Grothendieck Groups of Categories of Abelian Varieties 3

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Thus, the proof of Theorem 1.1 exploits the fact that certain elliptic curves have extra endomorphisms, and does not shed much light on the structure of the category A . For example, one consequence of Theorem 1.1 is that an abelian variety A and its dual Aˆ determine the same class in K0(A ). It is then natural to ask whether we can witness the relation [A] [Aˆ] in K (A ), using only = 0 short exact sequences that are intrinsic to A. In other words, we ask for short exact sequences involving only abelian varieties that can be con- structed from A or Aˆ in some natural way, and that don’t involve auxiliary abelian varieties such as CM elliptic curves. One way to formalize this question is as follows. Let A be a simple abelian variety of dimension g . Write C A for the category of abelian varieties B isogenous to An for some n 0, with mor- ≥ phisms as usual. We will write K0(A)for K0(C A), but noticethat this group depends only on the isogeny class of A. One can then ask whether it is true that [A] [Aˆ] in K (A). It will follow from our main result below that = 0 the answer is typically no. Write G(A) for the kernel of the map dim: K (A) Z sending [A] to 0 → dim A. The group G(A) measures the difference between the category C A and its isogeny category CÃ. We will describe G(A) in terms of the endomorphism algebra D End(A) Q, assuming k has characteristic 0. = ⊗ So let us assume chark 0 until further notice. Let F be the center of D, = 2 and write e [F : Q] and d [D : F ]. Also let F + be the set of non-zero = = elements of F which are positive at all the ramified real places of D. Then our main result is: Theorem 1.2. Let Q be the group of positive rational numbers under + multiplication. Then there is a canonical isomorphism 2g/de deg: G(A) Q NmF /Q F + , ≃ + which, for any two simple A1, A2 Ob(± C A), sends¡ ¢ the class of [A1] [A2] to ∈ − the class of the degree of any isogeny 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-isotypic part of A˜. Indeed: Corollary 1.3. The group G(A) is an infinite torsion group. 2g/de Proof. The group Q NmF /Q F + is torsion since every integer has + 2g/de a power which is a norm± from¡F . If¢ Q NmF /Q F + is finite, then we must have 2g de. By the classification+ of endomorphism algebras of = ± ¡ ¢ abelian varieties in characteristic 0, this implies that F is a CM field. But then NmF /Q(F ×) has infinite index in Q×, which contradicts the assump- tion that Q NmF /Q F + is finite. + Duality fits± nicely into¡ ¢ this picture as well: GROTHENDIECK GROUPS OF CATEGORIES OF ABELIAN VARIETIES 3

Theorem 1.4. The automorphism [B] [Bˆ] of K (A) is inversion on G(A). 7→ 0 Remark 1.5. Note that duality does not induce inversion on all of K0(A), as [A] [A] does not preserve dimension. 7→ − This lets us exhibit many cases where [A] [Aˆ] in K (A). 6= 0 Example 1.6. If A is an abelian surface with End(A) Z, then Theorem = 1.2 gives G(A) Q /Q4 Z/4Z, ≃ + + ≃ ℓ M with the direct sum over all primes ℓ. Suppose A admits an isogeny A 0 → A of degree ℓ from a principally polarized surface A ; in particular A 0 0 ≃ Aˆ . Then [A] [Aˆ] in K (A) if and only if [A ] [A] [A ] [Aˆ] in G(A). 0 = 0 0 − = 0 − By Theorem 1.4, this would mean that the class of [A0] [A] in G(A) is its − 4 3 own inverse. But the inverse of deg([A0] [A]) ℓ in Q Q is ℓ and not − = + ℓ,so we musthave[A] [Aˆ] in K (A). + 6= 0 ± On the other hand, Theorem 1.4 immediately yields the following positive result, giving a canonical class in degree two in K0(A):

Theorem 1.7. If A , A Ob(C ) are simple, then [A ] [Aˆ ] [A ] [Aˆ ] 1 2 ∈ A 1 + 1 = 2 + 2 in K (A). In other words, the class [A] [Aˆ] is independent of the choice of 0 + A in its isogeny class.

Theorem 1.2 and Corollary 1.3 can fail quite dramatically in characteristic p. The next two results show that G(A) need not be inﬁnite nor torsion in positive characteristic.

Theorem 1.8. Suppose chark p 0 and let A be a supersingular elliptic = > curve. ThenG(A) 0. In particular, K (A) Z. = 0 ≃ Theorem 1.9. Suppose chark p 0 and let A be an elliptic curve with = > End(A) Z. ThenG(A) contains an element of infinite order. = For general A in characteristic p, determining the structure of G(A) is somewhat subtle, and we hope to return to this question in future work. It would also be interestingto compute the groups G(A) when k is a finite 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 elliptic curve over a finite field, one can presumably deduce the answer from the results of the recent paper [JKP+]. The plan for the rest of the note is as follows. In Section 2, we prove Theorem 1.1. In Section 3, we prove a useful criterion for two isogenous abelian varieties B,B ′ Ob(C ) to determine the same class in K (A). In ∈ A 0 Section 4, we define the map deg in Theorem 1.2. In Section 5, we prove thatdeg is an isomorphismin characteristic0. In Section6, we determine G(E) for any elliptic curve E, in any characteristic. 4 ARI SHNIDMAN

2. THE GROTHENDIECK GROUP K0(A ) We let k be an algebraically closed ﬁeld and A the category of abelian varieties over k.

Theorem 2.1. There is an isomorphism K0(A ) Z[A], where the sum =∼ A is over representatives A of simple isogeny classes of abelian varieties. L Proof. By Poincaré reducibility, it is enough to show that if φ: A A′ is → an isogeny of abelian varieties, then [A] [A′] in K (A ). We immediately = 0 reduce to the case where φ is an ℓ-isogeny for some prime ℓ. Suppose A and B are abelian varieties, each containing an embedding of a group scheme G of order ℓ. Let C be the quotient of A B by a diag- × onal copy of G A A′. Then there are exact sequences ⊂ ⊕ 0 A C B/G 0, → → → → 0 B C A/G 0. → → → → We therefore have the relation [A] [A ] [B] [B ] − G = − G in K0(A ), where AG (resp. BG ) is any quotient of A (resp. B) by a group isomorphic to G. So to prove the theorem, it suffices to find, for every prime ℓ and for every group scheme G of order ℓ, a single abelian variety A and an endomorphism f End(A) such that ker f G. In fact, we will ∈ ≃ show that we can take A to be an elliptic curve E. Write p chark. If ℓ p, then G Z/ℓZ, and we may take E such = 6= ≃ that End(E) contains Z[p ℓ]. Such an elliptic curve exists over any alge- − braically closed field k, by the theory of complex multiplication. If ℓ p, = then there are three group schemes G to consider, but for all three we will take E with j -invariant lying in Fp . In this case, the Frobenius morphism F : E E (p) E → ≃ is an endomorphism. If E is supersingular, then kerF α , while if E is ≃ p ordinary, then kerF µ and kerFˆ Z/pZ. The number of supersingular ≃ p ≃ elliptic curves over Fp is related to a certain class number by a result of Deuring [De], and is always non-zero [C, Thm. 14.18]. On the other hand, the number of supersingular j -invariants over F¯p is less than p, so there are always j -invariants of both types in Fp . This concludes the proof. Corollary 2.2. Any additive function Ob(A ) G to an abelian group G is → an isogeny invariant. Corollary 2.2 can be used to show that certain functions are not additive: 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 GROTHENDIECK GROUPS OF CATEGORIES OF ABELIAN VARIETIES 5 see from Faltings’ isogeny formula [F, Lem. 5] that the height sometimes changes under isogeny.

3. A USEFUL CRITERION In this section, we let k be any algebraically closed ﬁeld. Let A be an abelian variety over k of dimension g , not necessarily simple. Lemma 3.1. Suppose π : A A and π : A A are isogenies such that 1 → 1 2 → 2 kerπ kerπ 0. Then 1 ∩ 2 = [A] [A ] [A ] [A ] K (A), = 1 + 2 − 3 ∈ 0 where A is the quotient A/(kerπ kerπ ). 3 1 + 2 Proof. For i 1,2, let π˜ : A A be the natural projection maps, so that = i i → 3 kerπ˜ π (kerπ ) and kerπ˜ π (kerπ ). 1 = 1 2 2 = 2 1 Then there is a short exact sequence

π1 π2 π˜ 1 π˜ 2 0 A × A A − A 0. → −→ 1 × 2 −→ 3 → To see this, we need to check that the kernel of φ : π˜ π˜ is contained = 1 − 2 in the image of π π . For concreteness, we argue on the level of points, 1 × 2 leaving to the reader the exercise of making this argument categorical.1 So suppose (P,Q) A A is in kerφ. Pick P¯ A such that π (P¯) P. ∈ 1 × 2 ∈ 1 = Then it suffices to show that Q π (P¯ R) for some R kerπ , because = 2 + ∈ 1 then (P,Q) (π (P¯ R),π (P¯ R)). = 1 + 2 + Now we compute π˜ (π (P¯) Q) π˜ (π (P¯)) π˜ (Q) π˜ (P) π˜ (Q) φ(P,Q) 0, 2 2 − = 1 1 − 2 = 1 − 2 = = showing that π (P¯) Q is contained in π (kerπ ) kerπ˜ , as desired. 2 − 2 1 = 2 Theorem 3.2. If A A and A A are isogenies of the same degree n, → 1 → 2 and if n is invertible in k, then [A ] [A ] in K (A). 1 = 2 0 Proof. First consider the case where the isogenies have prime degree ℓ. We may assume then that kerπ kerπ . By Lemma 3.1 we have 1 6= 2 [A] [A ] [A ] [A ], + 3 = 1 + 2 where A A/(kerπ kerπ ). But the same argument works for any two 3 = 1 + 2 distinct order ℓ subgroup schemes of kerπ kerπ A[ℓ]. Since ℓ is 1 + 2 ⊂ invertible in k, we can find a third such subgroup C kerπ kerπ , and ⊂ 1 + 2 we have: [A ] [A/C] [A] [A ] [A ] [A/C], 1 + = + 3 = 2 + and hence [A ] [A ]. 1 = 2 The general case proceeds by induction on the degree. If A A and → 1 A A have degree n, we can find a subgroup C A[n] of order n which → 2 ⊂ 1It is not enough to argue on the level of points if chark 0 and kerπ kerπ has > 1 + 2 order divisible by chark, but we will not actually use this case of the theorem. 6 ARI SHNIDMAN intersects non-trivially with kerπ and kerπ . Indeed, if n ℓa is a prime 1 2 = power then one can take C to contain any two points P1 and P2 of order ℓ in kerπ and kerπ , respectively. If n ℓai is not a prime power, 1 2 = i then one can take C to be generated by points of order ℓ and ℓ in kerπ Q 1 2 1 and kerπ , respectively. Then the isogenies A A and A A/C factor 2 → 1 → through isogenies A′ A and A′ A/C of degree n/ℓ for some prime ℓ. → 1 → By induction, [A ] [A/C] and similarly [A/C] [A ], so [A ] [A ]. 1 = = 2 1 = 2 4. SIMPLE ISOGENY CLASSES In this section we let A˜ be an isogeny class of simple abelian varieties of dimension g . Choose any A A˜ and set D End(A) Z Q. Then D ∈ = ⊗ is a division algebra whose isomorphism class depends only on A˜. The degree map End(A) Z extends to a multiplicative map deg : D× Q . → → + Remark 4.1. There may be elements of deg(D×) Z which are not of the ∩ form deg(α) for some α End(A). ∈ Lemma 4.2. If f : A A is an isogeny in A˜ of degree n, such that n is 1 → 2 invertible in k and n deg(β) for some β D×, then [A ] [A ] in K (A). = ∈ 1 = 2 0 1 Proof. Write β ab− with a,b End(A ) End(A ), and with deg(a) in- = ∈ 1 ∩ 2 vertible in k. Here we are thinking of End(A1) and End(A2) as abstract rings embedded in D. Then the composition

f b A A A 1 −→ 2 −→ 2 has the same degree as a : A A . By Theorem 3.2, [A ] [A ]. 1 → 1 1 = 2 To compute G(A) ker(K (A) Z), it is convenient to choose A A˜ = 0 → ∈ which is principally polarizable, so that A Aˆ. We write deg(D) for the ≃ submonoid deg(D×) Z of the monoid Z of positiveintegers under mul- ∩ + tiplication. Note that Z deg(D) is a group. + Lemma 4.3. Suppose f :±An An is an isogeny. Then deg(f ) deg(D). → ∈ Proof. The isogeny f can be thought of as an element M GL (D). If D ∈ n is commutative then one has the formula deg(f ) deg(detM). = In the general case, there is no well behaved determinant map GL (D) n → D×. Instead, we have 2g/de deg(f ) (Nm Q Nrd (M)) , = F / ◦ n where F is the center of D, d [F : Q], e2 [D : F ], and Nrd : GL (D) = = n n → F × is the reduced norm; see [Mu, §19]. On the other hand, for g D, we ∈ have 2g/de deg(g ) (Nm Q Nrd(g )) , = F / ◦ where Nrd : D× F × is the reduced norm. The lemma then follows from → Dieudonné’s result [Di] that Nrd (GL (D)) Nrd(D×). n n = GROTHENDIECK GROUPS OF CATEGORIES OF ABELIAN VARIETIES 7

Lemma 4.4. Let B C and let f : An B be an isogeny. Then the class of ∈ A → deg(f ) in Z deg(D) is independent of the choice of f . + Proof. Let φ ±: B Bˆ be a polarization on B. Then L → deg(φ ) deg(fˆφ f ) deg(f )2 deg(φ ). f ∗L = L = L n n n The isogeny φf L : A Aˆ A has degree in deg(D) by Lemma 4.3. ∗ → =∼ Hence 2 (4.1) deg(f ) deg(φL) 1 Z deg(D). = ∈ + If g : An B is another isogeny, then the composite± map → f φ gˆ ψ h : An B L Bˆ Aˆn An −→ −→ −→ −→ also has degree in deg(D). Here, ψ is any principal polarization. So modulo deg(D), we have 2 1 deg(f )deg(gˆ)deg(φ ) deg(f )deg(g )deg(g )− deg(f )/deg(g ), = L = = and therefore deg(f ) deg(g ) in Z deg(D). = + Deﬁnition 4.5. If B C A, then the class± of deg(f ) in Z deg(D), for any ∈ + isogeny f : An B, is denoted dist (B) and is called the distance of B → A ± from A. 1 Lemma 4.6. If B C , then dist (Bˆ) dist (B)− . ∈ A A = A Proof. From the sequence φ An B L Bˆ, −→ −→ 1 we obtain dist (Bˆ) dist (B)deg(φ ), which is equal to dist (B)− , by A = A L A (4.1). Lemma 4.7. Let B C and let g : B An be an isogeny. Then the class ∈ A → of deg(g ) in Z deg(D) is independent of the choice of g and is equal to + dist (B) 1. A − ± Proof. We have deg(g ) deg(gˆ), with gˆ : Aˆn Bˆ the dual isogeny. Since = → Aˆn An , the class of deg(g ) in Z deg(D) is independent of g and equal =∼ + to dist (Bˆ). Now use the previous lemma. A ± Deﬁnition 4.8. If B C A, then the class of deg(f ) in Z deg(D), for any ∈ + isogeny f : B An, is denoted deg (B). → A ± 1 Remark 4.9. We have degA(B) distA(B)− in Z deg(D). Context dic- = + tates which invariant is most convenient to use. ± C Remark 4.10. Themap degA : Ob( A) Z deg(D) depends on the choice → + of A. But note that if f : B B is an isogeny in C , then 1 → 2 ± A degA(B1)

degA(B2) is equal to the class of deg(f ) Z deg(D) and hence is independent of ∈ + the choice of A. ± 8 ARI SHNIDMAN

Corollary 4.11. If B C and α End(B) is any isogeny, then deg(α) ∈ A ∈ ∈ deg(D).

5. DETERMINATIONOF G(A) IN CHARACTERISTIC 0 Now we connect the notion of degree with the Grothendieck group. C Proposition 5.1. The function degA :Ob( A) Z deg(D) is additive. → + Proof. Let ± j π 0 B B B 0 → 1 −→ −→ 2 → be an exact sequence in C . Let φ : Bˆ B and φ : B Bˆ be polar- A M 1 → 1 L → izations and deﬁne h : B B B to be the map φ jˆφ π. From the → 1 × 2 M L × sequence of isogenies

h n1 n2 n B B1 B2 A A A , −→ × −→ × =∼ we conclude

degA(B) degA(B1)degA(B2)deg(h) Z deg(D). = ∈ + 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 Corollary 4.11, deg(h) deg(D). ∈

We therefore have a homomorphism degA : K0(A) Z deg(D). By → + Remark 4.10, the restriction of deg to the dimension 0 subgroup G(A) A ± ⊂ K0(A) is independent of A, so we write deg : G(A) Z deg(D). → + This homomorphism is surjective, since± we work over an algebraically closed ﬁeld. In fact:

Theorem 5.2. Suppose chark 0. Then the degree map deg : G(A) = → Z deg(D) is an isomorphism. + Proof.± 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 Lemma 3.1, we have, for any m,n Z : ∈ + (5.1) [A ] [A ] [A ] [A], nm = n + m − because we can always represent An and Am by cyclic quotients of A with non-intersecting kernels C and C (since chark 0), and A by the n m = mn quotient A/(C C ). m + n Note that K0(A) is generated by classes [A′] of simple A′Ob(C A). Thus, any β G(A), can be written as ∈ r β [An ] [Am ] [A r ] (r 1)[A] [A r ] (r 1)[A] [A′] [A′′], i i i ni i mi = i 1 − = + − − − − = − X= ¡ ¢ Q Q GROTHENDIECK GROUPS OF CATEGORIES OF ABELIAN VARIETIES 9 for certain A′, A′′ A˜ . If β is also in the kernel of the degree map G(A) ∈ → Z deg(D), then + deg (A′)/deg (A′′) deg(D). ± A A ∈ Equivalently, there is an isogeny f : A′ A′′ of degree deg(α)for some α → ∈ D. By Lemma 4.2, [A′] [A′′] and β 0, showing that G(A) Z deg(D) = = → + is injective, and hence an isomorphism. ± As a corollary, we obtain Theorem 1.4: Corollary 5.3. The additive function B Bˆ induces the inversion homo- 7→ morphism on G(A).

1 Proof. Since deg (Bˆ) deg (B)− , by Lemma 4.6 and Remark 4.9. A = A The following proposition gives a concrete description of Z deg(D). + Proposition 5.4. Let A be a simple abelian variety of dimension± g and D End(A) Q its endomorphism algebra, with center Z (D) F. Write = ⊗ = e [F : Q] and d 2 [D : F ]. Then = = 2g/de Z deg(D) Q NmF /Q(F +) , + ≃ + where F + is the set of non-zero± elements± of F which are positive at all the ramiﬁed real places of D.

Proof. We have seen already that deg : D× Q is given by the map 2g/de → + (Nm Q Nrd) . But the Hasse-Schilling-Maass theorem [R, Thm 33.15] F / ◦ states that the reduced norm on D surjects onto F +. Theorem 1.2 now follows from Theorem 5.2 and Proposition 5.4

6. THE CASE dim A 1 = In this section we determine G(E), for any elliptic curve E over any algebraically closed ﬁeld k, of any characteristic. The characteristic 0 cases can be read off from Theorem 1.2: 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 if D Q, G(E) + + = ≃ (Q /NmK /Q(K ×) if D K is imaginary quadratic. + = In theCM case, we can makethe group structureof G(E) more explicit: Proposition 6.2. If K is imaginary quadratic over Q, then 2 Q /NmK /Q(K ×) C/C Z/2Z, + ≃ ⊕ ℓ inert M where C Pic(O ) is the class group of K . = K 10 ARI SHNIDMAN

Proof. If ℓ Nm(α) for some α K ×, then ℓ is not inert in K and (α) 1 = ∈ = laa¯− for some ideal a of OK and some prime l above ℓ. It follows that [l] is a square in C. We therefore get a well defined map 2 Q /NmK /Q(K ×) C/C Z/2 + −→ ⊕ ℓ inert M by sending an inert prime ℓ to the generator of Z/2Z in the ℓth slot, and sending a non-inert prime ℓ to the class of [l] in C/C 2, where l is any prime above ℓ. This map is clearly surjective, and to prove injectivity, we need to show that if [l] is a square in C, then ℓ isanorm. If [l] is a square, 2 1 then l (α)a (β)aa¯− for some ideal a, so we see thatNm(β) ℓ. = = = Now suppose k has characteristic p 0. There are three cases to con- > sider, depending on the dimension of D End(E) Q over Q. = ⊗ Theorem 6.3. If E is supersingular, then G(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 number ℓ p such that there exist ℓ-isogenies E E and E E ′. Then 6= → → [E ′] [E] in K (E) by Theorem 3.2. = 0 Theorem 6.4. If D is isomorphic to an imaginary quadratic field K , then the degree map induces an isomorphism G(E) Q /NmK /Q(K ×). ≃ + Proof. We may assume that End(E) is isomorphic to the ring of integers OK . Let E ′ be an elliptic curve isogenous to E. By a result of Deuring [De, p. 263], the ring End(E ′) has index prime to p in OK and p is split in OK . Thus,by[Ka, Prop.40],theranktwoquadraticform deg: Hom(E,E ′) Z, → has discriminant prime to p. It follows that there is an isogeny E E ′ of → degree prime to p. Now we proceed exactly as in the proof of Theorem 5.2, but using the fact that [E ′] [E ] in K (E), for some n Z prime to = n 0 ∈ p. Theorem 6.5. If D Q, thenG(E) is isomorphic to a subgroup of index 2 = in Z Q Q2 . In particular, G(E) is not a torsion group. + + Proof.LWe± definean additivemap deg : Ob(C ) Z as follows. For any p,E E → isogeny f : B B ′ in C , let e(f ) denote the number of Jordan-Holder → E factors of ker f isomorphic to Z/pZ, let c(f ) denote the numberof factors isomorphic to µ ,andletdeg (f ) e(f ) c(f ). For B Ob(C ), we define p p = − ∈ E deg (B) deg (f ), where f is any isogeny f : B E n. Tocheckthatthis p,E = p → is well-defined we use: Lemma 6.6. If f : E n E n is an isogeny, then deg (f ) 0. → p = Proof. The case n 1 is clear since then f Z and E[p] Z/pZ µ . For = ∈ ≃ ⊕ p n 1, we may thinkof f as a matrix M Mat (Z). We mayassume M isin > ∈ n Smithnormalform, at the cost of choosing a new product decomposition GROTHENDIECK GROUPS OF CATEGORIES OF ABELIAN VARIETIES 11 for E n. Then ker f is the direct sum of kernels of endomorphisms of E and the lemma follows from the case n 1. = n n Now suppose f ′ : B E is another isogeny. If we let g ′ : E B be → → such that f ′g ′ [deg f ′] n , then deg (f ′) deg (g ′) 0 and = E p + p = deg (f ) deg (f ′) deg (f ) deg (g ′) deg (fg ′) 0, p − p = p + p = p = bythelemma. Sodeg (B) is well-defined. We also conclude that deg (f ) p,E p = 0 for any f End(B), just as in Corollary 4.11. ∈ One now checks that degp,E is additive, exactly as in Proposition 5.1. We therefore get an induced map K (E) Z which depends on E. But 0 → the restriction to G(E) gives a canonical map deg : G(E) Z sending p → [E ′] [E ′′] to deg (f ), where f : E ′ E ′′ is any isogeny. − p → For any isogeny f , we write degℓ(f ) for the prime-to-p part of deg(f ). Now consider the composite map (deg ,deg) p 2 tot: G(E) Z Q Q Z Hp −→ ⊕ + + −→ ⊕ 2 where Hp ℓ p Z/2Z is the prime-to-±p part of Q Q , and the map 2 ≃ 6= + + Q Q Hp is the canonical surjection. Then + + → L ± ± tot([E ] [E ]) (deg (f ),deg (f )) 1 − 2 = p ℓ for any isogeny f : E E . It follows that tot is surjective, and we will 1 → 2 show that it is injective as well. For every (a,n) Z Z with n prime to p, we let Ea,n be any ellip- ∈ ⊕ + tic curve admitting an isogeny f : E E such that deg (f ) a and → a,n p = deg (f ) n. Thentheclassof[E ]in K (E) is independentof the choice ℓ = a,n 0 of E . Indeed, the isogeny f : E E can be factored as a,n → a,n g E E E , −→ a −→ a,n a where Ea is the unique étale quotient of E of degree p , and g is an n- isogeny. Thus the class of [Ea,n ] is uniquely determined by Theorem 3.2. Using Lemma 3.1, we obtain the following relations in K0(E), for any integers a and b, and any positive integers n and m coprime to p: [E ] [E ] [E ] [E] a,n = a,1 + 0,n − [Ea b,1] [Ea,1] [Eb,1] [E] + = + − [E ] [E ] [E ] [E]. 0,nm = 0,n + 0,m − It follows that any β G(E) can be written as ∈ β [E ] [E ] [E ] [E ] = a,1 − b,1 + 0,n − 0,m for integers a and b, and positive integers n and m. If tot(β) 0, then we = must have a b and n md 2 for some rational number d. By Lemma = = 4.2, we must have [E0,n ] [E0,m], and so β 0. Thus, tot is an injection, = = 2 and hence an isomorphism. Since Z Hp embeds in Z Q Q as an ⊕ ⊕ + index 2 subgroup, Theorem 6.5 is proved. + ± 12 ARI SHNIDMAN

Remark 6.7. An example of a non-torsion class in G(E) is [E] [E (p)], − where E (p) is the Frobenius-transform of E, i.e. the elliptic curve with j - invariant j (E)p . 6.1. Acknowledgments. The author thanks Julian Rosen for suggesting the proof of Theorem 1.1 and for other conversations on this topic. He also thanks Taylor Dupuy and Yuri Zarhin for their helpful feedback.

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