On The j-invariants of CM-Elliptic Curves Defined Over Z p

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1 On The j-invariants of CM-Ellitic Curves Defined Over Z Andrew Fiori Deartment of Mathematics and Statistics, Queens University, Jeffery Hall, University Ave, Office 517, Kingston, ON, K7L 3N6, Canada Abstract We characterize the ossible reductions of j-invariants of ellitic curves which admit comlex multilication by an order O where the curve itself is defined over Z. In articular, we show that the distribution of these j-invariants deends on which rimes divide the discriminant and conductor of the order. Keywords: Comlex Multilication, Lifting, Ellitic Curves 1. Introduction There are several different ways of framing the results of this aer. Out main object of study will be CM-ellitic curves over Z which are suersingular at. The results we obtain will rimarily be directed towards trying to address the following three uestions: 1. When are there ellitic curves defined over Z with CM by an order O in a uadratic imaginary field K in which is inert and where does not divide the conductor of O?. What factors affect the ossible reductions of their j-invariants modulo amongst the set of all suersingular F -rational j-invariants? 3. Given an F -rational suersingular j-invariant which admits CM by O, when does there exist an ellitic curve defined over Z, with CM by O which reduces to it. One natural source of interest in these uestions is the following observation of Ernst Kani: Proosition. Every F ellitic curve with CM by O lifts to Z (with a lifting of its CM to Z ) if and only if does not divide the conductor of the ring Z[j(E 1 ),..., j(e n )] generated by the j invariants of all ellitic curves with CM by O. Remark 1.1. This ring Z[j(E 1 ),..., j(e n )] is a very natural order in the ring class field of O, its structure is mysterious. The results we obtain are somehow in contrast to the same uestion asked for ellitic curves over Z the unramified uadratic extension of Z. In articular, for the same uestions asked over Z, we have the following answers: 1. There are always CM-ellitic curves over Z with CM by O an order in a uadratic imaginary field K in which is inert, and where does not divide the conductor.. From the work of Cornut-Vatsal [CV05, CV07] and Jetchev-Kane [JK11] we have that the reductions of the j-invariants of ellitic curves with CM by O are euidistributed among the suersingular values in F (as we vary the conductors O subject to certain congruence conditions). Moreover, for each and all but finitely many O where is inert, the ma from ellitic curves with CM by O to suersingular j-invariants in F is surjective. 3. By the work of Deuring [Deu41] we know that given a suersingular ellitic curve E with CM by O there always exists a lift to an ellitic curve over Z with CM by O which reduces to E. The results we obtain are motivated by comutations, some of the data from which is resented in the Aendix, which gave results which seemed contrary to the above. In articular if we consider only the ellitic curves which are defined over Z then: address: afiori@mast.ueensu.ca (Andrew Fiori) This is a rerint May 18, 015

2 They are not always surjective onto suersingular F values as we vary O among maximal orders subject to certain congruence conditions on the discriminant; orders in a certain fixed K subject to certain congruence conditions on the conductor; orders subject to certain congruence conditions on the conductor and discriminant of K. The set of ossible values, and hence the overall distributions, deends on congruence conditions on both the discriminant of K and the conductor of O. For certain congruence conditions on discriminants and conductors there are irreducible factors which always aear together, in eual numbers. So the aearance of a given factor is not indeendent on the aearance of another. We should emhasize before roceeding that the above does not actually conflict with the aforementioned euidistribution results. Firstly, because the Z curves have density 0 among all curves, but moreover, because the data does suggest the following: Varying O subject to congruence conditions on the discriminants and/or conductors; the j-invariants which aear are very likely to following a simle distribution. The euidistribution results should work rimarily with conditions of the form l Df rather than l Df whereas the difference in behavior is rimarily from a comarison of these cases. Conseuently, it is ossible or erhas even likely that based on heuristic arguments and the recise tyes of families considered in the work of Cornut-Vatsal and Jetchev-Kane, some form of the euidistribution results one would have exected still hold for the Z -terms. The work of [CV07] actually describes circumstances in which there can be correlation between freuencies, however we should note that the correlations we observe are only aarent for Z values and not the Z values with the same reductions, so even if they have the same underlying exlanation, the henomenon is still somewhat distinct. This aer is organized as follows: In Section we introduce the relevant background. In Section 3 we state and rove our results In Section 4 we discuss two natural uestions our work leaves oen. In Aendix A we discuss the comutations and data on which are work is based.. Background In this section we will be introducing the results necessary to state and rove our theorems. Much of what we are saying is very well known, and can be found in many references on the theory of comlex multilication. Some results which are erhas less well known can be found in [Sch10], [Deu41], [Ibu8] or [Dor89]. We recall the following imortant facts: Theorem.1. If E is an ellitic curve over a field of characteristic 0 then either: End(E) = Z, this is the general case. End(E) = O, for O Q( D) an order in a uadratic imaginary field, this is the so-called CM-case. We will be interested in the CM or comlex multilication case in characteristic 0, where we have the following classification result: Theorem.. The ellitic curve E τ = C/(Z τz) has End(E) = O if and only if 1. τ Q( D), that is τ generates a (comlex) uadratic field, and. Z + τz Q( D) is a (rojective) O-module. Moreover, for any algebraically closed field C of characteristic 0 there is a bijective corresondence between ellitic curves over C with End(E) = O and Cl(O) the ideal class grou of O.

3 Remark.3. Note that there is an essentially euivalent bijection between Cl(O) and airs (E, ρ : O End(E)) of E and an isomorhism of O with End(E) with a fixed CM-tye. This bijection extends to characteristic where we consider instead certain otimal embeddings O End(E). In the definition of P O (X) below, it is concetually better to be considering the bijection of the theorem. Theorem.4. If E is an ellitic curve over a field of characteristic then either: End(E) = Z, this is the general case. End(E) = O, for O Q( D) an order in a uadratic imaginary field in which slits. End(E) = B, for B a maximal order in a uaternion algebra over Q ramified only at and. This is the so-called suersingular case. From the above we see that if ever we can reduce a CM ellitic curve E at a rime inert in O we will obtain a suersingular ellitic curve. In the characteristic setting it will be this case we are most interested in. Notation.5. Let m Z + be suare free so that K = Q( m) is the uadratic imaginary field of discriminant D, denote by O K its maximal order and O = O K,f = Z + fo K an order of conductor f Z. Denote by: P O (X) = a O(X j(c/a)). Denote by L the slitting field of P O (X) over K. The following facts are well known, for a reference see for examle [Sch10]. P O (X) Z[X] and is irreducible over K. L is abelian over K, with Gal(L/K) Cl(O), the action being the natural ermutation action of Cl(O) on the roots. L is galois over Q, the action of Gal(K/Q) on Cl(O) being g g 1 so that Gal(K/Q) is a generalized dihedral grou. The action of comlex conjugation on the ideals of K, agrees with the action on the set CM(O) which agrees with the action of Gal(K/Q). L/K is ramified only at rimes over f, whereas L/Q is ramified only at rimes over Df. We shall denote by N = Q(j) = Q[X]/(P O (X)) L. Based on the above we can conclude the following: If is inert in K and does not divide f (or euivalently that If ( Df ( Df ) = 1) then slits in L/K. ) = 1 then P O (X) factors as a roduct of uadratic and linear terms over Z. Remark.6. The above agrees with the fact that the reductions of these ellitic curves must be defined over F, as they are known to be suersingular. Proosition.7. If is inert in K, and E is an ellitic curve with CM by O, then the reduction of E modulo is suersingular. In articular, End(E) = B, where B is a maximal order in a uaternion algebra ramified only at and infinity. Moreover, there is a bijection between ellitic curves with CM by O and airs: (O B) of O with an otimal embedding into a maximal order B. Moreover, the natural action of the ideals of O by conjugation on such airs (O B) agrees with the action of the ideals on the collection of ellitic curves with CM by O. See [Dor89]. From now on we ( shall) be working in the setting where is slit in K and does not divide f. In articular we are assuming that Df = 1. 3

4 Proosition.8. If P O (X) has a linear factor over Z, the number of such linear factors is Gal(L/K)[] the size of the two torsion of the class grou. Proof. By basic algebraic number theory we must count the size of the conjugacy class of Frobenius. This is then a basic roerty to dihedral grous. Remark.9. If Gal(L/K)[] = 1 then P O (X) has a uniue linear factor over Z. Theorem.10 (Deuring). If E corresonds to the data (O B) then the reduction of E modulo is defined over F (rather than simly F ) if and only if B contains Z[ ]. See [Deu41]. In [Ibu8] Ibukiyama gives a comlete classification of the maximal orders B which contain Z[ ]. Notation.11. Fix and = 3 (mod 8) such that B = (, ) is the uaternion algebra ramified only at and. Fix α, β B such that α =, β = and αβ = βα. Choose r Z such that r + = m for some m Z. Denote: α(1 + β) O(,, r, m) = Z + Z + Z 1 + β (r + α)β + Z If = 3 (mod 4) choose r Z such that (r ) + = 4m for some m Z. Denote: O (,, r, m ) = Z + Z 1 + α + Zβ + Z (r + α)β. Theorem.1 (Ibukiyama). The sets O(,, r, m) (and O (,, r, m )) are maximal orders of B, their isomorhism classes deend only on and not on r or m. Moreover, all airs consisting of a maximal order in B with an embedding of Z[ ] are of the form O(,, r, m) (or O (,, r, m )) with the embedding taking ±α. The orders O(,, r, m) and O (,, r, m ) are only ever isomorhic if they corresond to the j-invariant 178. See [Ibu8]. Remark.13. In O(,, r, m) we may write: ( ) α(1 + β) α = ( (r + α)β ) + r. Remark.14. We can count the number of isomorhism classes of O(,, r, m) (resectively O (,, r, m )) by looking at the class numbers h for Z[ ] (and h for Z[(1 + )/]), we have the following standard formulas (for 3): The number of suersingular j invariants over F is n = ( 1)/1 +e 0 +e 178, where e x is 0 or 1 deending on if x is suersingular at. If = 7 (mod 8) then h = h and there are (h + 1)/ otions for both O(,, r, m) and O (,, r, m ). If = 3 (mod 8) then h = 3 h and there are (h + 1)/ otions for O (,, r, m ) and ( h + 1)/ otions for O (,, r, m ). If = 1 (mod 4) there are h / otions for O(,, r, m). Combining the above allows us to comute the number of F rational suersingular values in terms of h. More generally, if we fix K = Q( D) a uadratic imaginary field of discriminant D and class number h K. Fix an order O = Z + fo K and write f = ai i The class number of O is given by: ( ( )) D h O = ɛh K i ai 1 i i where ɛ = 1 unless D = 3 or D = 4. If D = 3 and the formula above is divisible by 3 then ɛ = 1 3. If D = 4 and the formula above is divisible by then ɛ = 1. i 4

5 Theorem.15 (Halter-Koch). If n is the number of rime divisors of Df and does not divide f then : n 1 Df odd n Df Cl(O)[] = n 1 4 Df. n 1 8 Df n 16 Df More recisely, the maximal -extension of the ring class field of O contains: Q( ( 1) ( 1)/ ) where is an odd rime factor of Df. If D = 8m then the maximal -extension of the ring class field of O contains: Q( ( 1) (m 1)/ ). If D = 4 (mod 8) and 4 f then the maximal -subextension of the ring class field of O contains: Q( ). If D is odd, and 8 f then the maximal -subextension of the ring class field of O contains: Q( ). If D = 4 (mod 8), or f and D, or D is odd and 4 f then the maximal -extension of the ring class field of O contains: Q( 1). The above fields generate the genus field F, that is the maximal -subextension of the ring class field of O. See [Sch10, Thm 6.1.4]. 3. Results In this section we will resent our main theorems. These are rimarily structured to address the entries in the data which we resent in Aendix A. We will begin by looking at certain conditions under which there can be no ellitic curves over Z at all. Theorem 3.1. Fix K = Q( ( ) D) of discriminant D. Fix an order O = Z+fO K and suose that Df = 1. There are no ellitic curves over Z with CM by O if any of the following occur: ( ) there is an odd rime factor of Df with = 1 = 1 (mod 4) and 16 Df. = 3 (mod 8) and 8 D. = 3 (mod 8) and 64 Df Otherwise there are exactly Cl(O)[] j-invariants for ellitic curves over Z with CM by O. ( ) Remark 3.. The condition that there is an odd rime factor of D with = 1 imlies in articular that the uaternion algebra (, D) is ramified at. Though this can be used to justify the condition for those D, we will not follow this strategy of roof, rather we give a roof which has a more natural connection to class field theory. The condition on odd rimes cannot be extended to even rimes by use of the Kronecker symbol, the deendence on the behavior at is more subtle. 5

6 We shall use the following two lemmas. Lemma 3.3. Fix K = Q( D) of discriminant D. Fix an order O = Z + fo K and suose that ( Df ) = 1. The olynomial P O (X) has a linear factor over Z if and only if N = Q(j(O)) has no uadratic subextension in which is inert. Proof. If there is a uadratic subextension of N which is inert, then all factors of in N have inertial degree, and thus there can be no linear factors. Conversely, suose every factor of in N has inertial degree. let be a rime of L over and let σ be a generator for the decomosition grou of and let τ be a generator of Gal(L/N). Then σ is indivisible with exact order, because this is true of Frob. σ and τ are not conjugate, since if τ were a conjugate of Frob the field N = L τ would have a non-inert rime. σ and τ commute since σ has order. στ is in Gal(L/K) as they both act non-trivially on K. It follows from the above, and the basic structure of dihedral grous, that στ is indivisible with exact order. Thus we may write: Gal(L/K) = στ H and thus Gal(L/Q) = σ (H τ ). We see that G = (H τ ) is a normal subgrou of Gal(L/Q), moreover, the field L G subextension of N. is an inert uadratic Lemma 3.4. The maximal -subextension of N is the totally real subfield M of F the genus field of L. Proof. It suffices to show that N has a real embedding since any comosite of uadratic extensions is either totally comlex or totally real. To see this we use the fact that: j(a) = j(a). It is thus sufficient to find a such that a = a, but indeed we may simly take a = O. roof of Theorem 3.1. The idea of the roof is to show that is inert in a uadratic subextension of the totally real subfield N of F if and only if one of the conditions of the theorem holds. To show this we must find a subextension of N defined by adjoining the suare root of a ositive integer which is not a suare modulo, in each of the following cases we describe how to find such a non-suare. Note that if = 3 (mod 4) then D N whereas if = 1 (mod 4) then N. ( Consider ) the case where = 1 (mod 4) and 4 D. In this case there exists odd rime factor of D with = 1. Moreover, D has a factor such that both ± are not suares mod. Suose there is an odd rime factor of Df with if = = 3 (mod 4) we obtain ( ) = 1. if = 3 (mod 4), = 1 (mod 4) and D we obtain if = 1 (mod 4) we obtain ( ) = 1 and thus D is not a suare mod. ( ) = 1 and thus D is not a suare mod. ( ) = 1 and thus and is not a suare mod. Suose = 3 (mod 8) and 8 D and D/8 = 3 (mod 4) then D has a factor d congruent to 3 (mod 4) which is not a suare mod. Suose = 3 (mod 8) and 8 D and D/8 = 1 (mod 4) then is not a suare mod. 6

7 Suose = 3 (mod 8) and 64 Df then is not a suare mod. Suose = 1 (mod 4) and 16 Df then D has a factor such that both ± are not suare mod. The above covers all of the cases of the theorem. To rove the converse we remark that if is inert in N it is inert in a uadratic subextension of one of the following tyes: Q( ) where fd or Q( 1 ) where both 1, = 3 (mod 4) and 1 fd. as such fields generate the genus field of N. Comleting the roof follows a similar case analysis to the above. We now shift to discussing a henomenon whereby certain F reductions are disallowed based on the differing behavior of. Remark 3.5. In the following theorem we will be distinguishing the suersingular j-invariants in F by identifying them as roots of P Z[ ] (X) or P Z[(1+ )/] (X). To understand the significance we recall the theorems above of Ibukiyama which asserted that this naturally divides the suersingular values into two almost disjoint sets. More recisely, we have that for = 3 (mod 4) these olynomials factor as (X 178) i (X α i) whereas for = 1 (mod 4) the factorization is i (X α i). In each case the α i are distinct in F. Furthermore, in the case = 3 (mod 4) the α i for P Z[ ] (X) are distinct from those for P Z[(1+ )/] (X). The olynomial for is recisely P Z[ ] (X) = X Theorem 3.6. Fix K = Q( ( ) D) of discriminant D. Fix an order O = Z + fo K of conductor f Z and suose that Df = 1. Let j be a Z root of P O (X). If = 7 (mod 4) If is unramified in K and f then j is a root of P Z[ ] (X). If is unramified in K and f but 8 f then j is a root of P Z[(1+ )/] (X). If is unramified in K and 8 f then j is a root of either P Z[ ] (X) or P Z[(1+ )/] (X). If is tamely ramified in K and f or 4 f then j is a root of P Z[ ] (X) or P Z[(1+ )/] (X). If is tamely ramified in K and f then j is a root of P Z[(1+ )/] (X). If is wildly ramified in K and f or 4 f then j is a root of P Z[(1+ )/] (X). If is wildly ramified in K and f or 4 f then j is a root of either P Z[ ] (X) or P Z[(1+ )/] (X). If = 3 (mod 8) If is unramified in K and f or 4 f then j is a root of P Z[ ] (X). If is unramified in K and f then j is a root of P Z[(1+ )/] (X). If is unramified in K and 8 f then there are no linear terms. If is tamely ramified in K and f then j is a root of P Z[ ] (X) or P Z[(1+ )/] (X). If is tamely ramified in K and f then j is a root of P Z[ ] (X). If is tamely ramified in K and 4 f then there are no linear terms. If is wildly ramified in K then there are no linear terms. If = 1 (mod 4) If is unramified in K and 4 f then j is a root of P Z[ ] (X). If is unramified in K and 4 f then there are no linear terms. If is tamely ramified then there are no linear terms. If is wildly ramified in K and f then j is a root of P Z[ ] (X). If is wildly ramified in K and f then there are no linear terms. To rove this we will make use of the following lemma. 7

8 Lemma 3.7. If E is an ellitic curve over Z with CM by O which corresonds to a datum (O B) then the Galois Frobenius Frob acting on E(Q ) over Z induces the endomorhism Frobenius Frob of E. Moreover we have: Frob, the Galois action of Frobenius on E, acts on O by x x. Frob, the endomorhism of E, satisfies Frob x = x Frob for x O. Frob, the Galois action of Frobenius on E, commutes with O. Frob, the endomorhism of E, satisfies Frob =. In articular Frob O is an element of norm. See [Sch10]. roof of Theorem 3.6. We must show, using the classification of maximal orders containing by Ibukiyama, that the only CM-orders in α are those satisfying the conditions of the theorem. We note that in selecting the values of, r and m we may assume by relacing r by r + a that 8 r. With this assumtion we have that = m (mod 8). When selecting, r and m we must have that r is odd, when = 3 (mod 8) this imlies that m is odd. We observe the following imortant facts about α in the various cases: 1. For the maximal orders of the form O (,, r, m ) we have that α contains no elements with odd trace.. For the maximal orders of the form O (,, r, m ) we have that all rimitive elements of Z[ ] are of the form: yβ + z (r + α)β for some choice of y and z corime. The suare of such an element is: y z m yzr. Notice that if = 3 (mod 8) this cannot be even. 3. For the maximal orders of the form O(,, r, m) we have that all rimitive elements of odd trace in Z[ ] are of the form: (r + α)β yβ + z for some choice of y and z corime, with z odd. The suare of such an element is: y z m yzr modulo 8 this becomes: (y z ). Notice that if this is odd, then y is even and (y z ) = (mod 8). Also, if it is even then y and z are both odd and it is divisible by (1 ) (y z ). By considering each of the cases of the theorem, the above allows us to conclude the result. Proosition 3.8. Suose there exists Z[ D] = O α, then P O (X) has Z roots. Proof. By the above argument we note that O α imlies the existence of a solution to: In the first case, multilying by we obtain: y + z m + yzr = D or y + z m + yzr = D. D = y + z ( + r ) + yzr = z + (y + rz). reducing modulo 8 and modulo all the odd rime factors of D we obtain the result. In the second case, multilying by 4 we obtain: 4D = 4y + z ( + r ) + yzr = z + (y + rz) and the result follows similarly. 8

9 Remark 3.9. Note that the above does not actually rove the converse to Lemma 3.7 though it would rovide for an alternate roof for one direction of Theorem 3.1. We now exlain the henomenon where in secific circumstances certain F reductions always occur with the same freuency. Based on [CV07] we should exect that this is caused by systematic collections of isogenies (coming from Hecke relations), and in our case we should exect -isogenies to lay a role. Lemma If α then O O(,, r, m) or O(,, r, m ) for some choice of r, m or r, m. Proof. By [Ibu8, Pro.1 and Rmk.] the conditions: 1 = z + (y 1 + rz) or 4 1 = z + (y 1 + rz) imly that 1 and satisfy O(, 1, r 1, m 1 ) O(,, r, m ) or resectively O (, 1, r 1, m 1) O (,, r, m ). The results then follow from the roof of Proosition 3.8. Lemma Fix = 3 (mod 4). Fix K = Q( ( ) D) of discriminant D. Fix an order O = Z+fO K and suose that Df = 1. Suose further that is tamely ramified in K but does not divide f. Suose that O is otimally embedded in O(,, r, m) and contained in α. Let a = () in O. Then ao(,, r, m)a 1 O (,, r, m ) is a maximal order with an otimal embedding of O. Conseuently, if E is an ellitic curve over Z with CM by O whose reduction has endomorhism ring O(,, r, m), then the reduction of a E has endomorhism ring O (,, r, m ) with the exact same choice of. Conversely, if E is an ellitic curve over Z with CM by O whose reduction has endomorhism ring O (,, r, m ), then the reduction of a E has endomorhism ring O(,, r, m) for some such that O (,, r, m ) O (,, r, m ). Proof. Let O = Z[γ = ]. It suffices to show that ao(,, r, m)a 1 contains both 1+α and β. We note that a = (, 1 + γ) and a 1 = (1, 1 γ ). It follows immediately that β ao(, 1, r, m)a 1. Now we may write γ = yβ + z r+α β with y and r even and z odd. Now, by observing that: ( ) ( ( ) ( )) ( ) 1 + α β r + α 1 γ = (1 + γ) ( zm + ry + 1) + (zm + ry) 1 (y + zr) β and that the right hand side is in ao(,, r, m)a 1 we conclude by Lemma 3.10 that ao(,, r, m)a 1 O (,, r, m ). Now suose we start with O otimal in O (,, r, m ). Attemting to reverse the above calculation cannot work in general as we no longer have r and m but r and m. However, we observe that: ( ( 1 + α (1 + γ) ) ( 1 γ ) ( 1 + γ 4 ) ) α ao (,, r, m )a 1 is erendicular to α and has odd trace. Hence, ao (,, r, m )a 1 O(,, r, m). The result now follows. Remark 3.1. Note, that we could not simly run the first art of the above argument in the oosite direction to go from O (,, r, m ) to O(,, r, m), in articular this would be imossible in any case where the class grous which classify O (,, r, m ) and O(,, r, m) are not in bijection. Theorem Fix = 7 (mod 4). Fix K = Q( ( ) D) of discriminant D. Fix an order O = Z + fo K and suose that Df = 1. Suose further that is tamely ramified in K but does not divide f. It we consider the set of suersingular values of F excet 178, each j-invariant J has a artner J such that, the freuency of the aearance of X J and X J as the reduction of irreducible linear factors of P O (X) modulo is the same. Proof. We first observe that if E is defined over Z then so too is a E. This follows by observing that the collection of endomorhisms in a is Galois stable. Moreover, in the case = 7 (mod 4) the ma from O(,, r, m) to O (,, r, m ) being injective imlies it is bijective as the collections have the same size. By Lemma 3.11 it now follows that O(,, r, m) and O (,, r, m ) must occur with the same freuency. We note that j-invariant 178 is the only one that can ever be identified with itself through this rocess, and in fact it must, because the class grou has odd order. 9

10 Remark For = 3 (mod 4) we obtain other less obvious relationshis between the counts for maximal orders of tye O and of tye O arising from the fact that the ma is generically 3 : 1. In articular, in general the freuency for those of tye O is the sum of the freuencies of a secific collection of three of orders of tye O. We note that there will be a curve which is -isogenous to the one with j-invariant 178. We should oint out that the F oints of the -torsion is well understood, that there is a uniue F rational -torsion oint is suggestive of the above results, but does not show that the association is between O(,, r, m) and O (,, r, m ) and certainly not that it resects. Theorem Fix = 1 (mod 4). Fix K = Q( ( ) D) of discriminant D. Fix an order O = Z + fo K and suose that Df = 1. Suose further that is wildly ramified in K but does not divide f. It we consider the set of suersingular values of F, each j-invariant J has a artner J such that, the freuency of the aearance of X J and X J as the reduction of irreducible linear factors of P O (X) modulo is the same. This artner J is indeendent of K and O and deends only on. Proof. Set a = () in O. In this case we have a = (, γ) and a 1 = (1, 1 γ). As in the revious case, we must only show that ao(,, r, m)a 1 is indeendent of O. Now set b = () in Z[ ]. We have that b = (, 1 + α). β = 1 (y + zr + zα)β with r even and both y and z odd. We claim that (1 + α) ao(,, r, m). Indeed, as β O(,, r, m) we have y + zr + zα = γβ ao(,, r, m). Since a the claim then follows immediately. Conversely, it is clear that γ bo(,, r, m). As is odd, and b we also have that γ bo(,, r, m). We thus have shown that ao(,, r, m) = bo(,, r, m). It follows that ao(,, r, m)a 1 = bo(,, r, m)b 1 is indeendent of O. We recall that we have γ = yβ + z r+α Remark In this case the uniueness of the F -rational -torsion oints is sufficient to conclude the result. 4. Further Questions Our results suggest the following natural uestions: Question 1. In Theorem 3.6 we gave necessary conditions for a datum (O B) to corresond to an ellitic curve over Z. Moreover, Proosition 3.8 gives the imression that this may be sufficient. It is natural to ask, if these conditions are in fact sufficient. (a) More recisely, given an ellitic curve over F, and an endomorhism (defined over some extension) when can we lift the curve to Z such that the endomorhism lifts to some extension? (b) Is it sufficient that the endomorhism be erendicular to Frobenious in the endomorhism algebra over F? A thorough answer to this uestion would shed light on the structure of Z[j(E 1 ),..., j(e n )] as remarked in the introduction. Question. Theorems 3.13 and 3.15 give situations in which there are automatic relationshis between certain roots of P O (X). As remarked a simalar result holds for the same reason when = 3 (mod 8). (a) It is natural to ask if there are other situations in such relationshis must exist? In articular are there situations where the role of can be relaced by some other rime? (b) The method of roof also suggests that we could anticiate relations between the roots of P O (X) between two different orders in the same field whose conductors differ by a factor of. Can the combinatorics of this be made more recise? Acknowledgements I would like to thank Prof. Eyal Goren for suggesting the comutations which led to the discovery of these results. I would like to thank Prof. Ernst Kani for some useful discussions as well as recommending several references. I would like to thank the many develoers of SAGE without which the comutations through which we uncovered these results would not be ossible. I would also like to thank the SAGE Notebook roject for the use of various comuting resources which they have made available through their various funding sources. 10

11 Aendix A. Data In this section we will resent a reresentative samle of the data, which forms the basis for how we discovered the theorems. Similar comutations have been done for all u to All of these comutation were erformed in SAGE. Data not contained can be obtained from the author. In all the data which follows, the freuencies resented reresent the total number of times each factor aears as the reduction modulo of an irreducible factor of P O (X) over Z for one of the orders under consideration. We will consider several families of orders, but in all cases we are considering all orders in the described class with class numbers strictly less than 40 (and discriminant of the base field less than 10 milliion, noting that there are no fields of class number less than 100 with discriminant between 3 and 10 million). Made recise, the aearance of the 199 in the first table indicates that there are exactly 199 different j-invariants congruent to 0 mod 71 for ellitic curves over Z 71 which admit CM by the maximal order of a uadratic imaginary field with odd class number less than 40 (and discriminant less than 10 million) in which 71 is inert. The 1 in the first table indicates that there is a uniue j-invariant over Z 71 congruent to 3 modulo 71 for which the associated ellitic curve admits CM by the maximal order of a uadratic imaginary field with odd class number less than 40 (and discriminant less than 10 million) in which 71 is inert. We remark that this uniue j-invariant is that of the curve with CM by Z[ ]. We should remark that ordering by class number is not ideal, in articular this aears to change the relative freuency of various congruence conditions. Heuristically this can be exlained for examle by noticing that if ramifies, this will tend to double the size of the class grou, whereas if slits this will tend to make it much larger. That is the slitting and ramification of small rimes tends to imact the size of the class grou as it is recisely these rimes which, by Minkowski theory, will be the generators. Moreover, adding factors of to the conductor will tyically double (or trile) the class number. The effect is that in the data which follows you should not try to comare data between columns without adjusting for the bias caused by the class number cutoffs. We should note, it is entirely ossible that the skew the class number ordering creates in the data is the only reason we were able to originally identify any of the underlying henomenon we have discussed. In articular, considering arity conditions on the class number is likely entirely unnatural. Aendix A.1. Data for = 71 This data is tyical for = 7 (mod 8), the choices of 5 and 7 are arbitrary but demonstrate contrasting behavior. Maximal Orders With Odd Class Number Inert at = 71 subject to arity condition on class number. Odd Class Number Even Class Number x x x x x x x x (x + 5) (x + 3) (x + 30) (x + 31) (x + 47) (x + 54) All Maximal Orders Innert at = 71 subject to conditions on discriminants. 11

12 All D 4 D 8 D 5 D 5 D 7 D 7 D x x x x x x x x (x + 5) (x + 3) (x + 30) (x + 31) (x + 47) (x + 54) All Orders Inert at = 71 subject to conditions on discriminants/conductors. All D D D D D 4 D 4 D 4 D 4 D 4 D 8 D 8 D 8 D 8 D 8 D 7 Df 7 Df All f f 4 f 8 f 16 f f f 4 f 8 f 16 f f f 4 f 8 f 16 f x x x x x x x x (x + 5) (x + 3) (x + 30) (x + 31) (x + 47) (x + 54) Aendix A.. Data for = 59 This data is tyical for = 3 (mod 8). All Orders Inert at = 59 subject to conditions on discriminants/conductors. All D D D D D 4 D 4 D 4 D 4 D 4 D 8 D 8 D 8 D 8 D 8 D All f f 4 f 8 f 16 f f f 4 f 8 f 16 f f f 4 f 8 f 16 f x x x x x x (x + 11) (x + 1) (x + 31) (x + 4) (x + 44) x Aendix A.3. Data for = 41 This data is tyical for = 1 (mod 4). All Orders Inert at = 41 subject to conditions on discriminants/conductors. All D D D D D 4 D 4 D 4 D 4 D 4 D 8 D 8 D 8 D 8 D 8 D All f f 4 f 8 f 16 f f f 4 f 8 f 16 f f f 4 f 8 f 16 f x x x x x (x + 9) (x + 13) (x + 38) Aendix A.4. Key Observations About Data In all of the data sets the freuency with which the roots aear aears to be euidistributed subject only to rescaling those j invariants for which the curves have automorhisms. The linear terms do not follow the same distribution as the underlying roots. It is not immediately clear if we restrict to maximal orders if the linear terms are euidistributed overall, however each family based on ramification at aears to be, and if we reweigh (to correct bias caused by class number bounds) and regrou it is ossible that the result is euidistribution. Secifying ramification conditions can have an effect on the resence or absence of linear factors. 1

13 All families where we account for the behavior at in the discriminant and conductor aears to satisfy a simle distribution. Those j-invariants with automorhisms may or may not be rescaled deending on the case. More secifically j = 0 is never aarently rescaled, whereas j = 178 may or may not be deending on discriminant and conductor. Some j-invariants may be favored desite no extra automorhisms. For examle, j = 44 for = 59 when 4 D and does not divide f. For = 3 (mod 4) there is a artitioning of j-invariants into two sets (with j = 178 the common intersection) where the distribution selects for one set or the other based on the conditions on discriminants and conductors. For = 7 (mod 8), 4 D and f there is an aarent bijection between these two sets (excluding 178) where the freuencies will be identical between the two sets. Note: Within the data this actually haens on the level of individual orders. For = 1 (mod 4), 8 D and f there is an aarent bijection between two sets where the freuencies will be identical between the two sets. Note: Within the data this actually haens on the level of individual orders. Based on heuristic reasoning on the effect on class number of changing conductors by and the aarent atterns and euidistribution in families one can reasonably exect euidistribution of the linear terms in the limit if we consider all conductors in a given field. That is, we know the relative sizes of the excetional sets and the effect on class numbers of increasing conductors by factors of. References [CV05] C. Cornut and V. Vatsal, CM oints and uaternion algebras, Doc. Math. 10 (005), [CV07] Christohe Cornut and Vinayak Vatsal, Nontriviality of Rankin-Selberg L-functions and CM oints, L- functions and Galois reresentations, London Math. Soc. Lecture Note Ser., vol. 30, Cambridge Univ. Press, Cambridge, 007, [Deu41] Max Deuring, Die Tyen der Multilikatorenringe ellitischer Funktionenkörer, Abh. Math. Sem. Univ. Hamburg 14 (1941), no. 1, [Dor89] David R Dorman, Global orders in definite uaternion algebras as endomorhism rings for reduced cm ellitic curves, Théorie des nombres (Quebec, PQ, 1987) (1989), [Ibu8] Tomoyoshi Ibukiyama, On maximal orders of division uaternion algebras over the rational number field with certain otimal embeddings, Nagoya Math. J. 88 (198), [JK11] Dimitar Jetchev and Ben Kane, Euidistribution of Heegner oints and ternary uadratic forms, Math. Ann. 350 (011), no. 3, [Sch10] Reinhard Schertz, Comlex multilication, New Mathematical Monograhs, vol. 15, Cambridge University Press, Cambridge,