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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 simplercategoryAõfabelianvarietiesuptoisogeny. 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 ellipticcurves.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 Aînsomenaturalway,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 characteristic 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 exact 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