Grothendieck groups of categories of abelian varieties

0.13 MB·

by  Ari Shnidman

GROTHENDIECKGROUPSOF CATEGORIESOF ABELIANVARIETIES
ARISHNIDMAN
ABSTRACT. We compute the Grothendieck group of the category of
abelianvarietiesoveranalgebraicallyclosedfieldk.Wealsocompute
7
theGrothendieckgroupofthecategoryofA-isotypicabelianvarieties,
1
0 foranysimpleabelianvarietyA,assumingkhascharacteristic0,and
2 foranyellipticcurveAinanycharacteristic.

n
a
J

3 1. INTRODUCTION
1

The purpose of this note is to determine the Grothendieck groups of
]
G variouscategoriesofabelianvarieties. IfC isanexactcategory,thenthe
A GrothendieckgroupK (C)isthequotientofthefreeabeliangroupgen-
0
. eratedbyisomorphismclassesinC modulotherelations[X] [Y] [Z],
h − +
t foranyexactsequence0 X Y Z 0inC.
a → → → →
m LetA bethecategoryofabelianvarietiesoveranalgebraicallyclosed
field k. The morphisms in A are homomorphismsof abelian varieties.
[
KernelsdonotnecessarilyexistinA,butcokernelsdoexist,andA isan
1
v exactcategory.
4 TocomputetheGrothendieckgroupK (A),itishelpfultoconsiderthe
0
4 simplercategoryA˜ofabelianvarietiesuptoisogeny. Thiscategoryhas
7
3 thesameobjectsasA,and
0
1. HomA˜(A,B)=Hom(A,B)⊗ZQ,
0 for A,B Ob(A˜). ThecategoryA˜issemisimplebyPoincaré’sreducibili-
7 ∈
1 itytheorem,sothat
v: K0(A˜)∼ Z[A],
=
i A
X M
where the direct sum is over representatives of simple isogeny classes
r
a of abelian varieties. We have a surjectivemap ι: K (A) K (A˜) which
0 0
→
sendsanabelianvarietytoitsisogenyclass. Infact:
Theorem1.1. The map ι is an isomorphism. In particular, any additive
functionOb(A) G toanabeliangroupG isanisogenyinvariant.
→
TheproofofTheorem1.1,whichwassuggestedtousbyJulianRosen,
usesatrickinvolvingnon-isotypicabelianvarietiestoreducetoshowing
thefollowingfact: foreveryfinitegroupschemeG overk ofprimeorder,
thereisanabelianvariety Aoverk withanendomorphismwhosekernel
2010MathematicsSubjectClassification. 11G10,14K99,19A99.
Keywordsandphrases. Abelianvarieties,Grothendieckgroups,ellipticcurves.
1
2 ARISHNIDMAN
is isomorphictoG. This fact is easily proved using the arithmeticof el-
lipticcurves.Thus,theproofofTheorem1.1exploitsthefactthatcertain
ellipticcurveshaveextraendomorphisms,anddoesnotshedmuchlight
onthestructureofthecategoryA.
Forexample,oneconsequenceofTheorem1.1isthatanabelianvari-
ety Aanditsdual Aˆ determinethesameclassinK (A). Itisthennatural
0
toaskwhetherwecanwitnesstherelation[A] [Aˆ]inK (A),usingonly
0
=
shortexact sequences thatareintrinsicto A. In otherwords, we askfor
short exact sequences involving only abelian varieties that can be con-
structedfrom Aor Aˆinsomenaturalway,andthatdon’tinvolveauxiliary
abelian varieties such as CM elliptic curves. One way to formalize this
questionisasfollows.
Let A be a simple abelian variety of dimension g. Write C for the
A
categoryofabelianvarietiesB isogenousto An forsomen 0,withmor-
≥
phismsasusual.WewillwriteK (A)forK (C ),butnoticethatthisgroup
0 0 A
depends only on the isogeny class of A. One can then ask whether it is
truethat[A] [Aˆ]inK (A). Itwillfollowfromourmainresultbelowthat
0
=
theansweristypicallyno.
WriteG(A) for the kernel of the map dim: K (A) Z sending [A] to
0
→
dimA. The group G(A) measures the difference between the category
C and its isogeny category C˜ . We will describe G(A) in terms of the
A A
endomorphismalgebraD End(A) Q,assumingk hascharacteristic0.
= ⊗
Soletusassumechark 0untilfurthernotice. LetF bethecenterofD,
=
andwritee [F :Q] andd2 [D :F]. Also letF betheset ofnon-zero
+
= =
elementsofF whicharepositiveatalltheramifiedrealplacesofD.Then
ourmainresultis:
Theorem 1.2. Let Q be the group of positive rational numbers under
+
multiplication. Thenthereisacanonicalisomorphism
2g/de
deg: G(A) Q NmF/Q F+ ,
≃ +
which,foranytwosimple A1,A2 O±b(CA),s¡end¢stheclassof[A1] [A2]to
∈ −
theclassofthedegreeofanyisogeny A A .
1 2
→
This shows that even though ι: K (A) K (A˜) is an isomorphism,
0 0
→
there is still a big difference between the A-isotypic part of A and the
A-isotypicpartofA˜. Indeed:
Corollary1.3. ThegroupG(A)isaninfinitetorsiongroup.
2g/de
Proof. ThegroupQ Nm F istorsionsinceeveryintegerhas
F/Q +
+ 2g/de
apowerwhichisanormfromF. IfQ Nm F isfinite,thenwe
± ¡ ¢ F/Q +
+
must have 2g de. By the classification of endomorphism algebras of
= ± ¡ ¢
abelianvarietiesincharacteristic0,thisimpliesthatF isaCMfield. But
thenNm (F )hasinfiniteindexinQ ,whichcontradictstheassump-
F/Q × ×
(cid:3)
tionthatQ Nm F isfinite.
F/Q +
+
Dualityfit±snicelyi¡nto¢thispictureaswell:
GROTHENDIECKGROUPSOFCATEGORIESOFABELIANVARIETIES 3
Theorem1.4. Theautomorphism[B] [Bˆ]ofK (A)isinversiononG(A).
0
7→
Remark 1.5. NotethatdualitydoesnotinduceinversiononallofK (A),
0
as[A] [A]doesnotpreservedimension.
7→−
Thisletsusexhibitmanycaseswhere[A] [Aˆ]inK (A).
0
6=
Example1.6. If A is an abeliansurface with End(A) Z, then Theorem
=
1.2gives
G(A) Q /Q4 Z/4Z,
≃ + +≃ ℓ
M
withthedirectsumoverallprimesℓ.SupposeAadmitsanisogeny A
0
→
A ofdegreeℓfroma principallypolarizedsurface A ; inparticular A
0 0
≃
Aˆ . Then [A] [Aˆ] inK (A)if and onlyif[A ] [A] [A ] [Aˆ] inG(A).
0 0 0 0
= − = −
ByTheorem1.4,thiswouldmeanthattheclassof[A ] [A]inG(A)isits
0
−
owninverse.Buttheinverseofdeg([A ] [A]) ℓinQ Q4 isℓ3andnot
0
ℓ,sowemusthave[A] [Aˆ]inK (A). − = + +
6= 0 ±
Ontheotherhand,Theorem1.4immediatelyyieldsthefollowingpos-
itiveresult,givingacanonicalclassindegreetwoinK (A):
0
Theorem1.7. If A ,A Ob(C )aresimple,then[A ] [Aˆ ] [A ] [Aˆ ]
1 2 A 1 1 2 2
∈ + = +
inK (A). Inotherwords,theclass[A] [Aˆ]isindependentofthechoiceof
0
+
Ainitsisogenyclass.
Theorem 1.2 and Corollary 1.3 can fail quite dramatically in charac-
teristic p. The next two results show thatG(A) need not be infinitenor
torsioninpositivecharacteristic.
Theorem1.8. Supposechark p 0andlet Abeasupersingularelliptic
= >
curve.ThenG(A) 0. Inparticular,K (A) Z.
0
= ≃
Theorem1.9. Suppose chark p 0 and let A be an elliptic curve with
= >
End(A) Z. ThenG(A)containsanelementofinfiniteorder.
=
Forgeneral A incharacteristicp,determiningthestructureofG(A)is
somewhatsubtle,andwehopetoreturntothisquestioninfuturework.
ItwouldalsobeinterestingtocomputethegroupsG(A)whenk isafinite
field. The answer should be related to Milne’s computation [Mi] of the
size of the Ext group of two abelian varieties over a finite field. When
A is an ellipticcurve over a finitefield, one can presumablydeduce the
answerfromtheresultsoftherecentpaper[JKP ].
+
The plan for the rest of the note is as follows. In Section 2, we prove
Theorem1.1. InSection3,weproveausefulcriterionfortwoisogenous
abelianvarietiesB,B Ob(C )todeterminethesameclassinK (A). In
′ A 0
∈
Section4,wedefinethemapdeginTheorem1.2. InSection5,weprove
thatdegisanisomorphismincharacteristic0.InSection6,wedetermine
G(E)foranyellipticcurveE,inanycharacteristic.
4 ARISHNIDMAN
2. THE GROTHENDIECK GROUP K0(A)
Weletk beanalgebraicallyclosedfieldandA thecategoryofabelian
varietiesoverk.
Theorem2.1. ThereisanisomorphismK (A) Z[A],wherethesum
0 ∼ A
=
isoverrepresentatives Aofsimpleisogenyclassesofabelianvarieties.
L
Proof. By Poincaré reducibility, it is enough to show that if φ: A A is
′
→
anisogenyofabelianvarieties,then[A] [A ]inK (A). Weimmediately
′ 0
=
reducetothecasewhereφisanℓ-isogenyforsomeprimeℓ.
Suppose AandB areabelianvarieties,eachcontaininganembedding
ofagroupschemeG oforderℓ. LetC bethequotientof A B byadiag-
×
onalcopyofG A A . Thenthereareexactsequences
′
⊂ ⊕
0 A C B/G 0,
→ → → →
0 B C A/G 0.
→ → → →
Wethereforehavetherelation
[A] [A ] [B] [B ]
G G
− = −
in K (A), where A (resp. B ) is any quotient of A (resp. B) by a group
0 G G
isomorphic to G. So to prove the theorem, it suffices to find, for every
primeℓandforeverygroupschemeG oforderℓ,asingleabelianvariety
A andanendomorphism f End(A)suchthatkerf G. Infact,wewill
∈ ≃
showthatwecantake AtobeanellipticcurveE.
Write p chark. If ℓ p, then G Z/ℓZ, and we may take E such
= 6= ≃
thatEnd(E)containsZ[p ℓ]. Suchanellipticcurveexistsoveranyalge-
−
braicallyclosedfieldk,bythetheoryofcomplexmultiplication.Ifℓ p,
=
thentherearethreegroupschemesG toconsider,butforallthreewewill
takeE with j-invariantlyinginF . Inthiscase,theFrobeniusmorphism
p
F: E E(p) E
→ ≃
isanendomorphism. If E is supersingular,thenkerF α , whileifE is
p
≃
ordinary,thenkerF µ andkerFˆ Z/pZ.Thenumberofsupersingular
p
≃ ≃
ellipticcurves over F is related to a certain class number by a result of
p
Deuring[De],andisalwaysnon-zero[C,Thm.14.18].Ontheotherhand,
thenumberofsupersingular j-invariantsoverF¯ islessthanp,sothere
p
(cid:3)
arealways j-invariantsofbothtypesinF . Thisconcludestheproof.
p
Corollary2.2. AnyadditivefunctionOb(A) G toanabeliangroupG is
→
anisogenyinvariant.
Corollary2.2can beusedtoshowthatcertainfunctionsarenotaddi-
tive:
Example 2.3. If k Q¯, then the stable Faltings height is additive under
=
taking products of abelian varieties. If it were additive under short ex-
act sequences, then it would be an isogeny invariant. But it is easy to
GROTHENDIECKGROUPSOFCATEGORIESOFABELIANVARIETIES 5
seefromFaltings’isogenyformula[F,Lem.5]thattheheightsometimes
changesunderisogeny.
3. AUSEFUL CRITERION
In this section, we let k be any algebraicallyclosed field. Let A be an
abelianvarietyoverk ofdimensiong,notnecessarilysimple.
Lemma3.1. Supposeπ : A A andπ : A A areisogeniessuchthat
1 1 2 2
→ →
kerπ kerπ 0. Then
1 2
∩ =
[A] [A ] [A ] [A ] K (A),
1 2 3 0
= + − ∈
where A isthequotient A/(kerπ kerπ ).
3 1 2
+
Proof. Fori 1,2,letπ˜ :A A bethenaturalprojectionmaps,sothat
i i 3
= →
kerπ˜ π (kerπ ) and kerπ˜ π (kerπ ).
1 1 2 2 2 1
= =
Thenthereisashortexactsequence
0 Aπ1×π2 A1 A2π˜1−π˜2 A3 0.
→ −→ × −→ →
Toseethis,weneedtocheckthatthekernelofφ: π˜ π˜ iscontained
1 2
= −
intheimageofπ π . Forconcreteness,weargueonthelevelofpoints,
1 2
×
leavingtothereadertheexerciseofmakingthisargumentcategorical.1
Sosuppose(P,Q) A A isinkerφ. PickP¯ A suchthatπ (P¯) P.
1 2 1
∈ × ∈ =
Thenit suffices to showthatQ π (P¯ R)for someR kerπ , because
2 1
= + ∈
then
(P,Q) (π (P¯ R),π (P¯ R)).
1 2
= + +
Nowwecompute
π˜ (π (P¯) Q) π˜ (π (P¯)) π˜ (Q) π˜ (P) π˜ (Q) φ(P,Q) 0,
2 2 1 1 2 1 2
− = − = − = =
showingthatπ (P¯) Q iscontainedinπ (kerπ ) kerπ˜ ,asdesired. (cid:3)
2 2 1 2
− =
Theorem3.2. If A A and A A are isogenies of the same degree n,
1 2
→ →
andifn isinvertibleink,then[A ] [A ]inK (A).
1 2 0
=
Proof. First consider the case where the isogenies have prime degree ℓ.
Wemayassumethenthatkerπ kerπ . ByLemma3.1wehave
1 2
6=
[A] [A ] [A ] [A ],
3 1 2
+ = +
where A A/(kerπ kerπ ). Butthesameargumentworksforanytwo
3 1 2
= +
distinct order ℓ subgroup schemes of kerπ kerπ A[ℓ]. Since ℓ is
1 2
+ ⊂
invertibleink,wecanfindathirdsuchsubgroupC kerπ kerπ ,and
1 2
⊂ +
wehave:
[A ] [A/C] [A] [A ] [A ] [A/C],
1 3 2
+ = + = +
andhence[A ] [A ].
1 2
=
Thegeneralcaseproceeds byinductiononthedegree. If A A and
1
→
A A havedegreen,wecanfindasubgroupC A[n]ofordern which
2
→ ⊂
1
Itisnotenoughtoargueonthelevelofpointsifchark 0andkerπ kerπ has
1 2
> +
orderdivisiblebychark,butwewillnotactuallyusethiscaseofthetheorem.
6 ARISHNIDMAN
intersectsnon-triviallywithkerπ andkerπ . Indeed,ifn ℓa isaprime
1 2
=
power then one can take C to contain any two points P and P of or-
1 2
derℓinkerπ andkerπ ,respectively. Ifn ℓai isnotaprimepower,
1 2 = i
thenonecantakeC tobegeneratedbypointsoforderℓ andℓ inkerπ
1 2 1
Q
andkerπ , respectively. Thentheisogenies A A and A A/C factor
2 1
→ →
throughisogeniesA A and A A/C ofdegreen/ℓforsomeprimeℓ.
′ 1 ′
→ → (cid:3)
Byinduction,[A ] [A/C]andsimilarly[A/C] [A ],so[A ] [A ].
1 2 1 2
= = =
4. SIMPLE ISOGENY CLASSES
Inthissectionwe let A˜ beanisogenyclass ofsimpleabelianvarieties
of dimension g. Choose any A A˜ and set D End(A) Q. Then D
Z
∈ = ⊗
is a division algebra whose isomorphism class depends only on A˜. The
degreemapEnd(A) Zextendstoamultiplicativemapdeg:D Q .
×
→ → +
Remark4.1. Theremaybeelementsofdeg(D ) Zwhicharenotofthe
×
∩
formdeg(α)forsomeα End(A).
∈
Lemma4.2. If f : A A is an isogeny in A˜ of degree n, such that n is
1 2
→
invertibleink andn deg(β)forsomeβ D ,then[A ] [A ]inK (A).
× 1 2 0
= ∈ =
Proof. Writeβ ab 1 witha,b End(A ) End(A ),andwithdeg(a)in-
− 1 2
= ∈ ∩
vertible in k. Here we are thinking of End(A ) and End(A ) as abstract
1 2
ringsembeddedinD. Thenthecomposition
f b
A A A
1 2 2
−→ −→
(cid:3)
hasthesamedegreeasa:A A . ByTheorem3.2,[A ] [A ].
1 1 1 2
→ =
To computeG(A) ker(K (A) Z), it is convenient to choose A A˜
0
= → ∈
which is principallypolarizable, so that A Aˆ. We write deg(D) for the
≃
submonoiddeg(D ) ZofthemonoidZ ofpositiveintegersundermul-
×
∩ +
tiplication.NotethatZ deg(D)isagroup.
+
Lemma4.3. Suppose f :±An An isanisogeny.Thendeg(f) deg(D).
→ ∈
Proof. Theisogeny f canbethoughtofasanelement M GL (D). IfD
n
∈
iscommutativethenonehastheformula
deg(f) deg(detM).
=
Inthegeneralcase,thereisnowellbehaveddeterminantmapGL (D)
n
→
D . Instead,wehave
×
deg(f) (Nm Nrd (M))2g/de,
F/Q n
= ◦
whereF isthecenterofD,d [F :Q],e2 [D :F],andNrd :GL (D)
n n
= = →
F isthereducednorm;see[Mu,§19]. Ontheotherhand,for g D,we
×
∈
have
deg(g) (Nm Nrd(g))2g/de,
F/Q
= ◦
whereNrd:D F isthereducednorm.Thelemmathenfollowsfrom
× ×
→ (cid:3)
Dieudonné’sresult[Di]thatNrd (GL (D)) Nrd(D ).
n n ×
=
GROTHENDIECKGROUPSOFCATEGORIESOFABELIANVARIETIES 7
Lemma4.4. LetB C andlet f :An B beanisogeny.Thentheclassof
A
∈ →
deg(f)inZ deg(D)isindependentofthechoiceof f.
+
Proof. LetφL±:B Bˆ beapolarizationonB. Then
→
deg(φ ) deg(fˆφ f) deg(f)2deg(φ ).
f L L L
∗ = =
The isogeny φ : An Aˆn An has degree in deg(D) by Lemma 4.3.
f L ∼
∗ → =
Hence
(4.1) deg(f)2deg(φ ) 1 Z deg(D).
L
= ∈ +
Ifg :An B isanotherisogeny,thenthecom±positemap
→
h:An f B φL Bˆ gˆ Aˆn ψ An
−→ −→ −→ −→
alsohasdegreeindeg(D). Here,ψisanyprincipalpolarization.Somod-
ulodeg(D),wehave
1 deg(f)deg(gˆ)deg(φL) deg(f)deg(g)deg(g)−2 deg(f)/deg(g),
= = =
(cid:3)
andthereforedeg(f) deg(g)inZ deg(D).
= +
Definition4.5. IfB CA,thenthec±lassofdeg(f)inZ deg(D),forany
isogeny f : An B,∈is denoted dist (B) and is called+the distance of B
→ A ±
from A.
Lemma4.6. IfB C ,thendist (Bˆ) dist (B) 1.
A A A −
∈ =
Proof. Fromthesequence
An B φL Bˆ,
−→ −→
we obtain dist (Bˆ) dist (B)deg(φ ), which is equal to dist (B) 1, by
A A L A −
= (cid:3)
(4.1).
Lemma4.7. LetB C and let g :B An be an isogeny. Thentheclass
A
∈ →
of deg(g) in Z deg(D) is independentof the choice of g and is equal to
dist (B) 1. +
A − ±
Proof. Wehavedeg(g) deg(gˆ),with gˆ: Aˆn Bˆ thedualisogeny. Since
= →
Aˆn An,theclassofdeg(g)inZ deg(D)isindependentofg andequal
∼
tod=ist (Bˆ). Nowusethepreviou+slemma. (cid:3)
A ±
Definition4.8. IfB C ,thentheclassofdeg(f)inZ deg(D),forany
A
isogeny f :B An,i∈sdenoteddeg (B). +
→ A ±
Remark 4.9. We have deg (B) dist (B) 1 in Z deg(D). Context dic-
A = A − +
tateswhichinvariantismostconvenienttouse.
±
Remark4.10. Themapdeg :Ob(C ) Z deg(D)dependsonthechoice
of A. Butnotethatif f : B A B isaAn→isog+enyinC ,then
1→ 2 ± A
deg (B )
A 1
deg (B )
A 2
isequaltotheclassofdeg(f) Z deg(D)andhenceisindependentof
∈ +
thechoiceof A.
±
8 ARISHNIDMAN
Corollary 4.11. If B C and α End(B) is any isogeny, then deg(α)
A
∈ ∈ ∈
deg(D).
5. DETERMINATION OFG(A)IN CHARACTERISTIC 0
NowweconnectthenotionofdegreewiththeGrothendieckgroup.
Proposition5.1. Thefunctiondeg :Ob(C ) Z deg(D)isadditive.
A A → +
Proof. Let ±
j π
0 B B B 0
1 2
→ −→ −→ →
bean exact sequence in C . Let φ :Bˆ B and φ :B Bˆ be polar-
A M 1 1 L
→ →
izationsanddefineh :B B B to bethemapφ jˆφ π. From the
1 2 M L
→ × ×
sequenceofisogenies
B h B B An1 An2 An,
1 2 ∼
−→ × −→ × =
weconclude
deg (B) deg (B )deg (B )deg(h) Z deg(D).
A = A 1 A 2 ∈ +
So it suffices to show that deg(h) is in deg(D). For±this, note that kerh
is isomorphic to the kernel of the isogeny φ jˆφ j End(B ). Then by
M L 1
∈ (cid:3)
Corollary4.11,deg(h) deg(D).
∈
We therefore have a homomorphism deg :K (A) Z deg(D). By
A 0 → +
Remark4.10,therestrictionofdeg tothedimension0subgroupG(A)
A ± ⊂
K (A)isindependentof A,sowewrite
0
deg:G(A) Z deg(D).
→ +
This homomorphism is surjective, sinc±e we work over an algebraically
closedfield. Infact:
Theorem 5.2. Suppose chark 0. Then the degree map deg : G(A)
= →
Z deg(D)isanisomorphism.
+
Pro±of. We write A for any A A˜ such that dist (A ) n. The class
n n A n
∈ =
[A ] K (A) is independent of the choice of A by Theorem 3.2. By
n 0 n
∈
Lemma3.1,wehave,foranym,n Z :
∈ +
(5.1) [A ] [A ] [A ] [A],
nm n m
= + −
becausewecanalwaysrepresentA andA bycyclicquotientsofAwith
n m
non-intersecting kernelsC and C (since chark 0), and A by the
n m mn
=
quotient A/(C C ).
m n
+
NotethatK (A)isgeneratedbyclasses[A ]ofsimple A Ob(C ). Thus,
0 ′ ′ A
anyβ G(A),canbewrittenas
∈
r
β= [Ani]−[Ami] =[A rini]+(r−1)[A]−[A rimi]−(r−1)[A]=[A′]−[A′′],
i 1
X= ¡ ¢ Q Q
GROTHENDIECKGROUPSOFCATEGORIESOFABELIANVARIETIES 9
forcertain A ,A A˜ . IfβisalsointhekernelofthedegreemapG(A)
′ ′′
∈ →
Z deg(D),then
+
deg (A )/deg (A ) deg(D).
± A ′ A ′′ ∈
Equivalently,thereisanisogeny f :A A ofdegreedeg(α)forsomeα
′ ′′
→ ∈
D. ByLemma4.2,[A ] [A ]andβ 0,showingthatG(A) Z deg(D)
′ ′′
= = → + (cid:3)
isinjective,andhenceanisomorphism.
±
Asacorollary,weobtainTheorem1.4:
Corollary5.3. TheadditivefunctionB Bˆ inducestheinversionhomo-
7→
morphismonG(A).
Proof. Sincedeg (Bˆ) deg (B) 1,byLemma4.6andRemark4.9. (cid:3)
A = A −
ThefollowingpropositiongivesaconcretedescriptionofZ deg(D).
+
Proposition 5.4. Let A be a simple abelian variety of dimensi±on g and
D End(A) Q its endomorphism algebra, with center Z(D) F. Write
= ⊗ =
e [F :Q]andd2 [D :F]. Then
= =
Z deg(D) Q Nm (F )2g/de,
F/Q +
+ ≃ +
whereF isthesetofn±on-zeroelem±entsofF whicharepositiveatallthe
+
ramifiedrealplacesofD.
Proof. We have seen already that deg : D Q is given by the map
×
(Nm Nrd)2g/de.ButtheHasse-Schilling-M→aass+theorem[R,Thm33.15]
F/Q
◦ (cid:3)
statesthatthereducednormonD surjectsontoF .
+
Theorem1.2nowfollowsfromTheorem5.2andProposition5.4
6. THE CASE dimA 1
=
InthissectionwedetermineG(E),foranyellipticcurveE overanyal-
gebraicallyclosedfieldk,ofanycharacteristic.Thecharacteristic0cases
canbereadofffromTheorem1.2:
Theorem 6.1. Suppose k k¯ has characteristic 0 and E/k is an elliptic
=
curve with endomorphism algebra D. Then the degree map induces an
isomorphism
Q /Q2 ifD Q,
G(E) + + =
≃(Q /NmK/Q(K×) ifD K isimaginaryquadratic.
+ =
IntheCMcase,wecanmakethegroupstructureofG(E)moreexplicit:
Proposition6.2. IfK isimaginaryquadraticoverQ,then
Q /Nm (K ) C/C2 Z/2Z,
K/Q ×
+ ≃ ⊕
ℓinert
M
whereC Pic(O )istheclassgroupofK.
K
=
10 ARISHNIDMAN
Proof. If ℓ Nm(α) for some α K , then ℓ is not inert in K and (α)
×
laa¯ 1 forso=meidealaofO and∈someprimelaboveℓ. Itfollowsthat[=l]
− K
isasquareinC.Wethereforegetawelldefinedmap
Q /Nm (K ) C/C2 Z/2
K/Q ×
+ −→ ⊕
ℓinert
M
bysendinganinertprimeℓtothegeneratorofZ/2Zintheℓthslot,and
sending a non-inert prime ℓ to the class of [l] in C/C2, where l is any
primeaboveℓ. Thismapisclearlysurjective,andtoproveinjectivity,we
l l
needtoshowthatif[ ]isasquareinC,thenℓisanorm.If[ ]isasquare,
thenl (α)a2 (β)aa¯ 1 forsomeideala,soweseethatNm(β) ℓ. (cid:3)
−
= = =
Nowsupposek hascharacteristicp 0. Therearethreecasestocon-
>
sider,dependingonthedimensionofD End(E) QoverQ.
= ⊗
Theorem6.3. IfE issupersingular,thenG(E) 0.
=
Proof. The isogeny class of E is the set of supersingular elliptic curves.
It is therefore enough to show that [E ] [E] in K (E) for any other su-
′ 0
=
persingular elliptic curve E . By [Ko, Cor. 77], we may choose a prime
′
numberℓ p suchthatthereexistℓ-isogeniesE E andE E . Then
′
6= → → (cid:3)
[E ] [E]inK (E)byTheorem3.2.
′ 0
=
Theorem6.4. IfD isisomorphictoanimaginaryquadraticfieldK,then
thedegreemapinducesanisomorphismG(E) Q /Nm (K ).
K/Q ×
≃ +
Proof. We mayassumethatEnd(E)is isomorphictotheringofintegers
O . LetE beanellipticcurveisogenoustoE. ByaresultofDeuring[De,
K ′
p.263],theringEnd(E )hasindexprimeto p inO and p issplitinO .
′ K K
Thus,by[Ka,Prop.40],theranktwoquadraticformdeg: Hom(E,E ) Z,
′
→
hasdiscriminantprimetop. ItfollowsthatthereisanisogenyE E of
′
→
degree primeto p. Now we proceed exactly as in the proof of Theorem
5.2, but using thefact that [E ] [E ] in K (E), for some n Z primeto
′ n 0
= ∈ (cid:3)
p.
Theorem6.5. IfD Q,thenG(E)isisomorphictoasubgroupofindex2
=
inZ Q Q2.Inparticular,G(E)isnotatorsiongroup.
+ +
ProoLf. We±defineanadditivemapdeg : Ob(C ) Zasfollows.Forany
p,E E →
isogeny f : B B in C , let e(f) denote the number of Jordan-Holder
′ E
→
factorsofkerf isomorphictoZ/pZ,letc(f)denotethenumberoffactors
isomorphictoµ ,andletdeg (f) e(f) c(f).ForB Ob(C ),wedefine
p p = − ∈ E
deg (B) deg (f),where f isanyisogeny f : B En.Tocheckthatthis
p,E = p →
iswell-definedweuse:
Lemma6.6. If f : En En isanisogeny,thendeg (f) 0.
→ p =
Proof. Thecasen 1isclearsincethen f ZandE[p] Z/pZ µ . For
p
= ∈ ≃ ⊕
n 1,wemaythinkof f asamatrixM Mat (Z).WemayassumeM isin
n
> ∈
Smithnormalform,atthecostofchoosinganewproductdecomposition