TORSION POINTS AND GALOIS REPRESENTATIONS ON CM ELLIPTIC CURVES

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1 TORSION POINTS AND GALOIS REPRESENTATIONS ON CM ELLIPTIC CURVES ABBEY BOURDON AND PETE L. CLARK Abstract. We prove severa resuts on torsion points and Gaois representations for compex mutipication (CM eiptic curves over a number fied containing the CM fied. The first, a cose reative of a resut of Stevenhagen [St01], computes the Weber function fied of the N-torsion subgroup of a CM eiptic curve over an arbitrary order, generaizing the cassica Main Theorem of Compex Mutipication. The second determines the degrees in which such an eiptic curve has a rationa point of order N. The third bounds the size of the torsion subgroup of an eiptic curve with CM by a nonmaxima order in terms of the torsion subgroup of an eiptic curve with CM by the maxima order. We give severa appications. Contents 1. Introduction 1.1. Overview 1.. Extending the First Main Theorem of Compex Mutipication 1.3. Appications of Theorem The Isogeny Torsion Theorem Acknowedgments 5. Preiminaries 5.1. Foundations 5.. Torsion Kernes 6.3. On Weber Functions 8 3. Proof of the Isogeny Torsion Theorem 8 4. The Projective Torsion Point Fied 9 5. Proof of Theorem 1.1 and Its Coroaries An Equaity of Cass Fieds Proof of Theorem Proof of Coroaries 1., 1.3, 1.4 and Appications SPY Divisibiities A Theorem of Franz The Fied of Modui of a Point of Prime Order Sharpness in the Isogeny Torsion Theorem Minima and Maxima Cartan Orbits Torsion over K(j: Part I Isogenies over K(j: Part I 1 7. The Torsion Degree Theorem 7.1. Statement and Preiminary Reduction 7.. Generaities The Case f The Case f Torsion over K(j: Part II Isogenies over K(j: Part II 7 References 8 1

2 ABBEY BOURDON AND PETE L. CLARK 1. Introduction 1.1. Overview. Let F be a fied of characteristic 0, and et E /F be an eiptic curve. We say E has compex mutipication (CM if the endomorphism agebra End 0 E = End(E /F Z Q is stricty arger than Q, in which case it is necessariy an imaginary quadratic fied K and O := End(E /F is a Z-order in K. This paper continues a program of study of torsion points and Gaois representations on CM eiptic curves defined over number fieds. Contributions have been made by Oson [O74], Siverberg [Si88], [Si9], Parish [Pa89], Aoki [Ao95], [Ao06], Ross [Ro94], Kwon [Kw99], Prasad-Yogananda [PY01], Stevenhagen [St01], Breuer [Br10], Lombardo [Lo15], Lozano-Robedo [LR], Gaudron-Rémond [GR18] and the present authors and our coaborators [CCS13], [CCRS14], [BCS17], [CP15], [BCP17], [BP16]. Á. Lozano-Robedo has informed us that he has aso done work on the image of the adeic Gaois representation in the CM case. Two ong-term goas of this program are on the one hand to competey understand the adeic Gaois representation on any CM eiptic curve defined over a number fied and on the other hand to determine a degrees of CM points on moduar curves associated to congruence subgroups of SL (Z. These two probems are cosey reated. An archetypica exampe is the foowing case of the First Main Theorem of Compex Mutipication (the fu statement is reproduced as Theorem.8: if K is an imaginary quadratic fied and E /K(j(E is an O K -CM eiptic curve, then for a N Z + the fied obtained by adjoining to K(j(E the Weber function of the N-torsion subgroup is K (N, the N-ray cass fied of K. For a N 3, we have (see Lemma.10 [K (N : K(j(E] = #(O K/NO K. #O K This impies that the mod N Gaois representation on an O K -CM eiptic curve E /K(j(E is as arge as possibe up to twisting, and we wi show there is an O K -CM eiptic curve E /K(j(E such that the mod N Gaois representation surjects onto the mod N Cartan subgroup (O/NO (see Coroary 1.5. This is a sharp version of Serre s Open Image Theorem in the O K -CM case. The corresponding resut on the moduar curve side is: the fied of modui of an O K -CM point on X(N /K(ζN is K (N. The above resuts restrict to the case of the maxima order O K, as does most of the cassica theory. Here we work in the context of an arbitrary order O, of conductor f, in an imaginary quadratic fied K. (This notation is fixed throughout the remainder of the introduction. Let F K be a number fied, and et E /F be an O-CM eiptic curve. For any positive integer N, we define the reduced mod N Cartan subgroup to be the quotient of C N (O = (O/NO by the image of O under the natura map q N : O O/NO. That is, C N (O = C N (O/q N (O. (The map q N : O (O/NO is injective when N 3; when N = its kerne is {±1}. We define the reduced Gaois representation to be the foowing composite homomorphism: ρ N : g F ρ N CN (O C N (O. 1.. Extending the First Main Theorem of Compex Mutipication. The reduced Gaois representation depends ony on j(e; it is independent of the F -rationa mode. Its fixed fied is the fied obtained by adjoining to K(j(E the vaues of a Weber function on E evauated at the points of order N. In particuar, it is a subextension of the (Nf-ray cass fied of K. The foowing resut determines it expicity and thus gives a generaization of the First Main Theorem of Compex Mutipication. Theorem 1.1. Let E /C be an O-CM eiptic curve, and et N Z +. Then the Weber function fied K(j(E(h(E[N] is the compositum of the N-ray cass fied K (N of K with the Nf-ring cass fied K(Nf of K. Moreover, we have [K(j(E(h(E[N] : K(j(E] = #C N (O.

3 TORSION POINTS AND GALOIS REPRESENTATIONS ON CM ELLIPTIC CURVES 3 Coroary 1.. For any O-CM eiptic curve E /K(j(E, the reduced mod N Gaois representation ρ N : g K(j(E C N (O is surjective. Coroary 1.3. (Uniform Open Image Theorem For a number fieds F K and a O-CM eiptic curves E /F we have [C N (O : ρ N (g F ] #O [F : K(j(E] 6[F : K(j(E]. Coroary 1.4. Let N Z +. There is a number fied F K and an O-CM eiptic curve E /F such that E[N] = E[N](F and [F : K(j(E] = #C N (O. Coroary 1.5. For a N Z +, there is an O-CM eiptic curve E /K(f such that ρ E,N (g K(f = C N (O. Theorem 1.1 is cosey reated to work of P. Stevenhagen [St01], as we now expain. Let O = O Z Ẑ be the profinite competion of O. Cass fied theory gives a canonica isomorphism (1 Ψ : Aut(K ab /K(f O /O [St01, (3.]. For N Z +, we have a quotient map from O /O to C N (O = (O/NO /q N (O, and thus via Ψ there is a finite Gaois extension H N,O /K(f, the N-ray cass fied of O, such that Aut(H N,O /K(f = C N (O. Stevenhagen uses Shimura s reciprocity aw to show that for any O-CM eiptic curve E, we have K(f(h(E[N] = H N,O. From this, Coroary 1. foows. (Indeed it foows that Ψ may be identified with the adeic reduced Gaois representation on any O-CM eiptic curve E /K(f. To deduce Theorem 1.1 one must show ( H N,O = K (N K(Nf. Thus Theorem 1.1 is reay a variant of work of Stevenhagen, though (rather curiousy our particuar neocassica formuation seems to be new. Stevenhagen s approach is cean and eegant. On the other hand, Gaois representations do not expicity appear in his work. Here we wi prove Theorem 1.1 by a different approach, proceeding via the observation that the two fieds K (N and K(Nf can be identified as subfieds of the Weber function fied K(f(h(E[N] the former by a connection to the O K -CM case and the atter by its reation to the projective torsion point fied, foowing work of Parish [Pa89]. Thus we get (3 K(f(h(E[N] K (N K(Nf. The formaism of reduced Gaois representations gives Moreover, combining ( and (3 we get [K(j(E(h(E[N] : K(j(E] #C N (O. [K (N K(Nf : K(j(E] = #C N (O, which gives Theorem 1.1. This proof is mosty independent of Stevenhagen s; the ony common ingredient is the purey cass fied theoretic ( Appications of Theorem 1.1. It foows from Coroary 1.3 that if E /F is an O-CM eiptic curve and F K is a number fied, the index of the image of the adeic Gaois representation on E in the Cartan subgroup C = O divides #O [F : K(j(E]. If F does not contain K, then the image of the adeic Gaois representation has index dividing [F : Q(j(E]#O in a subgroup of GL (Ẑ that contains the adeic Cartan C with index. This is cose to being a compete description of the adeic Gaois representation on any CM eiptic curve defined over a number fied. It fas short in two aspects: first, for a fixed N 3, to get a mod N Gaois representation with index #O in the mod N Cartan, our construction takes F to be a proper extension of the minima possibe ground fied K(j(E. Second, it shows that at any finite eve N the index of the mod N representation in the

4 4 ABBEY BOURDON AND PETE L. CLARK Cartan can be any divisor of #O but does not address whether this can happen for the adeic Gaois representation. These do not impact the second aspect of our program, which studies degrees of eve N structures of CM eiptic curves and fieds of modui of CM points on moduar curves and studies a pairs (E, L N /F for a eve N structure L N over a number fied F K(j(E, not just pairs in which the underying eiptic curve E arises from base extension of an eiptic curve E /K(j(E. We propose to use the above resuts to determine a degrees of CM points on moduar curves. To do so requires further work of a more agebraic nature: an anaysis of orbits of the mod N Cartan subgroup C N (O on eve N structures. To understand the reevance of this, et E /K(j(E be an O-CM eiptic curve. If P E[tors] is a point of order N, the fied of modui of the point (E, P on X 1 (N depends ony on the O orbit P of P. Since the reduced Gaois representation is surjective, the degree of this fied over K(j(E may be computed by determining the size of the orbit of C N (O on P. We give an anaysis of Cartan orbits on O/NO in 6 and 7. The agebra is much simper when O is maxima, and in this case our anaysis is compete. When O is nonmaxima we give substantia, but not fu, information on the structure of the Cartan orbits, enough to yied the foowing resut. Theorem 1.6. Let N Z. There is an integer T (O, N, expicity computed in 7, such that: (i if F K is a number fied and E /F is an O-CM eiptic curve with an F -rationa point of order N, then T (O, N [F : K(j(E], and (ii there is a number fied F K and an O-CM eiptic curve E /F such that [F : K(j(E] = T (O, N and E(F contains a point of order N. Theorem 1.6 shoud be compared to Theorem 6., a refinement of bounds of Siverberg [Si88], [Si9], Prasad-Yogananda [PY01] and Gaudron-Rémond [GR18]. Theorem 6. aso gives a divisibiity on [F : K(j(E] imposed by the existence of an F -rationa point of order N: in the current notation, Theorem 6. asserts ϕ(n #O T (O, N. This bound is homogeneous in the sense that it is a singe bound that hods in a cases. Theorem 1.6 gives the optima divisibiity in a cases. We give two other appications of our Cartan orbit anaysis: the determination of a possibe torsion subgroups of a K-CM eiptic curve E /K(j(E ( 6.6 and the set of N Z + for which there is a K(j(E-rationa cycic N-isogeny (Theorem S. Kwon gave a cassification of degrees of cycic isogenies rationa over Q(j(E [Kw99]. Our Theorem 6.18 is the anaogue over K(j(E The Isogeny Torsion Theorem. Athough we seek resuts which treat eiptic curves with CM by a nonmaxima order on an equa footing with the O K -CM case, in most cases the proofs use change of order functoriaities. Let E be an O-CM eiptic curve defined over a number fied F, and et f be a positive integer that divides f. Then by [BP16, Prop..] there is an eiptic curve (E f /F such that End E f is the order of conductor f in K and an F -rationa isogeny ι f : E E f that is cycic of degree f f. So the induced g F -modue map E[N] E f [N] is an isomorphism iff gcd( f f, N = 1; otherwise there is a nontrivia kerne. But nevertheess there are reations between the mod N Gaois representations on E and E f. Here is the ast main resut of this paper: Theorem 1.7. (Isogeny Torsion Theorem Let O be an order in an imaginary quadratic fied K, of conductor f, and et f be a positive integer dividing f. Let F K be a number fied, and et E /F be an O-CM eiptic curve. Let ι f : E E f be the F -rationa isogeny to an eiptic curve E f with CM by the order in K of conductor f, as described above. Then we have #E(F [tors] #E f (F [tors]. In particuar, taking f = 1, we see that #E(F [tors] is bounded by #E 1 (F [tors], where (E 1 /F is an eiptic curve with CM by the maxima order in K. We give exampes where the exponent of E f (F [tors] is stricty smaer than that of E(F [tors], showing in genera we cannot view E(F [tors] as a subgroup of E f (F [tors], and we prove that #E f #E(F [tors] can be arbitrariy arge (see Propositions 6.8 and 6.9. Moreover, the statement is fase if we do not require F K. Despite the fact that this (F [tors]

5 TORSION POINTS AND GALOIS REPRESENTATIONS ON CM ELLIPTIC CURVES 5 reationship is not as strong as one might hope, Theorem 1.7 has appications to determining fieds of modui of partia eve N structures ( 6., 6.3. A paper of R. Ross [Ro94] contains a resut reated to Theorem 1.7: in the notation of Theorem 1.7, Ross s assertion impies that the groups E(F [tors] and E f (F [tors] have the same exponent. This is fase: Proposition 6.8 gives counterexampes. Sti, Ross s work ed us to Theorem Acknowedgments. We are deepy indebted to R. Broker for making us aware of Stevenhagen s important work [St01]. Severa anonymous referees provided crucia critica feedback; in one case, this ed us to the understanding that our Theorem 1.1 is essentiay a variant of Stevenhagen s work.. Preiminaries.1. Foundations. We begin by setting some terminoogy for orders in imaginary quadratic fieds. Let K be an imaginary quadratic fied and O a Z-order in K. We put f = [O K : O], the conductor of O. Then O = Z + fo K, (O = f K. Conversey, for fixed K and f Z + there is a unique order O(f in K of conductor f. Thus an imaginary quadratic order is determined by its discriminant, a negative integer which is 0 or 1 moduo 4. Conversey, for any negative integer which is 0 or 1 moduo 4, we put τ = +, and then Z[τ ] is an order in K of discriminant. Throughout this paper we wi use the foowing terminoogica convention: by an order O we aways mean a Z-order O in an imaginary quadratic fied, which is determined as the fraction fied of O and denoted by K. We may specify an order O by giving its discriminant, which aso determines K. If K is aready given, then we specify an order O in K by giving the conductor f. For any O-CM eiptic curve E we have K(j(E = K(f, the ring cass fied of K of conductor f ([Co89, Thm. 11.1]. We may thus determine [K(j(E : K] via the foowing formua: Theorem.1. For N Z +, et K(N denote the N-ring cass fied of K. Then K(1 = K (1 is the Hibert cass fied of K, and for a N we have [K(N : K (1 ] = N Å Å ã ã K 1 1. w K p p p N Proof. See e.g. [Co89, Cor. 7.4]. For number fied F, a positive integer N, and E /F an eiptic curve, we denote by ρ N the homomorphism g F Aut E[N] = GL (Z/NZ, the moduo N Gaois representation. If E /F has CM by the order O in K, then E[N] = O O/NO (see [Pa89, Lemma 1], generaized in Lemma.4 beow, and provided F K we have ρ N : g F Aut O E[N] = GL 1 (O/NO = (O/NO. In other words, the image of the mod N Gaois representation ands in the mod N Cartan subgroup C N (O = (O/NO. Lemma.. Let O be an order of discriminant, and et N = p ar 1 par r Z +. a We have C N (O = r i=1 C p a i (O (canonica isomorphism. i b We have #C N (O = N Ä Ä ä ä Ä p N 1 1 p p 1 1 pä.

6 6 ABBEY BOURDON AND PETE L. CLARK Proof. a It suffices to tensor the Chinese Remainder Theorem isomorphism Z/NZ = r i=1 Z/pai i Z with the Z-modue O and pass to the unit groups. b By [CCS13], for any prime number p we have #C p (O = p Å1 Å p ã ã Å ã. p p The natura map C p a(o C p (O is surjective with kerne of size p a [CP15, p. 3]. Together with part a this shows that if N = p a1 1 par r then r Å Å ãã #C N (O = p ai i (p i 1 p i = N Å Å ã ã Å ã. p i=1 i p p p p N.. Torsion Kernes. Let E /C be an O-CM eiptic curve. For a nonzero idea I of O, we define the I-torsion kerne E[I] = {P E α I, αp = 0}. There is an invertibe idea Λ O such that If we put then we have (immediatey that Let I = #O/I. E = C/Λ. (Λ : I = {x C xi Λ} = {x K xi Λ} E[I] = {x C/Λ xi Λ} = (Λ : I/Λ. Lemma.3. Let I, J O be nonzero ideas and E /C be an O-CM eiptic curve. a If I J, then E[J] E[I]. b We have E[I] E[ I ]. In particuar #E[I] I. Proof. a This is immediate from the definition. b By Lagrange s Theorem, every eement of O/I is kied by I, so I I O I. Appy part a. Lemma.4. If I is an invertibe O-idea, then In particuar #E[I] = I = #O/I. E[I] = I 1 Λ/Λ = O O/I. Proof. An idea I is invertibe iff there is an O-submodue I 1 of K such that II 1 = O. If so, then for x K we have xi Λ xii 1 = xo I 1 Λ x I 1 Λ, giving E[I] = I 1 Λ/Λ. Because Λ is a ocay free O-modue, for a p Spec O we have Λ p = Op and thus (I 1 Λ/Λ p = (I 1 /O p = (O/Ip. Thus I 1 Λ/Λ is ocay free of rank 1 as an O/I-modue. But the ring O/I is semioca, hence has trivia Picard group: any ocay free rank 1 O/I-modue is isomorphic to O/I [CA, Cor ]. Lemma.5. Let R be a Dedekind domain, and et M be a cycic torsion R-modue, and et N M be an R-submodue. Then: a N is aso a cycic R-modue. b We have N = R/ ann N. Proof. Let I = ann M. Since M is a finitey generated torsion modue over a domain, we have I 0 and M = R/I. Thus N = I /I for some idea I I. The ring R/I is principa Artinian [CA, Thm. 0.11], so the idea I /I of R/I is principa. Thus N is a cycic, torsion R-modue, so N = R/ ann N.

7 TORSION POINTS AND GALOIS REPRESENTATIONS ON CM ELLIPTIC CURVES 7 Theorem.6. Let E /C be an O K -CM eiptic curve, and et M E(C be a finite O K -submodue. Then M = E[ann M] = O O/ ann M and thus #M = ann M. Proof. That M E[ann M] is a tautoogy. Because O = O K every nonzero O-idea is invertibe, so by Lemma.4 we have #E[ann M] = ann M. On the other hand, et t = #M. Then M E[t] = OK O K /to K, a finite cycic O K -modue. By Lemma.5 we have M = O K / ann M so #M = ann M. Thus M = E[ann M], hence Lemma.4 gives M = O/ ann M and #M = ann M. Remark.7. Let O be a nonmaxima order in K. There is nonzero prime idea p of O such that the oca ring O p is not a DVR. If p Z = (, then O/p = Z/Z. Since every idea of O can be generated by two eements, we have dim O/p p/p =. Thus #O/p = 3 and ( 3 p. It foows that in the quotient ring O/ 3 O the maxima idea p + 3 O is not principa. Let E /C be an O-CM eiptic curve, so E[ 3 ] = O O/ 3 O. So the O-submodue M = pe[ 3 ] of E[ 3 ] is not cycic and thus not isomorphic to O/ ann M. Now we reca an important cassica resut. Theorem.8. (First Main Theorem of Compex Mutipication Let E /C be an O K -CM eiptic curve, and et I be a nonzero idea of O K. Let h : E P 1 be a Weber function. Then: Proof. See e.g. [Si94, Thm. II.5.6]. K (1 (h(e[i] = K I. Combining Theorems.6 and.8, we get the cass-fied theoretic containment corresponding to any finite O K -submodue of E(F, for any O K -CM eiptic curve E defined over a number fied F K. Theorem.8 impies that whenever E is an O K -CM eiptic curve, K (1 (h(e[n] = K (N. In the case of CM by an arbitrary order in K, a containment has previousy been estabished. Theorem.9. [BCS17, Thm. F K. Then we have (4 F (h(e[n] K (N. 3.16] Let E be a K-CM eiptic curve defined over a number fied For convenience, we record here the formuas for [K I : K (1 ]. Here, ϕ denotes Euer s totient function and ϕ K (I the natura generaization for a nonzero idea I of O K. That is, ϕ K (I = #(O K /I = I Å 1 1 ã, p p I where I = #O K /I. Lemma.10. Let I be a nonzero idea of K, and et K I be the I-ray cass fied. We put U(K = O K and U I (K = {x U(K x 1 I}. a We have [K I : K (1 ϕ K (I ] = [U(K : U I (K]. b If K Q( 1, Q( 3, then c If K = Q( 1, then [K I : K (1 ϕk (I I ( ] = ϕ K (I I (. ϕ K (I I (1 + i [K I : K (1 ϕ ] = K (I I (1 + i and I (. ϕ K (I 4 I (

8 8 ABBEY BOURDON AND PETE L. CLARK d If K = Q( 3, then 1 I = (1 ϕ K (I [K I : K (1 ] = I (1 and I (ζ 3 1 ϕ K (I. 3 I = ( ϕ K (I 6 otherwise Proof. Parts b-d can be deduced from a, which appears as [Co00, Cor. 3..4]..3. On Weber Functions. Theorem.11. (Weber Function Principe Let N Z, et O be the order of conductor f in K, and et F = K(f. For an O-CM eiptic curve E /F, fix an embedding F C such that j(e = j(c/o. Define W (N, O = K(f(h(E[N]. #O N 3 a W (N, O is a subfied of F (E[N] and [F (E[N] : W (N, O] #O N =. b There is an eiptic curve E /F such that #O N 3 [F (E[N] : W (N, O] = #O N =. c As we range over a eiptic curves E /F with j(e = j(c/o, we have F (E[N] = W (N, O. E #O N 3 Proof. a Let w = #O N =. Let µ w be the image of O C N (O, a cycic group of order w. The fied F (E[N]/F is Gaois with Gaois group ρ N (g F C N (O. Since h(p = h(q for points P, Q on E if and ony if there is ξ O such that ξ(p = Q (e.g. [La87, Thm. I.7], it foows that Thus W (N, O = F (E[N] ρ N (g F µ w. [F (E[N] : W (N, O] w. b, c If E /F, E /F with j(e = j(e, then K(f(h(E[N] = K(f(h(E [N] by the mode independence of the Weber function. So W (N, O E F (E[N]. To see that equaity hods, et E /F have j(e = j(c/o. Let p be a prime of O F that is unramified in F = F (E[N]. By weak approximation, there is π p \ p. Put L = F (π 1 w, and et χ : g F µ w be a character with spitting fied F ker χ = L. Then L/F is totay ramified over p, so F and L are ineary disjoint over F. It foows that Thus ρ N,E χ(g F = (ρ N,E/F χ(g F = χ(g F = µ w. w = [F (E χ [N] : F (E[N] F (E χ [N]] [F (E χ [N] : W (N, O] w, so F (E χ [N] has degree w over W (N, O = F (E[N] F (E χ [N]. 3. Proof of the Isogeny Torsion Theorem For a positive integer d, we wi write O(d for the order in K of conductor d. There is a fied embedding F C such that E /C = C/O(f, (Ef /C = C/O(f and (ι f /C is the quotient map C/O(f C/O(f. Put τ K = K+ K, so O(f = Z[fτ K ] and O(f = Z[f τ K ]. For N Z +, et e 1,f (N := 1 N + O(f, e,f(n := fτ K N + O(f,

9 TORSION POINTS AND GALOIS REPRESENTATIONS ON CM ELLIPTIC CURVES 9 e 1,f (N := 1 N + O(f, e,f (N := f τ K N + O(f. Then ker(e[n] ι f E f [N] is cycic of order gcd(n, f N f, generated by e gcd(n, f f,f(n, and ι f (e 1,f (N = e 1,f (N. For finite commutative groups T 1 and T, we have #T 1 #T if and ony if #T 1 [ ] #T [ ] for a prime numbers. So it suffices to show: for a prime numbers, we have #E(F [ ] #E f (F [ ]. Write f = c1 f with gcd(f, = 1 and f = c f with gcd(f, = 1. Then we have #E(F [ ] = #E c 1 (F [ ], #E f (F [ ] = #E c (F [ ], so we may assume that f = c1 and f = c. Indeed, it is enough to treat the case c = c 1 1, since repeated appication of this case yieds the genera case. So suppose f = c for some c Z + and f = c 1. By (e.g. the Morde-Wei Theorem, there are integers 0 a b such that E(F [ ] = Z/ a Z Z/ b Z. In particuar, Q := 1 + O(f E(F and Q := ι a f (Q = 1 + O(f generates E a f [ a ] as an O(f - modue, so E f [ a ] = E f (F [ a ]. If a = b, it foows that #E(F [ ] #E f (F [ ], so we may assume b > a. Since ker(e[ ] ι f E f [ ] has order, we have Z/ a Z Z/ b 1 Z E f (F [ ]. Thus it suffices to show that E f (F has either a point of order b or has fu a+1 -torsion. Let P = E(F be a point of order b, and write P = αe 1,f ( b + βe,f ( b with α, β Z/ b Z. If α then ι f (P = αe 1,f ( b + βe,f ( b has order b and we are done, so we may assume that α, in which case β. With respect to the basis e 1,f ( b, e,f ( b of E[ b ], the image of the mod b Gaois representation on E consists of matrices of the form ñ ô (5 a b c K K 4 b a + b c with a, b Z/ b Z. K Since E(F has fu a -torsion, we have a 1 (mod a and b 0 (mod a. Thus ßï 1 + ρ a+1(g F a ò A 0 a B 1 + a A, B Z/ Z a+1. A Since b a 1 P = αe 1,f ( a+1 +βe,f ( a+1 is F -rationa, a such matrices in the image of Gaois satisfy ï 1 + a ò ï ò ï ò A 0 α α a B 1 + a =, A β β and thus a αb + β + a Aβ β (mod a+1. Since α, we get a Aβ a αb 0 (mod a+1, ï ò 1 0 and thus A and ρ a+1(g F consists of matrices of the form a for B Z/ B 1 a+1 Z. It foows that for a σ g F, there is B Z/ a+1 Z such that σ(ι f (e 1,f ( a+1 = ι f (e 1,f ( a+1 + B a ι f (e,f ( a+1 = ι f (e 1,f ( a+1 + B a (e,f ( a+1 = ι f (e 1,f ( a+1. Thus e 1,f ( a+1 = ι f (e 1,f ( a+1 E f (F. Since the O(f -submodue generated by e 1,f ( a+1 is E f [ a+1 ], we get Z/ a+1 Z Z/ a+1 Z E f (F, competing the proof of Theorem The Projective Torsion Point Fied Let F be a fied. For a positive integer N not divisibe by the characteristic of F and E /F an eiptic curve, we define the projective moduo N Gaois representation as the composite map Pρ N : g F The projective torsion fied is ρ N Aut E[N] = GL (Z/NZ PGL (Z/NZ := GL (Z/NZ/(Z/NZ. F (PE[N] = F ker Pρ N.

10 10 ABBEY BOURDON AND PETE L. CLARK Thus F (PE[N] is the unique minima fied extension of F on which the image of ρ N consists of scaar matrices. It foows that F (E[N]/F (PE[N] is a Gaois extension with automorphism group a subgroup of (Z/NZ. Observe that the projective Gaois representation and thus the projective torsion fied are unchanged by quadratic twists. If E /F has CM by an order of discriminant = f K 3, 4 and F K, then the projective N-torsion fied is a we-defined abeian extension of K(f. A resut of Parish identifies this projective torsion fied with a suitabe ring cass fied. When = 4 (resp. = 3 we have quartic twists (resp. sextic twists which can change the projective Gaois representation and the projective torsion fied. Theorem 4.1. Let O be an order of discriminant = f K. Let E be an O-CM eiptic curve defined over F = K(f. Let N. a We have F (PE[N] K(Nf. Thus we may put d(e, N = [F (PE[N] : K(Nf]. b If / { 3, 4}, then d(e, N = 1, i.e., F (PE[N] = K(Nf. c If = 4, then d(e, N. d If = 3, then d(e, N 3. Proof. For N Z +, et O(N be the order of conductor N in K. Thus O = O(f. Step 1: We show that F (PE[N] K(Nf in a cases. There is a fied embedding F C such that E /C = C/O. The C-inear map z Nz carries O(f into O(Nf and induces a cycic N-isogeny C/O(f C/O(Nf. Let C be the kerne of this isogeny, viewed as a finite étae subgroup scheme of E /C. Then C has a (unique minima fied of definition F (C F (E[N], hence of finite degree over F. The fied F (PE[N] is precisey the compositum of the minima fieds of definition of a order N cycic subgroup schemes C E /C, so F (C F (PE[N]. Since C is F (PE[N]-rationa, the eiptic curve E/C has a mode over this fied, and thus F (PE[N] K(j(E/C = K(Nf. Step : In view of Step 1, we have F (PE[N] K(Nf K(f = K(j(E, so we have F (PE[N] = K(Nf iff [F (PE[N] : K(f] [K(Nf : K(f]. We have [F (PE[N] : K(f] = #Pρ N (g F #(O/NO /(Z/NZ = N Å Å ã ã 1 1. p p p N Suppose f > 1. Using Theorem.1 to compute [K(Nf : K (1 ] and [K(f : K (1 ] gives [K(Nf : K(f] = [K(Nf : K(1 ] = N Å Å ã ã 1 1 = N Å Å 1 [K(f : K (1 ] p p p p N p N, p f because 1 Ä ä 1 p p = 1 for a p f. Thus d(e, N = 1 in this case. Suppose f = 1, so = K. Then [K(Nf : K(f] = [K(N : K (1 ] = w K N p N Å 1 Å p ã ã 1. p If / { 3, 4} then w K = 1, and again we get d(e, N = 1. If = 4 then w K = 1, so the cacuation shows d(e, N {1, }, and if = 3 then w K = 1 3, so the cacuation shows d(e, N {1, 3}. The foowing resut is an an anaogue of [BCS17, Thm. 5.6] for higher twists. Proposition 4.. (Higher Twisting at the Bottom For M Z +, we denote the mod M cycotomic character by χ M. ã 1 p ã,

11 TORSION POINTS AND GALOIS REPRESENTATIONS ON CM ELLIPTIC CURVES 11 a Let K = Q( 1 and et 5 (mod 8 be a prime number. There is a character Ψ : g K (Z/Z of order 1 4 and an O K -CM eiptic curve E /K such that the mod Gaois representation is ï ò Ψ(σ 0 σ ρ (σ = 0 Ψ 1. (σχ (σ b Let K = Q( 3 and et 7, 31 (mod 36 be a prime number. There is a character Ψ : g K (Z/Z of order 1 6 and an O K -CM eiptic curve E /K such that the mod Gaois representation is ï ò Ψ(σ 0 σ ρ (σ = 0 Ψ 1. (σχ (σ Proof. a Because 1 (mod 4, the Cartan subgroup C (O is spit, and for an O K -CM eiptic curve (E 1 /K, the mod Gaois representation has the form ï ò Ψ1 (σ 0 σ ρ (σ = 0 Ψ 1 1 (σχ (σ for a character Ψ 1 : g ï K (Z/Z ò. Under this isomorphism, the matrix representation of i O K z 0 is a diagona matrix 0 z 1, where z is a primitive 4th root of unity in Z/Z. A genera O K - CM eiptic curve over K is of the form E ψ 1 for a character ψ : g K µ 4 (Z/Z. Let Q 4 ( = (Z/Z /(Z/Z 4. Then the image of z in Q 4 ( has order 4: if not, there is w (Z/Z such that z = w, and then w has order 8 in (Z/Z, contradicting the assumption that 5 (mod 8. Thus the natura map µ 4 Q 4 ( given by i z (mod (Z/Z 4 is an isomorphism; we denote the inverse isomorphism Q 4 ( µ 4 by ι. Now take Ψ 1 1 ψ : g K (Z/Z q Q 4 ( ι µ 4. Let Ψ = ψψ 1. Then the twist E ψ 1 has mod Gaois representation ï Ψ (σ 0 σ ρ (σ = 0 Ψ 1 (σχ (σ The composite Ψ : g K (Z/Z Q 4 ( is trivia, so Ψ (g K has order c 1 4. Thus ( 1 #ρ,e ψ(g K c( 1 = [K ( : K (1 ] = [K ( : K]. 1 4 Because K(E ψ 1 [] K(, we have #ρ,e ψ(g K = ( and c = 1 4. b Since 1 (mod 3, we have a primitive 6th root of unity z in Z/Z. Since 7, 31 (mod 36, we have 4, 9 1, so z has order 6 in Q 6 ( = (Z/Z /(Z/Z 6. Aso ( 1 6 = [K ( : K (1 ]. The argument of part a carries over. Exampe 4.3. a Let K = Q( 1, and et 5 (mod 8. Let E /K be an O K -CM eiptic curve with mod Gaois representation as in Proposition 4.a. Then for a number fied L K, ρ gl has scaar image iff χ Ψ gl is trivia. Since χ : g K (Z/Z has order 1 that is, for a 1 k < 1, χ k 1 and Ψ has order dividing 1 4, the character χ Ψ has order 1. Thus [K(PE[] : K] = 1,whereas [K( : K] = 1. So d(e, =. b Let K = Q( 3, and et 7, 31 (mod 36. Let E /K be an O-CM eiptic curve with mod Gaois representation as in Proposition 4.b. As in part a, we have [K(PE[] : K] = 1 and [K( : K] = 1 3. So d(e, = 3. Remark 4.4. a Parts a and b of Theorem 4.1 are due to Parish [Pa89, Prop. 3]. However, Parish audes to a cacuation of the above sort rather than expicity carrying it out. Since Theorem 4.1 wi pay an important roe in the proof of Theorem 1.1, we have given a compete proof. b In [Pa89, Prop. 3], Parish assumes K Q( 1, Q( 3. Later on [Pa89, p. 63], he caims: If = 4 then F (PE[N] = K(N for a N 3, and ò.

12 1 ABBEY BOURDON AND PETE L. CLARK If = 3 then F (PE[N] = K(N for a N 4. As Exampe 4.3 shows, both caims are fase. Proposition 4.5. Let O be an order of discriminant = f K, and et N Z +. Then there is an O-CM eiptic curve E /K(Nf such that the mod N Gaois representation consists of scaar matrices. Proof. When / { 3, 4}, this is immediate from Theorem 4.1b: in that case, the eiptic curve has a mode defined over K(f. Thus we may assume that { 3, 4}, so f = 1. Let ζ O K be a primitive w K th root of unity. Let O be the order in K of conductor N, et Ẽ/K(N be an O- CM eiptic curve, and et ι : Ẽ E be the canonica K(N-rationa isogeny to an O K -CM eiptic curve E, et ι : E Ẽ be the dua isogeny, and et C be the kerne of ι. Identifying E[N] with N 1 O K /O K C/O K, ι : C/O K C/O is the map z + O K Nz + O, so C is the Z-submodue of C/O K generated by P 1 = 1 N + O K. Because C is stabe under the action of g K(N, this action is given by an isogeny character, say σ(p 1 = Ψ(σP 1. Let P = ζp 1. Then {P 1, P } is a Z/NZ-basis for E[N]. Moreover, for σ g K(N, σp = σζp 1 = ζσp 1 = ζψ(σp 1 = Ψ(σζP 1 = Ψ(σP. It foows that σ g K(N acts on E[N] via the scaar matrix Ψ(σ. 5. Proof of Theorem 1.1 and Its Coroaries 5.1. An Equaity of Cass Fieds. Let O and O be orders in an imaginary quadratic fied K of conductors f and Nf, respectivey. Here we prove (: H N,O = K (N K(Nf. We may assume that N. Via cass fied theory, it suffices to prove an equaity of open subgroups of ÔK /O K. We abbreviate O p := O Z p. Put A := {x O x 1 (mod N} = p N O p p N(1 + NO p, à := AO K, B := Ô = (O p, B := BO K, p C := {x ÔK x 1 (mod N} = K p N(O p (1 + N(O K p, C := CO K. p N Under cass fied theory, the fied H N,O corresponds to à (cf. [St01, p. 9], the fied K(Nf corresponds to B and the fied K (N corresponds to C, so H N,O = K (N K(Nf is equivaent to à = B C. Step 1: We show that A = B C. Writing A p, B p and C p for the components of p of each of these groups, it is enough to show that Case 1: Suppose p N. Then A p = B p C p for a primes p. A p = O p, B p = (O p = A p, C p = (O K p, so C p A p = B p and thus B p C p = A p. Case : Suppose p N. Write O K = Z1 + Zτ K, so O = Z1 + Zfτ K. We have A p = 1 + NO p = 1 + NZ p 1 + NfZ p τ K, B p = (1 + Z p 1 + NfZ p τ K,

13 TORSION POINTS AND GALOIS REPRESENTATIONS ON CM ELLIPTIC CURVES 13 C p = 1 + N(O K p = 1 + NZ p 1 + NZ p τ K, so indeed we have B p C p = A p. It foows that B C = BO K CO K AO K = Ã, so it remains to show that B C Ã. Step : Suppose K < 4, so O K = {±1}. Then B = B, so if z B C, then there is ɛ {±1} such that z B, z B and ɛz C, so ɛz B C = A and thus z Ã. Step 3: Suppose K = Q( 1 and et ζ be a primitive 4th root of unity, so O K = Z1 + Zζ and O K = {1, ζ, ζ, ζ 3 }. Suppose z B C. Then there are i, j {0, 1,, 3}, b B and c C such that z = ζ i b = ζ j c. We have z à iff ζ j z Ã, so we may assume that j = 0. If i is even we may argue as in Step, so assume that i {1, 3}, and thus we have either ζb = c or ζc = b. But we caim that there are no such eements b and c, which wi compete the argument in this case. Indeed, choose a prime p dividing N, and et b p and c p be the components at p. There is a reduction map (O K p O K Z/pZ = Z/pZ1 + Z/pZζ. Under this map, every eement of B p C p ands in Z/pZ1, so b p, c p Z/pZ1 whie ζb p, ζc p Z/pZζ. Thus we cannot have ζb p = c p or ζc p = b p. If K = Q( 3, then we et ζ be a primitive 6th root of unity, so O K = Z1 + Zζ and O K = {1, ζ, ζ, ζ 3, ζ 4, ζ 5 }, and the argument is very simiar: we cannot have ±ζb p = c p or ±b p = ζc p. 5.. Proof of Theorem 1.1. By Theorems.9 and 4.1a and (, we have K(f(h(E[N] K (N K(Nf = H N,O. For any O-CM eiptic curve E /K(f, the spitting fied K(f ker ρ N of the reduced mod N Gaois representation on E is K(f(h(E[N], so [K(f(h(E[N] : K(f] #C N (O. As described in the introduction, it is immediate from (1 and the definition of H N,O that and thus it foows that Aut(H N,O /K(f = C N (O, K(f(h(E[N] = K (N K(Nf Proof of Coroaries 1., 1.3, 1.4 and Proof of Coroary 1.. Coroary 1. is an immediate consequence of Theorem Proof of Coroary 1.3. Since the reduced moduo N Gaois representation is independent of the rationa mode, Coroary 1. impies that for any number fied F K(j(E and N Z +, we have and thus [C N (O : ρ N (g F ] [F : K(j(E] [C N (O : ρ N (g F ] #O [F : K(j(E] 6[F : K(j(E] Proof of Coroary 1.4. We may of course assume that N. Let w = #q N (O, so #O N 3 w = #O N =. Then we have an injection µ w C N (O. Let E /K(f be any O-CM eiptic curve. We may view G = Aut(K(f(E[N]/K(f as a subgroup of C N (O. Let H = G µ w and L = (K(f(E[N] H, so a suitabe twist (E /L of E /L has trivia mod N Gaois representation. As shown in the proof of Theorem.11, we have L = K(f(h(E[N], so by Theorem 1.1 we have [L : K(f] = #C N (O.

14 14 ABBEY BOURDON AND PETE L. CLARK Proof of Coroary 1.5. We may assume that N. Let q N : O C N (O be the natura homomorphism. By Theorem.11b, there is an eiptic curve E /K(f such that [K(f(E[N] : #O K(f(h(E[N]] = #q N (O N 3 = #O. By Theorem 1.1 we have [K(f(h(E[N] : K(f] = N = #C N (O. Thus ρ E,N (g K(f = C N (O SPY Divisibiities. 6. Appications Lemma 6.1. Let H, K be subgroups of a group G. If H is norma and H K = {1}, then #K [G : H]. Proof. The composite homomorphism K G G/H is an injection. The foowing resut extends [BCP17, Cor..5] from maxima orders to a imaginary quadratic orders, thereby confirming the expectation expressed in [BCP17, Remarks.]. Theorem 6.. Let O be an order in an imaginary quadratic fied K, and et E be an O-CM eiptic curve defined over a number fied F K. If E(F has a point of order N Z +, then ϕ(n #O [F : Q] # Pic O. Proof. Let I N = [C N (O : ρ N (g F ] be the index of the mod N Gaois representation in the Cartan subgroup. By Coroary 1.3 we have I N #O [F : K(j(E] = #O [F : Q] # Pic O. Since there is a rationa point of order N, ρ N (g F contains no scaar matrices other than the identity, so by Lemma 6.1 we have ϕ(n I N, and we re done. 6.. A Theorem of Franz. Let O be an order in K, of conductor f, and et E /K(f be an O-CM eiptic curve. Choose a fied embedding K(f C such that j(e = j(c/o and an isomorphism E /C C/O. This induces an isomorphism E(K(f[tors] C/O[tors], which we use to view (the image in C/O of τ K = K+ K as a point of E(K(f[tors] of order f. Theorem 6.3. (Franz [Fr35] With notation as above, we have K(f(h(τ K = K (f. Proof. As in the proof of Theorem 1.7, over C we may view the canonica isogeny as ι : C/O C/O K. We take e 1 = 1 f + O and e = τ K + O as a basis for E[f]. Then e generates ker(ι, a K(f-rationa cycic subgroup of order f, and there is a character Ψ : g F (Z/fZ such that ï ò Ψ(σ 0 ρ E,f (σ =. Ψ(σ If f, then K(f(h(τ K = K(f = K (f and the resut hods. Thus we may assume f 3. Let L := K(f(h(e. Since j(e 0, 178, the restriction Ψ gl : g L {±1} defines a quadratic character χ, and on the twist E χ of E /L the point e becomes L-rationa. As in the proof of Theorem 5.5 of [BCS17], et Ψ ± : g K(f (Z/fZ /{±1} denote the composition of Ψ with the natura map (Z/fZ (Z/fZ /±1. Then L (K(f ker Ψ±, so [L : K(f] ϕ(f. If ι : Eχ E is the canonica isogeny, then the proof of Theorem 1.7 shows that ι(e 1 is an eement of E (L which generates E [f] as an O K -modue. Thus E has fu f-torsion over L, so by Theorem.8, K (f L. So and thus K(f(h(e = L = K (f. [L : K(f] [K (f : K(f] = ϕ(f [L : K(f],

15 TORSION POINTS AND GALOIS REPRESENTATIONS ON CM ELLIPTIC CURVES The Fied of Modui of a Point of Prime Order. In the introduction, we discussed a program to determine fieds of modui of a CM points on moduar curves. Theorem 1.1 carries out this program for the curves X(N. In this section we wi obtain a resut on the curves X 1 (N. Let K Q( 1, Q( 3 be an imaginary quadratic fied, and et O K be the order of conductor f. Here we use Theorem 1.7 to determine the smaest fied F K for which there exists an O-CM eiptic curve E /F with an F -rationa point of order >. Lemma 6.4. Let K be an imaginary quadratic fied, et f Z +, and et > be prime. K ( K(f = K(. Then Proof. Let = f K. The statement is immediate if f = 1, so suppose f > 1. By Theorem.1, Å ã [K(f : K(f] =. Since [K ( K(f : K(f] = #C (O/ by Theorem 1.1, we have in both cases that [K ( K(f : K(f] = #C (O [K(f : K(f] = 1 ( 1. Thus [K ( : K ( K(f] = [K ( K(f : K(f] = 1 ( 1. As we have K( K( K(f and [K ( : K(] = 1 ( 1, the resut foows. Theorem 6.5. Let K Q( 1, Q( 3 be an imaginary quadratic fied, and et O be the order of conductor f in K. Let F K. a Let E /F be an O-CM eiptic curve such that E(F contains a point of prime order >. Then there is a prime p of O K ying over such that K(fK p F. b If ( 1, then there is a prime p is a prime of OK ying over and an O-CM eiptic curve E /K(fK p such that E(K(fK p has a point of order. If ( = 1, then an O-CM eiptic curve E/F with an F -rationa point of order must have fu -torsion (see [BCS17, Thm. 4.8] or Lemma 6.1. In this case, K(fK ( F by Theorem 1.1. The existence of an eiptic curve E /K(fK ( with fu -torsion is guaranteed by Coroary 1.4. Proof. a Let F K and E /F be an O-CM eiptic curve with an F -rationa point of order. By Theorem 1.7, there is an O K -CM eiptic curve E /F with an F -rationa point P of order. If M is the O K -submodue of E (F generated by P, then M = E [ann M] and #M = ann M by Theorem.6. Since #M, we must have p ann M for some prime p of O K above. By Theorem.8 we have K(fK p K(fK ann M = K(j(EK (1 (h(e [ann M] F. b If ( 1, then an O-CM eiptic curve E/K(f possesses a K(f-rationa cycic subgroup of order. (See e.g. [CCS13, p.13]. This is aso a specia case of Theorem By [BCS17, Thm. 5.5], there is an extension L/K(f of degree ( 1/ and a quadratic twist (E 1 /L such that E 1 (L has a point of order. By part a, there is a prime p of O K ying over such that K(fK p L, so it wi suffice to show that [K(fK p : K(f] 1. If f, then is unramified in K(f. Thus K(f, K p are ineary disjoint over K (1, and we have [K(fK p : K(f] = [K p : K (1 ] = 1 ( 1 since ( K ( = 1. If f, then by Lemma 6.4 we have Thus K p K(f = K p K(, so K p K(f K ( K(f = K(. [K(fK p : K(f] = [K p : K p K(f] = [K p : K p K(] = [K(K p : K(] and it is enough to show that [K(K p : K(] 1.

16 16 ABBEY BOURDON AND PETE L. CLARK ( K = 1: We wi prove that K p K( = K (1 using CM eiptic curves. Let (E 0 /K (1 be an O K -CM eiptic curve. Then E 0 [p] is stabe under the action of g K (1 and generated by a point P of order. By [BCS17, Thm. 5.5], there is an extension L/K (1 of degree ( 1/ and a quadratic twist (E 1 /L such that P becomes L-rationa. By Theorem.8 we have K p L, and K p = L since [K p : K (1 ] = 1 ( 1. Over K(Kp, the curve E 1 has a rationa point of order, and the mod Gaois representation is scaar by Theorem 4.1. Thus E 1 has fu -torsion over K(K p, and K ( K(K p. This impies 1 ( 1 [K(Kp : K(] = [K p : K p K(]. Since [K p : K (1 ] = 1 ( 1, we have Kp K( = K (1, and [K(fK p : K(f] = [K p : K (1 ] = 1 ( 1. ( K = 1: In this case, K p = K (, so K p K( = K(. This impies [K(fK p : K(f] = [K p : K(] = 1 ( 1. ( K = 0: Since [K( : K (1 ] = and [K p : K (1 ] = 1 ( 1, we have Kp K( = K (1. Thus [K(fK p : K(f] = [K p : K (1 ] = 1 ( 1. Remark 6.6. Assume the setup of Theorem 6.5 but take K = Q( 1 or K = Q( 3. Then the assertion of Theorem 6.5b is fase. Indeed, if 5 and ( 1, we have [K(fK p 1 : K(f] w K ( 1. (See Lemma.10. Suppose F K, and et E /F be an eiptic curve with CM by the order in K of conductor f. If E(F contains a rationa point of order, then Theorem 6. impies 1 ( 1 [F : K(f]. Thus F must propery contain K(fK p Sharpness in the Isogeny Torsion Theorem. The foowing resut was estabished during the proof of Theorem 1.7. Lemma 6.7. Let E be an O-CM eiptic curve defined over a number fied F containing the CM fied K, and for a positive integer f dividing the conductor f of O, et ι : E E be the canonica F -rationa isogeny to an eiptic curve E with CM by the order in K of conductor f. Write where s e and s e. Then s s. E(F [tors] = Z/sZ Z/eZ, E (F [tors] = Z/s Z Z/e Z, In [Ro94, 4], Ross caims that if E is a CM eiptic curve defined over a number fied F containing the CM fied, then the exponent of the finite group E(F [tors] is an invariant of the F -rationa isogeny cass. In the setting of Lemma 6.7, this woud give e = e, and combining this with the concusion of Lemma 6.7 we woud get an injective group homomorphism E(F [tors] E(F [tors]. This concusion is stronger than that of Theorem 1.7. Unfortunatey Ross s caim is fase: in the setup of Lemma 6.7 one can have e < e (in which case there is no injective group homomorphism E(F [tors] E (F [tors], as the foowing resut shows. Proposition 6.8. Let > 3 be a prime number, et K = Q(, et n Z 3, et O be the order in K of conductor f = n, and et F = K(f. For any O-CM eiptic curve E /F, there is an extension L/F of degree ϕ( n such that E(L ( has a point of order n, and no O K -CM eiptic curve has an L-rationa point of order k for k > 1 n n (hence no L-rationa point of order n. Proof. Let E /F be an O-CM eiptic curve. As in (5 we may choose a basis {e 1, e } for E[ n ] so that the image of the mod n Gaois representation consists of matrices ñ ô a bf K K 4 a, b Z/ n Z. b a + bf K Since ramifies in K and f = n, we have ord (bf K K 4 = 1 + n n, so the matrices have the form ï ò a 0 a, b Z/ n Z. b a + bf K

17 TORSION POINTS AND GALOIS REPRESENTATIONS ON CM ELLIPTIC CURVES 17 The action of g F on e gives a character Φ : g F (Z/ n Z. Take M = (F ker Φ. Then [M : F ] ϕ( n and Φ gm is trivia. Thus there exists an extension L/F with [L : F ] = ϕ( n such that E(L contains e. Let E /L be an O K-CM eiptic curve, and suppose E (L contains a point P of order k. Let p be the prime idea of O K such that O K = p. We caim that the O K -submodue M = P OK of E (L generated by P contains E[p k 1 ] and thus, by Theorem.8, that K pk 1 L. Indeed, by Theorem.6, we have M = E[I] for some idea I of O K such that (O K /I, + has -power order and exponent k. Since ramifies in O K, this forces I to be of the form p a for some a Z +, and the smaest a such that (O K /p a, + has exponent k is a = k 1, estabishing the caim. Thus n ord ([K pk 1 : K (1 ] = k ord ([L : K (1 ] = + n 1, so k 1 (n n. In the setting of Theorem 1.7, one wonders whether #E(F [tors] = #E (F [tors]. In fact #E (F [tors] #E(F [tors] can be arbitrariy arge: Proposition 6.9. Let be an odd prime, et K Q( 1, Q( 3 be an imaginary quadratic fied, et O be the order in K of conductor, and et F = K(. For any O-CM eiptic curve E /F there is an extension L/F such that if ι : E E is the canonica isogeny to an O K -CM eiptic curve E, then #E (L[tors] #E(L[tors]. Proof. Let E /F be an O-CM eiptic curve. As above, there is a basis {e 1, e } for E[] such that ßï ò a 0 ρ (g F a, b Z/Z b a and there is an extension L/F with [L : F ] = 1 such that E(L contains e. In fact, E(L[ ] = Z/Z. Indeed, E does not have fu -torsion over L since Theorem 1.1 woud impy K ( K( L and 1 ( 1 = [K( K( : K(]. In addition, E has no point of order by Theorem 6.. Let ι : E E be the canonica L-rationa isogeny from E /L to E /L, where E has O K -CM. Since e E(L, the proof of Theorem 1.7 shows ι(e 1 E (L, and ι(e 1 generates E [] as an O K -modue. In other words, Z/Z Z/Z E (L[tors]. It foows that #E (L[tors] #E(L[tors]. Finay, Theorem 1.7 requires K F. This hypothesis cannot be omitted: Proposition Let > 3 be a prime with 3 (mod 4 and et n Z 3. Let K = Q(, and et O be the order in K of conductor f = n. Let F = Q(j(C/O. There is an eiptic curve E /F and an extension L/F of degree ϕ(n such that: (i L K, (ii E(L has a point of order n, and (iii for every O K -CM eiptic curve E /L we have n #E (L[tors]. Proof. Let E /F be an O-CM eiptic curve. By [Kw99, Coroary 4.], E has an F -rationa subgroup which is cycic of order n. It foows from [BCS17, Theorem 5.6] that there is a twist E 1 of E /F and an extension L/F of degree ϕ( n / such that E 1 (L has a point of order n. Note [L : Q] = h K n ϕ( n is odd (see [Co89, Proposition 3.11], so K L. Let E /L be an O K-CM eiptic curve. Since [L : Q] is odd, E (L[ ] must be cycic, as fu k - torsion woud impy Q(ζ k L by the Wei pairing. As in the proof of Proposition 6.8, E (LK contains no point of order n. Hence E (L contains no point of order n, and n #E (L[tors].

18 18 ABBEY BOURDON AND PETE L. CLARK 6.5. Minima and Maxima Cartan Orbits. Let O be an order, et N Z +, and et P O/NO be a point of order N. Since C N (O contains a scaar matrices, if P O/NO has order N, then the orbit of C N (O on P has size at east ϕ(n. On the other hand, the orbit of C N (O on P is certainy no arger than the number of order N points of O/NO. In this section we wi find a pairs (O, N for which there exists a Cartan orbit of this smaest possibe size and aso a pairs for which there exists a Cartan orbit of this argest possibe size. We introduce the shorthand H(O, N to mean: there is a point P of order N in O/NO such that the C N (O-orbit of P has size ϕ(n. Lemma Let O be an order, and et N = a1 1 ar r hods for a 1 i r. Proof. This is an easy consequence of the Chinese Remainder Theorem. Z +. Then H(O, N hods iff H(O, ai i Lemma 6.1. Let O be the order of discriminant, a prime number and a Z +. a If ( = 1, there is an O-submodue of O/ a O with underying Z-modue Z/ a Z. b If ( = 1, then C a(o acts simpy transitivey on the order a eements of O/ a O. Proof. a if ( = 1, then O/O = OK /O K = Z/Z Z/Z, so O Z is isomorphic as a ring to Z Z (see e.g. [Ei, Cor. 7.5] and thus O/ a O is isomorphic as a ring to Z/ a Z Z/ a Z. b If ( = 1, then O Z = O K Z is a compete DVR with uniformizer, so the ring O/ a O is finite, oca and principa with maxima idea. An eement of O/ a O has order a iff it ies in the unit group C a(o. Lemma Let O be the order of discriminant, and et N Z +. The foowing are equivaent: (i If N, then ( 1. (ii The Z/NZ-subagebra of O/NO generated by C N (O is O/NO. Proof. Using the Chinese Remainder Theorem we reduce to the case of N = a a power of a prime number. Let B be the Z/ a Z-subagebra generated by C a(o, so #B = b for some b a. (i = (ii: Since 0 B \ C a(o, we have Å = a 1 1 ã Å 1 Å #B #C a(o a + 1 > a 1, if 3 ã ã Thus b = a and B = O/ a O. (i = (ii: If = and ( = 1, then ßï ò O/ a O α 0 = 0 β 1 a + 1 > a 1, if = and (. 1 α, β Z/ a Z and C a(o consists of the set of such matrices with α, β (Z/ a Z. Thus C a(o is contained in the subagebra ßï ò α 0 B = α, β Z/ a Z and α β (mod 0 β of order a 1, so B B O/ a O. 1 Lemma For an order O and N Z +, the foowing are equivaent: (i There is an idea I of O with O/I = Z/NZ. (ii There is an O-submodue of O/NO with underying commutative group Z/NZ. (iii H(O, N hods. 1 Since #B #C a (O + 1 = a + 1 > a, in fact we have B = B.

19 TORSION POINTS AND GALOIS REPRESENTATIONS ON CM ELLIPTIC CURVES 19 Proof. (i (ii: Step 1: Let Λ be a free, rank Z-modue, and et Λ be a Z-submodue of Λ containing NΛ. By the structure theory of modues over a PID, there is a Z-basis e 1, e for Λ and positive integers a b such that ae 1, be is a Z-basis for Λ. Thus Λ/Λ = Z/aZ Z/bZ, Λ /NΛ = Z/(N/bZ Z/(N/aZ. It foows that Λ/Λ = Z/NZ Λ /NΛ = Z/NZ. Step : If I is an idea of O with O/I = Z/NZ, then I NO, so I/NO = Z/NZ by Step 1. Let M be an O-submodue of O/NO with underying Z-modue Z/NZ. Then M = I/NO for an idea I of O, and by Step 1 we have O/I = Z/NZ. (ii = (iii: Let P O/NO have order N such that the subgroup generated by P is an O-submodue. For a g C N (O, gp = a g P for a g (Z/NZ. Conversey, since C N (O contains a scaar matrices, the orbit of C N (O on P has size ϕ(n. (iii = (ii: Case 1: Suppose N or ( 1. Let P O/NO be a point of order N with C N (O-orbit of size ϕ(n. There is a Z/NZ-basis e 1, e of O/NO ßï with e ò 1 = P, and our hypothesis a b gives that with respect to this basis C N (O ies in the subagebra a, b, d Z/NZ of upper 0 d trianguar matrices. By Lemma 6.13, O/N O aso ies in the subagebra of upper trianguar matrices, and thus P is an O-stabe submodue with underying Z-modue Z/NZ. Case : Suppose N and ( = 1, and write N = a N with N. By Lemma 6.1 and the equivaence of (i and (ii, there is an idea I 1 in O with O/I 1 = Z/ a Z, and by Case 1 there is an idea I in O with O/I = Z/N Z. By the Chinese Remainder Theorem, O/I 1 I = Z/NZ. Since (i (ii, this suffices. Theorem Let O be an order of discriminant, and et N Z +. The foowing are equivaent: (i H(O, N hods. (ii is a square in Z/4NZ. Proof. Using the Chinese Remainder Theorem and Lemma 6.11, we reduce to the case in which N = a is a power of a prime number. Case 1 ( is odd: Since gcd(4, a = 1, we may put D = 4 Z/a Z. Then is a square in Z/4 a Z iff D is a square in Z/ a Z, and (6 O/ a O = Z/ a Z[t]/(t D. If there is s Z/ a Z such that D = s, then O/ a O = Z/ a Z[t]/((t + s(t s, so if I is the idea t + s, a of O, then O/I = Z/ a Z. By Lemma 6.14, H(O, a hods. Conversey, suppose H(O, a hods, so by Lemma 6.14 there is an idea I of O with O/I = Z/ a Z. Since a I, we may regard I as an idea of O/ a O such that (O/ a O/I = Z/ a Z. In other words, we have a Z/ a Z-agebra homomorphism f : Z/ a Z[t]/(t D Z/ a Z. Then f(t = D Z/ a Z, so D is a square in Z/ a Z. Case ( =, is odd: Then ( = ±1. If ( = 1, then 1 (mod 8; by Hense s Lemma, is a square in Z/ a Z. On the other hand, by Lemmas 6.1a and 6.14, H(O, a hods. If ( = 1, then 5 (mod 8, so is not a square moduo 8 and thus not a square moduo 4 a. On the other hand, by Lemma 6.1b H(O, a does not hod. Case 3: ( =, is even: Again we may put D = 4 Z/a Z, and again (6 hods. The argument of Case 1 shows that H(O, a hods iff D is a square moduo Z/ a Z iff is a square moduo Z/4 a Z. Proposition Let O be an order, and et N Z +. The foowing are equivaent: (i C N (O acts simpy transitivey on order N eements of O/NO.