An Unexpected Trace Relation of Cm Points 3

Home , Benedict Gross

arXiv:1806.11337v2 [math.NT] 22 Apr 2019 nr,i oshv Mpit fconductor of points CM have does it inert, n usinaie aual:i Tr is naturally: arises conductor question One of order the to associated oua curve modular eertdGosZge oml G8]sosta hspiti non is point this that when shows [GZ86] formula Gross-Zagier celebrated aia naie bla xeso of extension abelian unramified maximal onsaduniformizes and points oml,wscridotb hn [Zha01]. Zhang by out carried was formula, O htisiaeudrtemdlrprmtiainis parametrization modular the under image its that mgnr udai edadlet and field quadratic imaginary aaerzto rmtecasclmdlrcurve modular classical the from parametrization a K nsieof spite In Kwsspotdb nAeadrvnHmod ototrlfello postdoctoral Humboldt von Alexander an by supported was DK 2010 Let h lsia hoyo ege ons[r8]sple on on point a supplies [Gro84] points Heegner of theory classical the , L NUEPCE RC EAINO MPOINTS CM OF RELATION TRACE UNEXPECTED AN E/ ahmtc ujc Classification. Subject Mathematics h ucinleuto of equation functional the N uv fnnsltCra ee at level Cartan non-split of curve d rm o dividing not prime odd fteC onsw banpit on points obtain we points CM the of ftelclsg of non-tr sign is that local compatibility the trace if a satisfy curves Shimura different igcasfilso conductor of fields class ring Abstract. locnie Mpito conductor of point CM a consider also atnlvlat level Cartan ′ ( na mgnr udai field quadratic imaginary an in ) E/K, Q ea litccreo conductor of curve elliptic an be X 1) X 0 ( 0 p .When 0. 6= Let ( 2 p o ossigHenrpit fcnutr1whenever 1 conductor of points Heegner possessing not ) p 2 hs uvsamtprmtiain to parametrizations admit curves These . ,hwvr h o-pi atncurve Cartan non-split the however, ), E/ E/ E Q hs oehrwt eeaiaino h Gross-Zagier the of generalization a with together This, . Q ea litccreo conductor of curve elliptic an be M at Let . E/K p p s+1. is sietin inert is f O O Introduction and AILKOHEN DANIEL is K f H H p − eteodro conductor of order the be p p K eisrn fitgr.I h prime the If integers. of ring its be n Mpito conductor of point CM a and .Ascae oteedt hr saShimura a is there data these to Associated 1. P pf rmr:1F2,Scnay 10 . 11G05 Secondary: , 11F32 Primary: inside E p nwhich in .W rv htteepit rsn from arising points these that prove We ). pf ∈ endover defined 1 K K naohrSiuacre sn split a using curve, Shimura another on E Consider . ( p O ed o aeHenrpit nthe in points Heegner have not do we H p K endover defined p 2 non-torsion? ) Since . aeaysc point such any Take . sietadsc httesg of sign the that such and inert is H f X P and N P E 1 0 E ( K = p n aigteimages the taking and ∈ f H 2 smdlr hr exists there modular, is Tr = p to ) pf H rltvl rm to prime (relatively E wship. 2 X M p ( epciey(the respectively f h igcasfield class ring the , va fadonly and if ivial H ns K H where ni.W can We it. on E ( ,where ), P p trinprecisely -torsion Let . ot Heegner hosts ) 1 ∈ p P X E san is p p 0 ( ∈ ssltin split is K ( K H p E 2 .The ). such ) ean be sthe is ( H p p is ). 2 DANIEL KOHEN

Theorem. (Special case of Proposition 3.3 and Theorem 4.3) Let E/Q be an elliptic curve of conductor p2 and let K be an imaginary quadratic field such that p is inert in K. If the local Atkin-Lehner sign wp(E) of E at p is 1 Hp Hp − then TrH Pp is zero. If wp(E)=+1, then TrH Pp E(H) is non-torsion if and only if P E(H) is non-torsion. ∈ 1 ∈ The theme of Heegner points constructions for more general orders (including non- split Cartan orders) was also studied in [CCL18], [KP16], [KP18] and [LRdVP18]. An immediate application of the present article is that, if wp(E) = +1, we can explicitly compute Heegner points associated to imaginary quadratic fields in which p is inert in a much more efficient way than [KP16], as we can avoid working with a non-split Cartan curve and computing the explicit modular parametrization to the elliptic curve E. One of the main features of Heegner points (or CM points in general) is that they come in families satisfying trace compatibilities (see for example [BD96, Section 2.4],[CV07, Section 6], [Dar04, Proposition 3.10],[Gro91, Proposition 3.7],[Nek07, Proposition 4.8]). These compatibilities are a crucial tool in order to bound Selmer groups and construct p-adic L-functions. Cornut and Vatsal introduced the notion of good CM points which are essentially the CM points that satisfy such compatibility relations. Our CM points of conductor p are not good (they are of type III in the sense [CV07, Section 6.4]), thus the main interest of this paper is that it provides an (unexpected!) trace relation of CM points living in two distinct modular curves. A parallel of this situation is the work of Bertolini and Darmon [BD96] that relies in understanding the relation between CM points in distinct curves. Their situation is deeper and more delicate since there is a sign change phenomenon and they have to study CM points living in Shimura curves with different ramification sets. 2 The modular curve X0(p ) is isomorphic to Xs(p), the modular curve associated to a split Cartan group modulo p. The new part of its Jacobian is isogenous over Q (in a manner compatible with the Hecke operators) to the Jacobian of the curve Xns(p), providing a modular parametrization from Xns(p) to E. This was first proved by Chen [Che98] comparing the traces of the Hecke operators for these curves. Af- terwards, de Smit and Edixhoven [dSE00] gave another (more geometric) argument using the representation theory of GL2(Fp). The result was later reproved in the aforementioned paper of Zhang resorting to the theory of automorphic representations. We would also like to mention the related nice article [Gro18] which was kindly brought to our attention by Benedict Gross. The paper focus on the action of SL (F )/ 1 on the regular differentials on X(p) and its relation with Hecke’s 2 p ± Q −1 subspace associated to forms with complex multiplication by ( ( p )p). q AN UNEXPECTED TRACE RELATION OF CM POINTS 3

If wp(E)= 1 the proof that the trace of the point of conductor p is 0 (Proposition 3.3) follows easily− by studying the interplay between the Galois action and the Atkin- Lehner involution. Suppose now that wp(E) = +1. In that case, we can work with the normalizers of the Cartan groups (both split and non-split), obtaining the isogenous Jaco- new bians Jns+(p) and Js+ (p). Chen [Che00] was able to provide an explicit isogeny between these two Jacobians (the other proofs mentioned where non-constructive). The isogeny is given in very simple terms as a double coset operator, that can be thought as a trace between the modular curves. The isogeny was also illustrated, using a nice combinatorial description of the moduli interpretation of the non-split Cartan curve, by Rebolledo and Wuthrich [RW18]. In the recent paper [CS19] Chen and Salari Sharif also gave a simpler proof and even showed an explicit isogeny be- new tween Jns(p) and Js (p) (unluckily, the explicit isogeny is less nice, as it is given by a linear combination of several double coset operators). The results of this paper boil down to prove that, when wp(E) = +1, the double coset operator giving Chen’s isogeny applied to the CM point on the non-split Cartan curve essentially gives the trace of the CM point of conductor p on the split Cartan curve (Proposition 3.4). In addition, we prove that the isogeny is equivariant for the Hecke operators (Proposition 4.1). Combining these results we prove the main result of the article (Theorem 4.3). Although the reader should have in mind this case for intuition, we will actually prove a more general result. Namely, we will work with an elliptic curve of conductor N = p2M where p is an odd prime number that does not divide M, and with K an imaginary quadratic field such that p is inert in K, the sign of E/K is 1 and such that if q is prime and q2 M, then q is unramified in K. Under these− hypotheses, there exists an embedding| of K into a certain indefinite quaternion algebra B and we consider Shimura curves Xns and Xs by choosing certain orders inside B that, locally at p, look like the non-split and split Cartan orders respectively. For f relatively prime to N we consider points P E(H ),P E(H ) obtained as the f ∈ f pf ∈ pf images of specific special points under the parametrizations from Xns and Xs to E Hpf respectively. We prove that if wp(E) = 1, then Tr Ppf = 0 (Proposition 3.3). − Hf Hpf Conversely, if wp(E) = +1, we prove that the point Tr Ppf E(Hf ) is non-torsion Hf ∈ if and only if P E(H ) is non-torsion (Theorem 4.3). f ∈ f Acknowledgments: I would like to thank Ariel Pacetti and Matteo Tamiozzo for their helpful comments and discussions. 4 DANIEL KOHEN

Notation Let p be an odd prime number and let ε be a non-square modulo p. By x we • will denote the reduction modulo p of x, whenever it makes sense. We define the non-split Cartan order as • a b Mns = M2(Zp): a d,bε c mod p . ( c d ! ∈ ≡ ≡ ) We also define a b Mns+ = Mns M2(Zp): a d, bε c mod p . ∪ ( c d ! ∈ ≡ − − ≡ ) We define the split Cartan order as • a b Ms = M2(Zp): b c 0 mod p , ( c d ! ∈ ≡ ≡ ) and we consider a b Ms+ = Ms M2(Zp): a d 0 mod p . ∪ ( c d ! ∈ ≡ ≡ ) For ? ns, ns+,s,s+ we denote the corresponding Cartan group • ∈{ } × C? = M : M M? GL2(Fp). ∈ ⊂ × The group Cns is a commutative subgroup of GL2(Fp) isomorphic to Fp2 . The group Cns+ (resp. Cs+) is the normalizer of Cns (resp. Cs) and the quotients Cns+/Cns,Cs+/Cs are both of size 2. Given a Z-module A and a prime v, we set A = A Z . Also, set Z = Z • v ⊗ v v v and A = A Z. Q Given a quaternion⊗ algebra B (respectively an order R B) web denote by • ⊂ B1 (resp.b R1)b the elements of B (resp. R) of reduced norm 1.

1. Optimal embeddings of Cartan orders Let E/Q be an elliptic curve of conductor N = p2M, where p is an odd prime relatively prime to M. Let K be an imaginary quadratic ﬁeld such that the sign ǫ(E/K) of the functional equation of E/K is equal to 1 and the prime p is inert in K. For simplicity we assume that if q is prime and q2−M, then q is unramiﬁed in K. The sign ǫ(E/K) decomposes as the product of local| signs ǫ (E/K) 1. Let v ∈ ± AN UNEXPECTED TRACE RELATION OF CM POINTS 5

η = v ηv be the character associated to K via class field theory. For each place v let ǫ(Bv) be the unique sign such that Q ǫ (E/K)= η ( 1)ǫ(B ). v v − v Let S be the set of places such that ǫ(Bv)= 1. Since ǫ(E/K)= 1, S has odd cardinality. Moreover, because K is imaginary, − S [Gro88, Proposition− 6.5]. Let B/Q be the unique quaternion algebra with ramification∞ ∈ set S . Let f be a − {∞} positive integer relatively prime to N and let f be the unique order of conductor f inside , the ring of integers of K. O OK Proposition 1.1. There exists an order R B of discriminant M and an embedding ι : K B such that ⊂ 0 → ι (K) R = ι ( ). 0 ∩ 0 Of Proof. The existence of local orders and embeddings satisfying the required property is proven in [Gro88, Propositions 3.2, 3.4]. The arguments of [Ibid. Section 9] show that we can find a global order with the required properties. Remark 1.2. Since p2 divides N exactly, p / S [Gro88, Proposition 6.3 (2)], and thus B M (Q ). As R has discriminant relatively∈ prime to p we can (and do) assume p ≃ 2 p that the local maximal order Rp is M2(Zp).

Our next goal is to change the embedding ι0 in a fashion more suitable for our computations. For ? ns, ns+,s,s+ we deﬁne R? = x R : xp M? . The orders Rns and ∈ { } 2 { ∈ ∈ } Rs are both of discriminant N = p M. We have the following proposition. Proposition 1.3. There exists an embedding ι : K B such that: → (1) The order embeds optimally into R , that is, Of ns ι(K) R = ι( ). ∩ ns Of (2) The order embeds optimally into R , that is, Opf s ι(K) R = ι( ). ∩ s Opf Proof. Let ωf K be such that f = Z + ωf Z. Consider its characteristic polyno- 2∈ O O 2 mial χωf = X tX+n of discriminant D = t 4n. Embedding B into Bp = M2(Qp) we get − − ι (ω ) = a0 b0 = A M (Z ), 0 f p c0 d0 0 ∈ 2 p where the characteristic polynomial of A is equal to χ . Since p is inert in K, D is 0 ωf not a square modulo p and the reduction modulo p of A0 is conjugate (in GL2(Fp)) to t/2 √D/4ε Cns, ε√D/4ε t/2 ∈ 6 DANIEL KOHEN since both matrices have the same characteristic polynomial (with simple roots). In addition, because det : C F× is surjective and C is commutative, they can ns → p ns be conjugated by a matrix γ SL2(Fp). As a consequence of strong approximation, 1 1 ∈ the map R (Rp/pRp) SL2(Fp) is surjective [Voi18, Corollary 28.4.11], where the superscript→ 1 denotes the≃ elements of reduced norm 1. Lastly, we take a lifting 1 γ R of γ and we conjugate ι0 by γ to obtain the desired ι. ∈Statement (1) is clear from the construction. For statement (2), note that we only changed the order at p, so we just need to check locally at p. Let ι(ω ) = ( a b ) f p c d ∈ (Rns)p. The local order ( pf )p is equal to Zp+pwf Zp and thus ι(( pf )p) (Rs)p. For the other inclusion, noteO that since p is inert in K we must have thatO p does⊆ not divide bc. Consequently, if we take an element (ι(x + x pw )) (R ) with x , x Q , 1 2 f p ∈ s p 1 2 ∈ p looking at the (2, 1) entry we obtain x2 Zp and looking at the (1, 1)-entry we get x Z , as we wanted. ∈ 1 ∈ p 2. Shimura curves For ? ns, ns+,s,s+ consider the (open) Shimura curve whose complex points are given∈{ by the double quotient}

× × × Y?(C)= B (C R) B /R? , × \ − × where B acts on C R via the action of B∞ GL2(R) by Mbius transformations × − × ≃ b c and by left multiplication on B and R? acts trivially on C R and by right × × − multiplication on B . For z C R and b B we will denote by [z, ˆb] the class of an element in this double quotient.∈ b− c ∈ We consider itsb compactiﬁcation X?(C),b givenb by

X (C)= Y (C) cusps . ? ? ∪{ } The set of cusps is empty if B = M (Q) and otherwise equal to the finite set 6 2 × GL (Q) P1(Q) GL (Q)/R , 2 \ × 2 ? where the action of GL (Q) on P1(Q) is given by Möbius transformations. These 2 b curves admit the following, more classical, description.c × 1 Proposition 2.1. Let Γ? be the congruence subgroup defined as Γ? = R? B . Let ∗ be the union of the upper half plane and the set of cusps. Then the∩ map H H c Γ ∗ X (C), [z] [z, 1], ?\H → ? 7→ is an isomorphism. AN UNEXPECTED TRACE RELATION OF CM POINTS 7

Proof. The only subtle point is to check that this map is surjective. Since B is an indeﬁnite quaternion algebra, strong approximation [Vig80, Theorem 4.3] yields (applying the determinant map)

× × × B× 1 B×/R Q × Q×/det(R ) Z×/det(R ). \ {± } × ? ≃ >0 \ ? ≃ ? In addition, as every prime q such that q2 N is unramiﬁed in K, it is easy to see × b c b ×c b c × × | × that det(R? )= Z and thus B 1 B /R? consists of a single point, which immediately proves that the map\ is {± surjective.} × c b b c

Let J? be the Jacobian variety of X?. We can embed X? ֒ i? J? using the so-called Hodge class [Zha01, Section 6.2],[YZZ13, Section 3.1.3],→ that is, the unique class ξ P ic(X ) Q of degree 1 such that T ξ = (ℓ + 1)ξ for every prime number ? ∈ ? ⊗ ℓ ? ? ℓ not dividing N. In the case B = M2(Q) the Hodge class is a (rational) linear combination of cusps. The embedding i? is given by sending a point P to the class of P ξ?. − 0 Let π = Hom (X , E), that is, the morphisms in Hom(X , E) Z Q sending a E,? ξ? ? ? ⊗ divisor representing ξ? to zero. In other words,

π = Hom(J , E) Z Q. E,? ? ⊗

This is the space of parametrizations from X? to the elliptic curve E. Let fE be the newform of level N corresponding to E by the modularity theorem.

Proposition 2.2.

(1) The spaces πE,ns and πE,s are 1-dimensional. Non-zero elements Φns πE,ns and Φ π have the same eigenvalues as f for the good (i.e. not dividing∈ s ∈ E,s E N) Hecke operators and the Atkin-Lehner involutions. The element ΦE,s p−new factors through Js , since the corresponding form is new at p. (2) Furthermore, if wp(E) = +1 we have the induced non-zero elements Φns+ π and Φ π such that the following diagrams are commutative: ∈ E,ns+ s+ ∈ E,s+ ins Φns is p−new Φs Xns Jns E Xs Js E

i + Φ + ns ns is+ p−new Φs+ Xns+ Jns+ E, Xs+ Js+ E.

Proof. This is proved by [Zha01, Theorem 1.3.1] (see also [GP91, Proposition 2.6] and [CST14, Propositions 3.7,3.8]). 8 DANIEL KOHEN

3. Explicit Galois action on special points The embedding ι : K B induces an action of K×/K× Gal(Kab/K) on the → ≃ curves X?(C) by the formula b a [z, b] := [z, ˆι(a)b]. · Recall that ω satisfies = Z + ω Z. Let z be the unique fixed point in f ∈ OK Of f 0 of ι∞(ω ) GL (R) under the actionb given byb Möbius transformations. Let H H f ∈ 2 f (resp. Hpf ) denote the ring class field of conductor f (resp. pf), characterized by × × × the fact that Gal(Hf /K) (resp. Gal(Hpf /K)) is identified with K /K f (resp. × × × O K /K pf ). O b c Proposition 3.1. b d The point z0 gives rise to Qf = [z0, 1] Xns(Hf ). • The point z ∈ H gives rise to Q = [z , 1]∈ X (H ). • 0 ∈ H pf 0 ∈ s pf Proof. Proposition 1.3 tells us that Qf Xns(K) (resp. Qpf Xs(K)) is fixed under × × ∈ ∈ the action of f (resp. pf ). O O c d The main task of this section is to understand the action Gal(Hpf /Hf ) on Qpf . Looking at the p-th component we get

× × Gal(H /H ) K× /K× ( ×) /( × ) . pf f ≃ Of Opf ≃ Of p Opf p Reducing modulo p we further obtain c d ( ×) /( × ) ( /p )×/F× P1(F ). Of p Opf p ≃ Of Of p ≃ p More explicitly, this last isomorphism is given by sending the class of x1 + x2ωf 1 (with xi Zp) to the element [x1 : x2] P (Fp). The element [1 : 0] is the identity ∈ 1 ∈ and multiplication on P (Fp) is given by the rule [x : 1][y : 1] := [xy n : x + y + t], − where, as before, X2 tX +n is characteristic polynomial of ι(ω ) =( a b ) (R ) . − f p c d ∈ ns p 1 Lemma 3.2. The only element of order 2 in P (Fp) is [ a : 1], and in that case the × × − corresponding matrix ι( a + ωf )p belongs to Ms+ Ms . Conversely, if ι(x + −x ω ) M , then [x : x\] [ a : 1], [1 : 0] P1(F ). 1 2 f p ∈ s+ 1 2 ∈{ − } ⊂ p 1 Proof. An element [x : 1] has order 2 in P (Fp) if and only if 2x t mod p. a b ≡ − Because ( c d ) (Rns)p has trace t, then t = a + d 2a mod p, as we wanted. By the same argument,∈ ι( a + ω ) has both its diagonal≡ elements congruent to 0 − f p AN UNEXPECTED TRACE RELATION OF CM POINTS 9 modulo p (and the anti-diagonal elements not congruent to 0) so it lies in M × M ×. s+ \ s Conversely, suppose that ι(x + x ω ) = x1+x2a x2b M , with x , x Z . If 1 2 f p x2c x1+x2d ∈ s+ 1 2 ∈ p x 0 mod p, then we obtain the point [1 : 0]. On the other hand, if x 0 mod p, 2 ≡ 2 6≡ as b, c are not divisible by p this implies that both x1 + x2a and x1 + x2d are divisible by p, and we get the point [ a : 1], as we wanted to prove. −

The local Atkin-Lehner involution at p of Xs is given by right multiplication by × × 0 1 any element in (R ) (R ) . In particular we can take the element ( − ) . s+ p \ s p 1 0 p Hpf Proposition 3.3. If wp(E)= 1, then Tr Φs(Qpf )=0. − Hf Proof. Let b0 be the element of Gal(Hpf /Hf ) corresponding to a + ωf . We can group the elements of Gal(H /H ) into pairs of the form b, bb −. Consider pf f { 0}

bb0 Qpf = [z0, ˆι(b)ˆι(b0)]. × × By Lemma 3.2 the element ˆι(b0) belongs to Rs+ Rs and thus it can be written × 0 1 \ as a product ( −1 0 )p rˆs, wherer ˆs Rs . Therefore we get, in Xs, · ∈ d c bb0 0 1 0 1 b Qpf = [z0, ˆι(b)( − ) crˆs] = [z0, ˆι(b)( − ) ]= wpQpf . 1 0 p · 1 0 p As w (E)= 1, p − Φ (Qbb0 ) = Φ (w (E)Q b)= Φ (Q b), s pf s p pf − s pf and the result follows. In view of the above proposition, from now on we will study the case where w (E) = +1. Consider the double coset operator Γ 1 Γ . It gives rise to p s+ · · ns+ a correspondence between the Shimura curves Xns+ and Xs+. The following result is the heart of this article.

Proposition 3.4. The following identity holds on Div(Xs+):

σ 2(Γs+ 1 Γns+)Qf = Qpf . · · ∈Gal σ X(Hpf /Hf ) × Proof. Take a set of representatives bi of Gal(Hpf /Hf ) in f . The element × { } O × ri = ι(bi) belongs to ι( f ) and, therefore, it also belongs to Rns by Proposition O × c 1.3. We want to write r as a product γ r where r R and γ Γ . In order to i i i i ∈ s+ i ∈ ns+ do that,b we need to modifyb c the component at p. Suppose that (dri)p has determinant × d Zp . d ∈ 10 DANIEL KOHEN

−1 2 µ 0 If d = µ , consider (r ) −1 SL (F ). By strong approximation, • i p 0 µ ∈ 2 p −1 1 µ 0 we take a lifting of this element to an element γ R . Since −1 i ∈ 0 µ ∈ Cns+ Cs+, we get the decomposition ∩ r = γ ((γ )−1r ), i i · i i that has the desired properties. −1 2 0 µ If d = εµ , consider (r ) −1 SL (F ) and do the same as the • i p −εµ 0 ∈ 2 p 0 µ−1 previous case, using that −1 C C . It is worth noticing that −εµ 0 ∈ ns+ ∩ s+ this case shows the necessity of working with the normalizers, in concordance with Proposition 3.3. By construction we get

b b Q i = [z , 1] i = [z , r ] = [z ,γ r ] = [γ −1 z , 1] X (H ), pf 0 0 i 0 i i i · 0 ∈ s+ pf −1 which corresponds to the point γi z0 in the upper half plane of the model ∗ · Γs+ = Xs+. Note\H that the correspondence Γ 1 Γ is of degree s+ · · ns+ 2(p + 1) (p + 1) [Γ :Γ Γ ] = [C : C C ]= = . ns+ ns+ ∩ s+ ns+ ns+ ∩ s+ 4 2

Since γi Γns+, as we vary over all p + 1 elements of Gal(Hpf /Hf ) we obtain the ∈ −1 −1 −1 corresponding coset Γs+γi Γs+ 1 Γns+. The cosets Γs+γi and Γs+γj are equal if and only if ⊂ · ·

γ −1γ Γ . i j ∈ s+ By the deﬁnition of γi and γj we obtain

−1 −1 −1 ri rj = ri γi γjrj. × Since r ,r R we know that i j ∈ s+ × −1 −1 d γi γj Γs+ ri rj Rs+ . ∈ ⇐⇒ ∈ × Using the second part of Lemma 3.2 we see that r −1r R if and only if the id j s+ element r −1r corresponds to either [1 : 0] or [ a : 1] in P1∈(F ). i j − p Consequently, as we vary i, we run through every elementd of (Γs+ 1 Γns+)Qf exactly twice, as we wanted to prove. · · AN UNEXPECTED TRACE RELATION OF CM POINTS 11

4. Chen’s explicit isogeny

Chen [Che00, Theorem 2] proved that an explicit isogeny between Jns+(p) and − J p new(p) is given by the double coset operator Γ (p) 1 Γ (p). In fact, the s+ s+ · · ns+ nature of the proof shows that it extends to an isogeny Γs+ 1 Γns+ between Jns+ p−new · · and Js+ .

Proposition 4.1. The map Γs+ 1 Γns+ is Hecke equivariant for the good Hecke operators. · · Proof. Let ℓ be a prime that does not divide N. Since the reduced norm gives a surjective map R× Z× we choose β R× of reduced norm ℓ. We perform the s → ℓ ∈ s same construction as in Proposition 3.4 but for βℓ instead of ri and we obtain the −1 same matrix γ (thereb denotedb by γi). This matrixb has the property that αℓ := γ βℓ × × belongs to Rs+ Rns+ and has reduced norm equal to ℓ. Thus the ℓ-th Hecke operator in the split side∩ (resp. non-split side) is given by Γ α Γ (resp. Γ α Γ ). s+ · ℓ · s+ ns+ · ℓ · ns+ Using [Shi71,b Lemmab 3.29 (4)] it is clear that

(Γ 1 Γ ) (Γ α Γ )=Γ α Γ = (Γ α Γ ) (Γ 1 Γ ), s+ · · ns+ · ns+ · ℓ · ns+ s+ · ℓ · ns+ s+ · ℓ · s+ · s+ · · ns+ and thus Γs+ 1 Γns+ commutes with the good Hecke operators as desired. · · Remark 4.2. This proof is also in concordance with Proposition 3.3 and with the fact that the natural map Γs 1 Γns does not always give an isogeny between Jns and p−new · · Js . We can only prove that it commutes with the Hecke operators Tℓ with ℓ a square modulo p, and given a newform f of level divisible by p2, the space of forms with the same eigenvalues as f for these Hecke operators has (usually) dimension 2. As the correspondence Γ = 2(Γ 1 Γ ) is Hecke equivariant and of degree p+1 s+ · · ns+ it sends the Hodge class ξns+ to (p + 1)ξs+. Therefore, we can define Γ p−new Jns+ Js+ Φns+ = Φs+ Γ πE,ns+, e Φs+ ◦ ∈ Φns+ e E. Theorem 4.3. Let E/Q be an elliptic curve of conductor N = p2M where p is an odd prime not dividing M and such that wp(E) = +1. Let K be an imaginary quadratic field such that p is inert in K, the sign of E/K is 1 and such that if q is prime with q2 M, then q is unramified in K. Let f be relatively− prime to N | and consider the points Ppf = Φs+(Qpf ) E(Hpf ), Pf = Φns+(Qf ) E(Hf ) and Hpf ∈ ∈ P˜f = Φns+(Qf ) E(Hf ). Then, Tr Ppf = P˜f E(Hf ) and P˜f is non-torsion if ∈ Hf ∈ and only if Pf is non-torsion. e 12 DANIEL KOHEN

Hpf ˜ Proof. First, note that TrHf Ppf = Pf follows immediately from Proposition 3.4.

The second statement follows from the fact that the elements Φns+, Φns+ lie in the 1-dimensional space π = Hom(J , E) Z Q and Φ is non-zero since it is the E,ns+ ns+ ⊗ ns+ composition of the non-trivial Φs+ and the isogeny Γ. e ′ e Remark 4.4. For f = 1, P1 is non-torsion if L (E/K, 1) = 0 by Zhang’s generalization of the Gross-Zagier formula [Zha01, Theorem 1.2.1]. 6

References [BD96] M. Bertolini and H. Darmon. Heegner points on Mumford-Tate curves. Invent. Math., 126(3):413–456, 1996. [CCL18] Li Cai, Yihua Chen, and Yu Liu. Heegner points on modular curves. Trans. Amer. Math. Soc., 370(5):3721–3743, 2018. [Che98] Imin Chen. The Jacobians of non-split Cartan modular curves. Proc. London Math. Soc. (3), 77(1):1–38, 1998. [Che00] Imin Chen. On relations between Jacobians of certain modular curves. J. Algebra, 231(1):414–448, 2000. [CS19] Imin Chen and Parinaz Salari Sharif. An explicit correspondence of modular curves. Journal of Number Theory, 200:185–204, 2019. [CST14] Li Cai, Jie Shu, and Ye Tian. Explicit Gross-Zagier and Waldspurger formulae. Algebra Number Theory, 8(10):2523–2572, 2014. [CV07] Christophe Cornut and Vinayak Vatsal. Nontriviality of Rankin-Selberg L-functions and CM points. In L-functions and Galois representations, volume 320 of London Math. Soc. Lecture Note Ser., pages 121–186. Cambridge Univ. Press, Cambridge, 2007. [Dar04] Henri Darmon. Rational points on modular elliptic curves, volume 101 of CBMS Re- gional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Prov- idence, RI, 2004. [dSE00] Bart de Smit and Bas Edixhoven. Sur un r´esultat d’Imin Chen. Math. Res. Lett., 7(2- 3):147–153, 2000. [GP91] Benedict H. Gross and Dipendra Prasad. Test vectors for linear forms. Math. Ann., 291(2):343–355, 1991. [Gro84] Benedict H. Gross. Heegner points on X0(N). In Modular forms (Durham, 1983), Ellis Horwood Ser. Math. Appl.: Statist. Oper. Res., pages 87–105. Horwood, Chichester, 1984. [Gro88] Benedict H. Gross. Local orders, root numbers, and modular curves. Amer. J. Math., 110(6):1153–1182, 1988. [Gro91] Benedict H. Gross. Kolyvagin’s work on modular elliptic curves. In L-functions and arithmetic (Durham, 1989), volume 153 of London Math. Soc. Lecture Note Ser., pages 235–256. Cambridge Univ. Press, Cambridge, 1991. [Gro18] Benedict H. Gross. On Hecke’s decomposition of the regular diﬀerentials on the modular curve of prime level. Res. Math. Sci., 5(1):Paper No. 1, 19, 2018. [GZ86] Benedict H. Gross and Don B. Zagier. Heegner points and derivatives of L-series. Invent. Math., 84(2):225–320, 1986. AN UNEXPECTED TRACE RELATION OF CM POINTS 13

[KP16] Daniel Kohen and Ariel Pacetti. Heegner points on Cartan non-split curves. Canad. J. Math., 68(2):422–444, 2016. [KP18] Daniel Kohen and Ariel Pacetti. On Heegner points for primes of additive reduction ramifying in the base field. Trans. Amer. Math. Soc., 370(2):911–926, 2018. [LRdVP18] Matteo Longo, V´ıctor Rotger, and Carlos de Vera-Piquero. Heegner points on Hijikata– Pizer–Shemanske curves and the Birch and Swinnerton-Dyer conjecture. Publ. Mat., 62(2):355–396, 2018. [Nek07] Jan Nekováˇr. The Euler system method for CM points on Shimura curves. In L- functions and Galois representations, volume 320 of London Math. Soc. Lecture Note Ser., pages 471–547. Cambridge Univ. Press, Cambridge, 2007. [RW18] Marusia Rebolledo and Christian Wuthrich. A moduli interpretation for the non-split Cartan modular curve. Glasg. Math. J., 60(2):411–434, 2018. [Shi71] Goro Shimura. Introduction to the arithmetic theory of automorphic functions. Pub- lications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo; Princeton University Press, Princeton, N.J., 1971. KanôMemorial Lectures, No. 1. [Vig80] Marie-France Vignéras. Arithmétique des algèbres de quaternions, volume 800 of Lecture Notes in Mathematics. Springer, Berlin, 1980. [Voi18] John Voight. Quaternion algebras. https://math.dartmouth.edu/ jvoight/quat-book.pdf, 2018. [YZZ13] Xinyi Yuan, Shou-Wu Zhang, and Wei Zhang. The Gross-Zagier formula on Shimura curves, volume 184 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 2013. [Zha01] Shou-Wu Zhang. Gross-Zagier formula for GL2. Asian J. Math., 5(2):183–290, 2001.

Universitat¨ Duisburg-Essen, Fakultat¨ fur¨ Mathematik E-mail address: [email protected]