurable groups, and hence the notion of an arithmetic subgroup doesn’t depend on the embedding. Let
.N /D fA2G.Q/j A2Mn.Z/; AI mod.N /g:
Then .N / is a subgroup of finite index in 0, and so it is arithmetic. An arithmetic subgroup of this type is said to be aprincipal congruence subgroup.18
(c) A representation ofG on a vector spaceV is a homomorphismG ! GL.V / of algebraic groups. We can regardV itself as an algebraic variety (the choice of a basis forV determines an isomorphismV An,nDdim.V //, and we are given mapping of algebraic varieties
GV !V.
Ifvis an element ofV, then the orbitGvis the image of the map G fvg !V,g 7!gv:
It is a constructible set, but it need not be closed in general. To check that the orbit is closed, one needs to check that
X.kal/D fgvjg 2G.kal/g
is closed inV ˝kal.An/. One should interpreteL\X asL\X.kal/.
We give three applications of (15.5).
APPLICATION 15.7. LetG DSLn, and let DSLn.Z/. ThenGacts in a natural way on the spaceV of quadratic forms innvariables with rational coefficients,
V D fX
aij XiXj jaij 2Qg D fsymmetric nnmatrices, coeffs inQg;
— ifq.X /DXAXtr, then.gq/.X /DX gAgtrXtr— and preserves the latticeLof such forms with integer coefficients. Letq be a quadratic form with nonzero discriminant d, and let X be the orbit of q, i.e., the image G q of G under the map of algebraic varieties g 7! gqWG ! V. The theory of quadratic forms shows that X.Qal/is equal to the set of all quadratic forms (with coefficients inQal/of discriminantd. Clearly this is closed, and so the theorem shows that X \Lcontains only finitely manySLn.Z/-orbits:
the quadratic forms with integer coefficients and discriminantd fall into a finite number of proper equivalence classes.
APPLICATION 15.8. With the notations of (15.4), there exists an algebraic groupG over QwithG.Q/DEwhich is automatically reductive (this only has to be checked overQal; butE˝Qalis a product of matrix algebras, and soGQalis a product ofGLns). Take to be the arithmetic subgroupRofG.Q/,V to beEwithGacting by inner automorphisms, andL to beR. Then the idempotents inE form a finite set of orbits underG, and each of these orbits is closed. In proving these statements we may again replaceQby Qal and assumeE to be a product of matrix algebra; in fact, we may takeE D Mn.k/. Then the argument in the proof of (15.3) shows that
fidempotents in Eg=E' fdirect factors of kng=:
But, up to isomorphism, there is only one direct factor of kn for each dimension n.
Thus, each idempotent is conjugate to one of the form e D diag.1; :::; 1; 0; :::; 0/. If r is the number of1s, then the orbit ofe underE corresponds to the set of subspaces of
18The congruence subgroup problem asks whether every arithmetic subgroup contains a congruence sub-group. It has largely been solved — for some groupsGthey do; for some groupsGthey don’t.