We writeFqfor a finite field withqelements,Ffor an algebraic closure ofFq, andFqmfor the unique subfield ofFwithqm elements. Thus the elements ofFqm are the solutions of cqm Dc.
For a varietyV overFq, the Frobenius mapVWV ! V is defined to be the identity map on the underlying topological space ofV and is the mapf 7!fqonOV. For example, ifV DPnDProj.kŒX0; :::; Xn/, thenV is defined by the homomorphism of rings
Xi 7!XiqWkŒX0; :::; Xn!kŒX0; :::; Xn and induces the map on points
.x0W Wxn/7!.x0q W Wxnq/WPn.F/!Pn.F/:
For any regular map'WW !V of varieties overFq, it is obvious that'ıW DV ı'.
Therefore, ifV ,!Pnis a projective embedding ofV, thenV induces the map .x0W:::Wxn/7!.xq0 W:::Wxnq/
onV .F/. ThusV .Fq/is the set of fixed points ofVWV .F/!V .F/:
LetAbe an abelian variety overFq. ThenAmaps0to0(because0 2 V .Fq//, and so it is an endomorphism of A. Recall that its characteristic polynomial P is a monic polynomial of degree2g,g Ddim.A/, with coefficients inZ.
THEOREM1.1. WriteP.X /DQ
i.X ai/, and letNmDˇ
ˇA.Fqm/ˇ ˇ. Then (a) NmDQ2g
iD1.1 ami /for allm1, and (b) (Riemann hypothesis)jaij Dq12. Hence
jNm qmgj 2gqm.g 12/C.22g 2g 1/qm.g 1/:
PROOF. We first deduce the inequality from the preceding statements. Takem D1in (a) and expand out to get
ˇˇA.Fq/ˇ
ˇDa1 a2g 2g
X
iD1
a1 ai 1aiC1 a2gC
75
The first term on the right is an integer, and in fact a positive integer because it isP.0/D deg./, and (b) shows that it has absolute valueqg. Hence it equalsqg (actually, it easy to prove directly that deg./ D qg). The Riemann hypothesis shows that each term a1 ai 1aiC1 a2g has absolute value D qg 12, and so the sum has absolute value 2g qg 12. There are .2g 2g 1/ terms remaining, and each has absolute value qg 1, whence the inequality. We first prove (a) in the casemD1. The kernel of
idWA.F/!A.F/ isA.Fq/. I claim that the map
.d /0WTgt0.A/!Tgt0.A/
is zero — in fact, that this is true for any variety. In proving it, we can replaceAwith an open affine neighbourhoodU, and embedU intoAmsomemin such a way that0maps to the origin0. The map.d /0onTgt0.U /is the restriction of the map.d /0onTgt0.Am/.
But WAn ! An is given by the equations Yi D Xiq, and d.Xiq/ D qXiq 1 D 0 (in characteristicp). We now find that
d. id/0 D.d /0 .d.id //0D 1:
Hence 1is ´etale at the origin, and so, by homogeneity, it is ´etale at every point — each point in the kernel occurs with multiplicity1. Therefore,
ˇˇA.Fq/ˇ
ˇDdeg. id/:
But, from the definition ofP, we know that
deg. id/DP.1/;
and this isQ
.1 ai/. When we replace withmin the above argument, we find that ˇˇA.Fqm/ˇ
ˇDPm.1/:
Recall (10.20) that a1; :::; a2g can be interpreted as the eigenvalues of acting onT`A.
Clearlymhas eigenvaluesam1; :::; a2gm, and so Pm.X /DY
.X aim/,Pm.1/DY
.1 ami /
which proves (a) for a generalm. 2
Part (b) follows from the next two lemmas.
LEMMA1.2. Letbe the Rosati involution onEnd.A/˝Qdefined by a polarization of A; thenAıADqA.
PROOF. LetDbe the ample divisor onAdefining the polarization; thus.a/DŒDa D.
We have to show that
_ıı Dq:
Recall that, on points,_is the map
ŒD07!ŒD0WPic0.A/!Pic0.A/:
1. THE ZETA FUNCTION OF AN ABELIAN VARIETY 77 LetD0 be a divisor onA(or, in fact any variety defined over Fq/. If D0 D div.f /near .P /, then, by definition,D0 Ddiv.f ı/nearP. But.P /DP andf ı Dfq (this was the definition of/, anddiv.fq/ Dqdiv.f /; thusD0 DqD. Next observe that, for any homomorphism˛WA!Aand any pointaonA,
˛ıta.x/D˛.aCx/D˛.a/C˛.x/Dt˛.a/ı˛.x/:
We can now prove the lemma. For anya2A.F/, we have ._ıı/.a/DŒD.a/ D
DŒt.a/ D D
DŒ.t.a/ı/D D
DŒ.ıta/D D
DŒtaD D
DŒtaqD qD
Dq.a/;
as required. 2
LEMMA1.3. LetAbe an abelian variety over a fieldk(not necessarily finite). Let˛be an element ofEnd.A/˝Qsuch that˛ı˛is an integerr; for any rootaofP˛inC,jaj2Dr.
PROOF. Note that QŒ˛is a commutative ring of finite-dimension over Q; it is therefore an Artin ring. According to (Atiyah and Macdonald 1969, Chapter 8)1, it has only finitely many prime ideals m1; :::;mn each of which is also maximal, every element of T
mi is nilpotent, andQŒ˛=T
mi is a product of fields QŒ˛=T
mi DK1 Kn; Ki DQŒ˛=mi: We first show that T
mi D0, i.e., that QŒ˛ has no nonzero nilpotents. Note that QŒ˛
is stable under the action of . Let a ¤ 0 2 QŒ˛. Then b Ddf a a ¤ 0, because Tr.aa/ > 0. Asb Db,Tr.b2/ DTr.bb/ > 0, and sob2 ¤0. Similarly,b4 ¤0, and so on, which impliesb is not nilpotent, and so neither isa. Any automorphism of QŒ˛permutes the maximal idealsmi; it therefore permutes the factors Ki, i.e., there is a permutation off1; 2; :::; ngand isomorphismsiWKi ! K .i /such that .a1; :::; an/ D .b1; :::; bn/withb .i / Di.ai/. In the case that D, must be the trivial permutation, for otherwise Tr.a a/ would not always be positive (consider .a1; 0; 0; :::/ if .1/ ¤ 1). Hence preserves the factors of QŒ˛, and is a positive-definite involution on each of them. The involution extends by linearity (equivalently by continuity) to a positive-definite involution ofQŒ˛˝R. The above remarks also apply toQŒ˛˝R: it is a product of fields, andpreserves each factor and is a positive-definite involution on each of them. But now each factor is isomorphic toRor toC. The fieldRhas no nontrivial automorphisms at all, and somust act on a real factor ofQŒ˛˝Ras the identity map. The fieldChas only two automorphisms of finite order: the identity map and complex conjugation. The identity
1In fact, it is easy to prove this directly. Letf .X /be a monic polynomial generating the kernel ofQŒX ! QŒ˛. ThenQŒX =.f .X // ' QŒ˛, and the maximal ideals ofQŒ˛correspond to the distinct irreducible factors off .X /.
map is not positive-definite, and somust act on a complex factor as complex conjugation.
We have shown: for any homomorphismWQŒ˛! C,.˛/D.˛/:Thus, for any such homomorphism,r D.˛˛/ D j.˛/j2, and so every root of the minimum polynomial of˛inQŒ˛=Qhas absolute valuer12. Now (10.24) completes the proof. 2 REMARK1.4. We have actually proved the following: QŒis a product of fields, stable under the involution; under every mapWQŒ!C, ./D ./, andj j Dq12.
The zeta function of a varietyV overkis defined to be the formal power series Z.V; t /Dexp
PROOF. Take the logarithm of each side, and use (1.1a) and the identity (from calculus) log.1 t /Dt Ct2=2Ct3=3C:::: 2
REMARK1.6. (a) The polynomialPr.t /is the characteristic polynomial of acting on Vr
[A future version of the notes will include a complete proof of the Honda-Tate theorem, assuming the Shimura-Taniyama theorem (proved in the next section).]
For a fieldk, we can consider the following category:
objects:abelian varieties overkI
morphisms:Mor.A; B/DHom.A; B/˝Q:
This is called thecategory of abelian varieties up to isogeny,Isab.k/, overkbecause two abelian varieties become isomorphic inIsab.k/if and only if they are isogenous. It is Q-linear category (i.e., it is additive and theHom-sets are vector spaces overQ) and (10.1) implies that every object in Isab.k/ is a direct sum of a finite number of simple objects.
In order to describe such a category (up to a nonunique equivalence), it suffices to list the isomorphism classes of simple objects and, for each class, the endomorphism algebra. The