Torsion points on elliptic curves with complex multiplication
aa r X i v : . [ m a t h . N T ] J u l TORSION POINTS ON ELLIPTIC CURVES WITH COMPLEXMULTIPLICATION
PETE L. CLARK, BRIAN COOK, AND JAMES STANKEWICZ
Abstract.
We present seven theorems on the structure of prime order torsionpoints on CM elliptic curves defined over number fields. The first three resultsrefine bounds of Silverberg and Prasad-Yogananda by taking into account theclass number of the CM order and the splitting of the prime in the CM field. Inmany cases we can show that our refined bounds are optimal or asymptoticallyoptimal. We also derive asymptotic upper and lower bounds on the least degreeof a CM-point on X ( N ). Upon comparison to bounds for the least degree forwhich there exist infinitely many rational points on X ( N ), we deduce that,for sufficiently large N , X ( N ) will have a rational CM point of degree smallerthan the degrees of at least all but finitely many non-CM points. Introduction
Notation.
For d ∈ Z + , we define the following quantities: T ( d ): the supremum of the orders of the groups E ( K )[tors] as K ranges overall number fields of degree d and E ranges over all elliptic curves defined over K . N ( d ): the supremum of all orders of K -rational torsion points P ∈ E ( K ), with K and E varying as above. P ( d ): the supremum of all prime orders of K -rational torsion points P ∈ E ( K ),with K and E varying as above.We shall have occasion to consider analogues T ∗ ( d ), N ∗ ( d ), P ∗ ( d ) of the abovequantities, which are defined by restricting to some subset of elliptic curves E /K .Specifically we will be interested in the set of all elliptic curves with integral mod-ulus j ( E ) and also the set of all elliptic curves with complex multiplication.1.2. Background on torsion.
Since the torsion subgroup of an elliptic curve over a number field is a finite abeliangroup with at most two generators, we have(1) P ( d ) ≤ N ( d ) ≤ T ( d ) ≤ N ( d ) . The uniform boundedness theorem of L. Merel [Mer96] asserts T ( d ) < ∞ forall d ∈ Z + . Using (1), the finiteness of P ( d ) and N ( d ) follows immediately.Merel’s proof gives an explicit upper bound on T ( d ), which was then improved by work of Merel, Oesterl´e and Parent. For instance, Parent showed [Par99] that ifa power p a of a prime p > d number field, then p a ≤ d − d ) . However, it is a “folk conjecture” that there exists a constant α such that T ( d ) = O ( d α ): thus it seems that Merel’s bounds are a full exponential away from thetruth. In fact, we record here a more precise conjecture: Conjecture 1.
There is a C > such that T ( d ) ≤ C d log log d for all d ∈ Z + . Conjecture 1 is very close to being the most ambitious conceivable one: we shallshow (Theorem 6) that there is a positive constant C and a strictly increasingsequence { d n } ∞ n =1 of positive integers such that T ( d n ) > C d n √ log log d n for all n .Unfortunately it is not currently tenable to seek numerical confirmation for Con-jecture 1a). The only values of d for which any of T ( d ), N ( d ), P ( d ) are known are: T (1) = 16, N (1) = 12, P (1) = 7 ([Maz77]). T (2) = 24, N (2) = 18, P (2) = 13 ([Kam86], [Kam92], [KM88]). P (3) = 13 ([Par03]).Since further direct computation of these quantities is out of current reach, it seemsthat one must find some more tractable sub-problem and examine the extent towhich it is representative of the general case.One approach is to concentrate on the case of elliptic curves with algebraic inte-gral j -invariant (henceforth integral modulus ). In this case we write T IM ( d ) , N IM ( d ), P IM ( d ) for the order, exponent and largest prime dividing the order of an ellipticcurve E with integral modulus defined over any number field of degree d . For suchcurves the uniform boundedness is much easier to prove. Moreover, in the integralmodulus case the computation of all possible torsion subgroups over Q was done byG. Frey in 1977 [Fre77]. Analogous computations in higher degree are significantlymore difficult and have been the subject of several papers of H. Zimmer and hiscollaborators: the 1976 paper [Zim76] lays foundations by giving a generalization ofthe Lutz-Nagell restrictions on torsion points to arbitrary number fields; the 1989paper [MSZ89] enumerates the torsion subgroups of elliptic curves with integralmodulus over quadratic fields ( d = 2); special kinds of cubic fields ( d = 3) wereconsidered in 1990 [FSWZ90] and the case of a general cubic field was completedin 1997 [PWZ97]; only a very restricted class of quartic fields has ever been consid-ered, so already the case d = 4 seems to be out of reach.However, Hindry and Silverman have shown [HS99] that(2) ∀ d ∈ Z + , T IM ( d ) ≤ d log d, (3) ∀ d ≥ , T IM ( d ) ≤ d log d. Another idea is to search for all finite groups which arise as the torsion subgroupof infinitely many elliptic curves defined over number fields of degree d . In this ORSION POINTS ON ELLIPTIC CURVES WITH COMPLEX MULTIPLICATION 3 case the computations in degree up to d = 4 have been done by Jeon, Kim, Parkand Schweizer [JKS04], [JK06], [JKP06], and reasonably good asymptotic boundscan be obtained by applying theorems of Faltings and Abramovich. This work isdescribed in some detail below.In this paper we shall usually restrict to elliptic curves with complex multiplication.This is a very special subclass of the class of integral moduli curves, comprising foreach degree d only finitely many j-invariants (but infinitely many nonisomorphic twists for a given j -invariant). Accordingly, we are able to derive more preciseresults than in the general case. We also take up the task of relating the specialcase of CM points to the general case – not definitively, of course, but in a depthand level of detail which we feel deserves a place in the literature on the subject.1.3. Prior results.
Let F be a field of characteristic 0 and E /F an elliptic curve. We say that E has complex multiplication (henceforth CM ) if the ring End E of endomorphismsof E defined over an algebraic closure F of F is strictly larger than Z . In this case,End ( E ) := End( E ) ⊗ Z Q is an imaginary quadratic field Q ( √ D ) and End( E ) isan order in End ( E ).As alluded to above, we write T CM ( d ), N CM ( d ), P CM ( d ) for, respectively, thelargest order, exponent and prime dividing the order of any CM elliptic curve de-fined over any number field of degree d .The j -invariant of a CM elliptic curve is an algebraic integer [Sil94, Thm. II.6.1],so that (2), (3) we have E ( F )[tors] = O ( d log d ). If we restrict to the order of asingle torsion point – i.e., to N CM ( d ) rather than T CM ( d ) – we can do qualitativelybetter: one knows that N CM ( d ) = o ( d log d ). More precisely: Theorem. (Silverberg [Sbg88] , Prasad-Yogananda [PY01] ) Let F be a number fieldof degree d , and let E /F be an elliptic curve with complex multiplication by an order O in the imaginary quadratic field K . Let w = w ( O ) = O × (so w = 2 , or )and let e be the maximal order of an element of E ( F )[tors] . Then:a) ϕ ( e ) ≤ wd ( ϕ is Euler’s totient function).b) If F ⊇ K , then ϕ ( e ) ≤ w d .c) If F does not contain K , then ϕ ( E ( F )[tors]) ≤ wd . Applying the theorem necessitates separate consideration of three cases:Case 1: O = Z [ √− ], of discriminant −
3, which has w ( O ) = 6. We get(4) ϕ ( e ) ≤ d. Case 2: O = Z [ √− −
4, which has w ( O ) = 4. We get(5) ϕ ( e ) ≤ d. Case 3: For every other order we have w ( O ) = 2. We get(6) ϕ ( e ) ≤ d. Let us call (4), (5) and (6) the
SPY bounds . PETE L. CLARK, BRIAN COOK, AND JAMES STANKEWICZ
Recall the classical result ϕ ( N ) ≫ N log log N (e.g. [HW, Thm. 328]). From thisand the SPY bounds we deduce that there exists a constant C such that(7) N CM ( d ) ≤ Cd log log d. This improves upon what one gets by applying (2): N CM ( d ) ≤ N IM ( d ) ≤ T IM ( d ) ≤ d log d. Theorem 6 below asserts N CM ( d ) = o ( d √ log log d ), so that our understanding ofthe true lower order of magnitude of N CM ( d ) is rather good. On the other hand, itis vexing that we cannot get any improvement on T CM ( d ) ≤ T IM ( d ) = O ( d log d )by applying the methods of SPY, or indeed by any other means that we know.1.4. Computational results.
We briefly report on some calculations done by the University of Georgia Num-ber Theory VIGRE Research Group, which has implemented an algorithm (c.f.[Cla04]) to do the following: given a positive integer d , compute the complete listof isomorphism classes of finite abelian groups which arise as the full torsion sub-group of some CM elliptic curve with defined over any number field of degree d .This algorithm requires knowledge of the CM j-invariants (more precisely, theirminimal polynomials) of degree d ′ strictly dividing d , so in full generality requiresan enumeration of the set of imaginary quadratic fields with any given class number,i.e., an effective solution of the Gauss class number problem . Work of Watkins[Wat04] gives a solution to this problem up to class number 100, so the data from ibid. enable us, in theory, to run the algorithm for all degrees up to d = 201. But infact this is much more class number data than we have been able to use: one of thesteps in our algorithm is the computation of an explicit polynomial P N ( x, y ) = 0which (birationally) defines the modular curve X ( N ), a computation which be-came prohibitively expensive for us around N = 79. The complete list of possibletorsion subgroups of CM elliptic curves defined over any degree d number field hasbeen computed by our VIGRE research group for 1 ≤ d ≤
13 (but will be describedelsewhere). The case of d = 1 is a 1974 result of L. Olson [Ols74]. For d = 2 and 3the results are subsumed by the calculations of [MSZ89], [PWZ97]. To the best ofour knowledge the cases 4 ≤ d ≤
13 had not been computed before.Upon restriction from T CM ( d ) to P CM ( d ), the above problem can be rephrasedas follows: for a fixed d , find all prime numbers N such that the modular curve X ( N ) has a CM point of degree d . It is natural to consider also the following“converse problem”: for fixed prime N , find the smallest degree of a CM point on X ( N ). Our algorithm works equally well on this converse problem, and we presentthe solution, for all N ≤
79, in the following table: TABLE 1 N = 2: d = 1, D = − , − , − , − , − , − , − N = 3: d = 1, D = − , − , − Some preliminary calculations were done by the first author. The calculations were recheckedand completed by Steve Lane, who also pointed out – several times – an error in the preliminarycalculations at N = 11, which turned out to be very interesting and significant. ORSION POINTS ON ELLIPTIC CURVES WITH COMPLEX MULTIPLICATION 5 N = 5: d = 2, D = − N = 7: d = 2, D = − N = 11: d = 5, D = − N = 13: d = 4, D = − N = 17: d = 8, D = − N = 19: d = 6, D = − N = 23: d = 22, D = − , − , − , − , − , − N = 29: d = 14, D = − N = 31: d = 10, D = − N = 37: d = 12, D = − N = 41: d = 20, D = − N = 43: d = 14, D = − N = 47: d = 46, D = − , − , − , − , − N = 53: d = 26, D = − N = 59: d = 58, D = − , − , − , − N = 61: d = 20, D = − N = 67: d = 22, D = − N = 71: d = 70, D = − , − , − , − , − N = 73: d = 24, D = − N = 79: d = 26, D = − h ( D ) as well as giving a much largerlower bound in case ( DN ) = −
1. Theorem 3 gives conditions under which one getsan extra factor of 2 in the SPY-type bounds. Moreover, for N sufficiently largecompared to D , the bounds of Theorem 3 are optimal.1.5. Theoretical results I: Optimal bounds on prime order torsion points.Theorem 1. a) For every prime N ≡ , there exists an elliptic curve E over a numberfield K of degree N − , with j ( E ) = 0 , and with a K -rational N -torsion point.b) There exists an absolute constant N such that for all primes N ≥ N :(i) if X ( N ) has a CM point of degree d , then d ≥ N − ;(ii) if X ( N ) has a CM point of degree d < N − then d = N − and j ( E ) = 0 . Remark 1.1: The data suggests that it may be possible to take N = 5. Theorem 2.
Let O K be the maximal order in K = Q ( √ D ) , F a number field, and E /F an elliptic curve with O K multiplication. Let w ( K ) = O × K . Suppose that E ( F )[tors] contains an element of odd prime order N . Define δ ( F, K ) to be if K is contained in F and otherwise.a) ( DN ) = 1 , then ( N − · δ ( F, K ) h ( K ) w ( K ) | [ F : Q ] . b) If ( DN ) = 0 , then ( N − · (3 − δ ( F, K )) h ( K ) w ( K ) | [ F : Q ] . PETE L. CLARK, BRIAN COOK, AND JAMES STANKEWICZ c) If ( DN ) = − , then ( N − · h ( K ) w ( K ) | [ F : Q ] . It is interesting to compare this with the SPY-bounds. Our Theorem 2 is morespecial in that it only applies to the case of torsion points of odd prime order(although we believe the methods should generalize to arbitrary N ). In the case ofprime N , it does not strengthen the SPY-bound – indeed, both bounds agree in thecase when N | D , but it significantly refines the SPY-bounds, making clear thatthey are in some sense a “worst case scenario.” Theorem 3.
Let O be an order in the field K = Q ( √ D ) , w ( O ) be the cardinalityof its unit group and h ( O ) = O ) its class number. Then:a) For every odd prime N which splits in K , there exists an O -CM elliptic curvedefined over a number field of degree N − · h ( O ) w ( O ) with a rational N -torsion point.b) There is an N = N ( D ) such that for N ≥ N , the least degree of an O ( D ) -CMpoint on X ( N ) is N − · h ( O ) w ( O ) if N splits in K and (cid:0) N − (cid:1) h ( O ) w ( O ) otherwise. Remark 1.2: Taking O to be the quadratic order of discriminant − Theoretical results II: CM points of small degree on X ( N ) . Throughout this section N denotes a prime number different from 2 and 3.Define d CM ( N ) to be the least degree of a CM point on X ( N ).Theorem 1 shows that the smallest (resp. second smallest) possible degree of aCM point on X ( N ) is N − (resp. N − ), and shows that this degree can be at-tained iff N ≡ N ≡ N ranges overall primes N which are not
11 (mod 12), the least degree of a CM point on X ( N )is linear in N . Notice that the excluded set of primes N ≡
11 (mod 12) has den-sity in the set of all primes. By Theorem 2, the problem of bounding the upperorder of d CM ( N ) as N ranges over prime numbers, comes down to finding, for agiven prime N , an imaginary quadratic field Q ( √ D ) such that ( DN ) = − h ( D ) as small as possible. By applying what is known about theseelementary – but difficult! – analytic problems, we arrive at the following result. Theorem 4. a) For any ǫ > , there exists C ǫ such that for any prime N , the curve X ( N ) has a CM point of degree at most C ǫ N c/ ǫ , where c/ e − ≈ . .b) Assuming the Generalized Riemann Hypothesis (GRH), the least degree of a CMpoint on X ( N ) is O ( N log N log log N ) . However, d CM ( N ) is not bounded by a linear function of N . Theorem 5.
For any
C > , there is a positive density set P of prime numberssuch that for all N ∈ P , the least degree of a CM point on X ( N ) exceeds CN . ORSION POINTS ON ELLIPTIC CURVES WITH COMPLEX MULTIPLICATION 7
Theorem 6. a) There exists
C > such that for any F/ Q with [ F : Q ] = d andany CM elliptic curve E /F , one has exp( E ( F )[tors]) ≤ Cd log log d .b) There exists a sequence F n of number fields, of degree d n = [ F n : Q ] tending toinfinity, and CM elliptic curves E n /F n such that exp( E n ( F n )[tors]) ≫ d n p log log d n . We have already seen that part a) is a consequence of the SPY bounds; we repeat ithere for the sake of parallelism. Neither is part b) very difficult: all in all Theorems4 and 5 seem to lie significantly deeper.1.7.
Theoretical results III: small degree points on X ( N ) : comparisonwith non-CM case. The overarching problem is to understand all points of degree d on the familyof modular curves X ( N ). Merel’s theorem asserts that for fixed d the set of allsuch points on X ( N ) is finite, so it is natural to enumerate this list. Conversely,one can fix N and ask for the least degree of a noncuspidal point on X ( N ). Inthe previous section we presented results giving rather tight estimates on the leastdegree of a noncuspidal CM point. Therefore the key issue is: how many non -CMpoints are there of small degree?The next result gives a precise sense in which d ≈ N is the threshold betweensmall degree and large degree: Theorem 7.
Let
N > be a prime number. Then:a) The set of points of X ( N ) of degree less than ⌈ ( N − ⌉ is finite. AssumingSelberg’s eigenvalue conjecture the bound can be improved to ⌈ ( N − ⌉ .b) The set of points of X ( N ) of degree at most N − N +1112 is infinite. Remark 1.3: The proof of part a) uses deep theorems of Faltings, Frey and Abramovich,but the deduction itself is now routine. Essentially the same result appears as[JKS04, Cor. 1.4], the only difference being that we get a sharper bound by restric-ing to prime N . Part b) is much more elementary. Nevertheless, it is in the spiritof this paper to pursue quantitative rather than just qualitative results, and in thisregard the fact that we can compute the “threshold” value of d sharply to withina factor of 32 seems interesting. For instance, it raises the question of whether thetruth lies closer to N or to N .Remark 1.4: Selberg’s eigenvalue conjecture states that for a modular curve Y (Γ) :=Γ \H associated to a congruence subgroup Γ ⊂ P SL ( Z ), the least positive eigen-value λ of the hyperbolic Laplacian on Y (Γ) satisfies λ ≥ . Selberg himselfshowed λ ≥ ; in 1994, Luo, Rudnick and Sarnak showed λ ≥ ; this thebound we use in our unconditional estimate. As of this writing, the best knownestimate on λ is due to Kim and Sarnark: λ ≥ > . N = 127 the least degree of a rational CM point is 42, whereas– assuming Selberg’s eigenvalue conjecture – the bound of Theorem 7a) gives that PETE L. CLARK, BRIAN COOK, AND JAMES STANKEWICZ there are only finitely many points (if any, of course!) on Y (127) of any smallerdegree. For all larger N ≡ N ≤
13. When N = 17 the boundensures infinitely many points of degree at most 8, and the table above shows thatthe least degree of a rational CM point is 8. But in fact there exists a degree 4map from X (17) to the projective line, so that there are infinitely many rationalpoints of degree at most 4. This suggests that there is room for improvement inthe bound of Theorem 7b).Write d CM ( N ) for the least degree of a CM point on X ( N ) and d ∞ ( N ) for the leastdegree d such that X ( N ) has infinitely many points of degree at most d . Then byTheorem 4, d CM ( N ) = O ( N . ... ) whereas d ∞ ( N ) ≥ ⌈ ( N − ⌉−
1. It followsthat there exists a prime N such that d CM ( N ) ≥ d ∞ ( N ) and d CM ( N ) < d ∞ ( N )for all N > N . In other words, for all sufficiently large primes, there are onlyfinitely many points on X ( N ) of degree smaller than that of any CM point.The prime N of the previous paragraph is effectively computable. Indeed, B.Cook and A. Rice are engaged in such a computation. Their preliminary workshows that one can take – unconditionally – N = 5 . × . This N is smallenough to allow case-by-case analysis, and we believe that the final result will bemore like N ≈ Dramatis Personae and Acknowledgments.
The 2007-2008 UGA VIGRE research group in number theory included:Group leaders (year long):Pete L. Clark (assistant professor), Patrick Corn (postdoc)Graduate students (year long):Steve Lane, Jim Stankewicz, Nathan Walters, Steve Winburn, Ben WyserGraduate students (spring semester only): Brian CookUndergraduate student (year long): Alex Rice.For a 21st century paper on elliptic curves, the theory we need here is relativelymiddlebrow and classical: most of the results we need go back, in some form, toDeuring or even Weber. Each of the individual results we use can be picked up bya hard-working second year graduate student, but to master them all in a limitedamount of time while doing research including substantial computer programmingis a taller order. Part of the goal of this project was indeed to foster learning bydoing, and we have aimed for an exposition which maximizes accessibility to thestudents in the seminar and other early career graduate students.Many of the participants were assigned specific subproblems which they wroteup formally and have been incorporated into this paper. Specifically, we wish toacknowledge the contributions of Steve Lane in computing Table 1, of Alex Rice in § .
4, of Jim Stankewicz in § . § ORSION POINTS ON ELLIPTIC CURVES WITH COMPLEX MULTIPLICATION 9 enlightening and stimulating experience; this paper represents a substantial ad-vancement of his prior work in this area, which would probably not have been donewere it not for the interest and involvement of the students.2.
Background on elliptic curves and complex multiplication
Some facts about elliptic curves with complex multiplication.
Let E be an elliptic curve over any field K . A K -rational endomorphism of E is amorphism of K -varieties ϕ : E → E such that ϕ ( O ) = O . Then ϕ induces anendomorphism (i.e., self-homomorphism) on the group E ( L ) of L -rational points,for any field extension L of K . By definition, the endomorphism ring of E isthe set of all K -rational endomorphisms of E , endowed with the structure of aring under pointwise addition and composition. As for any ring, there is a naturalhomomorphism ι : Z → End( E ), in which the image of n is the multiplication by n map on E , traditionally denoted [ n ].In all cases ϕ is an injection and End( E ), as an abelian group, is a free Z -moduleof rank 1, 2 or 4. When End( E ) has rank 4, the endomorphism ring is noncom-mutative, an order in a definite rational quaternion algebra. Such an elliptic curveis said to be supersingular ; supersingular elliptic curves over K exist iff K haspositive characteristic. So if K has characteristic 0, we have either End( E ) = Z ,or End( E ) ∼ = Z as a free abelian group; in the latter case End( E ) is isomorphic toan order O of an imaginary quadratic field Q ( √− n ), and “thus” we say that E has complex multiplication . More precisely, we say E has O -CM if End( E ) ∼ = O .Since the ring O has exactly one nontrivial automorphism – complex conjugation– if End( E ) ∼ = O , there are two such isomorphisms.Let D be a fundamental imaginary quadratic discriminant , i.e., the discrim-inant of the full ring of integer of some imaginary quadratic field. More concretely, D is a negative integer which is either (i) congruent to 1 (mod 4) and squarefree,or (ii) congruent to 0 (mod 4) and such that D is squarefree. Every imaginaryquadratic order O in Q ( √− D ) is of the form Z [ f τ n ] for a uniquely determined f ∈ Z + , the conductor of O . Thus an order is determined by its fundamentaldiscriminant D – the discriminant of the full ring of integers of O ⊗ Q – and f .On the other hand, an order is also determined by its discriminant D = f D .This means that for any imaginary quadratic discriminant D – i.e., an integer D with D < D ≡ , O ( D ) of discriminant D .For any integral domain R , one may consider its Picard group
Pic( R ), of rankone locally free R -modules under tensor product. Otherwise put, Pic( R ) is thequotient of the group of invertible fractional R -ideals by the subgroup of principal R -ideals. The class number h ( R ) is the cardinality of Pic( R ). For an arbitrarydomain R , the class number may well be infinite, but it is finite when R is an orderin any algebraic number field, so in particular when R = R ( n, d ) is an imaginaryquadratic order. When R is a Dedekind domain all nonzero fractional ideals areinvertible, and Pic( R ) = Cl( R ) is the usual ideal class group.We abbreviate h ( O ( D )) to h ( D ), and if K = Q ( D ) is an imaginary quadratic field, then the class number of K , denoted h ( K ), means the class number of themaximal order O K of K .Until further notice we fix an imaginary quadratic order O , of discriminant D ,and with quotient field K = Q ( √ D ). Fact 1. a) There exists at least one complex elliptic curve with O -CM.b) Let E , E ′ be any two complex elliptic curves with O -CM. The j -invariants j ( E ) and j ( E ′ ) are Galois conjugate algebraic integers. In other words, j ( E ) is a root ofsome monic polynomial with Z -coefficients, and if P ( t ) is the minimal such poly-nomial, P ( j ′ ( E )) = 0 also.c) Thus there is a unique irreducible, monic polynomial H D ( t ) ∈ Z [ t ] whose rootsare the j -invariants of the various non-isomorphic O -CM complex elliptic curves.d) The degree of H D ( t ) is the class number h ( O ) = h ( D ) of the order O , so when O is the full ring of integers of its quotient field K , deg( H D ( t )) = h ( K ) , the classnumber of K .e) Let F D := Q [ t ] /H D ( t ) . Then F D can be embedded in the real numbers, so in par-ticular is linearly disjoint from the imaginary quadratic field K . Let K D denote thecompositum of F D and K . Then K D /K is abelian, with Galois group canonicallyisomorphic to Pic( O ) . Moreover, K D / Q is Galois and the exact sequence → Gal( K D /K ) → Gal( K D / Q ) → Gal( K/ Q ) → splits, i.e., Gal( K D / Q ) is up to isomorphism the semidirect product of Pic( O ) withthe cylic group Z of order , where the map Z → Aut(Pic( O )) takes the nontrivialelement of Z to inversion: x x − . References for this fact include: Cox [Cox89] and Silverman II [Sil94].This fact has many implications. First, it follows that one can define an O -CMelliptic curve over a number field F iff F ⊃ F D . In particular, it follows that onecan define an O -CM elliptic curve over Q iff h ( D ) = 1, which by the Heegner-Baker-Stark theorem is known to occur for exactly 13 values of D : D = − , − , − , − , − , − , − , − , − , − , − , − , − . Let E : y = x + Ax + B be a complex elliptic curve in Weierstrass form. Wedefine a Weber function h on E , as: h ( x, y ) = x if AB = 0, h ( x, y ) = x if B = 0, h ( x, y ) = x if A = 0.(The point of the Weber function is to make explicit the quotient map E → E/ Aut( E ) ∼ = P . See [Sil94, Ch. II] for more details.)If E is defined over some subfield K of C , let K ( E [ N ]) be the field extensionof K obtained by adjoining the coordinates of all the N -torsion points on E .The following is a celebrated classical result. ORSION POINTS ON ELLIPTIC CURVES WITH COMPLEX MULTIPLICATION 11
Theorem 8. (Weber) Let D be an imaginary quadratic order, and E /F D an O K -CM elliptic curve. For any positive integer N , the field Q ( √− D, j ( E ) , h ( E [ N ])) isthe N -ray class field of K = Q ( √− D ) . Proof: See e.g. [Sil94, Thm. II.5.6].
Corollary 9.
Let K = Q ( √ D ) be an imaginary quadratic field, and let E /F ( D ) be an elliptic curve with O K -CM. Let N be an odd prime. Then [ Q ( p D , j ( E ) , h ( E [ N ])) : Q ( p D , j ( E ))] = (cid:18) N − w ( K ) (cid:19) (cid:18) N − (cid:18) D N (cid:19)(cid:19) . Proof: We deduce the corollary from the theorem using the description of the N -rayclass field K ( N ) of K provided by class field theory. Namely, consider the N - ringclass field L ( N ), a subextension of K ( N ) /K . Putting D = N · D , we haveGal( L ( N ) /K ) ∼ = Pic( O ( D )) , whereas Gal( K ( N ) /L ( N )) ∼ = ( Z /N Z ) × / ± . Recall the relative class number formula [Cox89, Thm. 7.24] h ( N D ) h ( D ) = N − (cid:0) D N (cid:1) [ O × K : O × ] , Thus [ Q ( p D , j ( E ) , h ( E [ N ]) : Q ( p D , j ( E ))] = [ K ( N ) : K (1)]= [ K ( N ) : K ][ K (1) : K ] = h ( N D )( N − h ( D ) = N − w ( K ) · (cid:18) N − (cid:18) D N (cid:19)(cid:19) . The Galois representation.
Let F be a field of characteristic 0, E /F anelliptic curve, and N a positive integer. Let σ ∈ Gal F = Aut( F /F ). Let E [ N ] bethe set of N -torsion points on E over F ; the action of Gal F is seen to be Z /N Z -linear, so E [ N ] may naturally be viewed as a Z /N Z [Gal F ]-module. Recall that,as a Z /N Z -module (or equivalently, as an abelian group), E [ N ] ∼ = Z /N Z × Z /N Z [Sil86]. It is notationally convenient to choose such an isomorphism – i.e., to choosean ordered Z /N Z -basis e , e of E [ N ]. The Z /N Z [Gal F ]-module structure is thengiven by a homomorphism ρ N : Gal F → GL ( Z /N Z ) , which we call the mod N Galois representation associated to E . Let M = F ( E [ N ]) be the field extension obtained by adjoining to F the x and y coordinatesof all the N -torsion points. Then the kernel of ρ N is nothing else than Gal( F /M ) =Gal M , so ρ N factors through to give an embedding ρ N : Gal( M/F ) ֒ → GL ( Z /N Z ) . There is “a piece” of ρ N which is well understood in all cases. Namely, composingwith the determinant map det : GL ( Z /N Z ) → ( Z /N Z ) × , we get a homomorphismdet( ρ N ) : Gal( M/F ) → ( Z /N Z ) × . This homomorphism evidently cuts out an abelian extension of F , so can be viewedas a “character” of the group Gal( M/F ). More precisely:
Theorem 10.
We have det( ρ N ) = χ N , where χ N is the mod N cyclotomiccharacter , defined as follows: χ N : Gal F → Gal( F ( ζ N ) /F ) → ( Z /N Z ) × , where σ ∈ Gal F σ ∈ Gal( F ( ζ N ) /F ) , an automorphism which is determined byits effect on a primitive N th root of unity: ζ N σ ( ζ N ) = ζ χ N ( σ ) N , for a uniquely determined element χ N ( σ ) ∈ Z /N Z × . Proof: See [Sil86, Ch. III].
Corollary 11.
We have det( ρ N (Gal F )) = 1 iff F contains the N th roots of unity. The following is a special case of an extremely important theorem of Serre:
Theorem 12. (Serre’s Open Image Theorem, non-CM Case [S72] ) Let E be anelliptic curve defined over a number field F , and suppose that E does not havecomplex multiplication.a) For all sufficiently large prime numbers ℓ , ρ ℓ : Gal F → GL ( Z /ℓ Z ) is surjective.b) There exists a fixed number B such that for all N ∈ Z + , ρ N ) := GL ( Z /N Z ) ρ N (Gal F ) ≤ B. In other words, part b) says the failure of all the maps ρ N to be surjective can bemeasured by a single finite quantity. Since GL ( Z /ℓ · · · ℓ r Z ) ∼ = GL ( Z /ℓ Z ) × · · · × GL ( Z /ℓ r Z ) , this in fact implies part a). Note also that we must allow some finite amount ofnonsurjectivity, because we are considering an elliptic curve E defined over anynumber field. So for instance, start with E over Q and take F = Q ( E [ N ]) to be theextension obtained by adjoining all the coordinates of the N -torsion points. Forthis E/F one tautologically has ρ N (Gal F ) = 1. Serre himself noted that there is noelliptic curve over Q for which all the mod N Galois representations are surjective.2.3.
Galois representation in the CM case.
Our interest here is in the fact that this result fails in the presence of CM.We assume that N is an odd prime .Suppose first that E/F is a O ( D )-CM elliptic curve and that F contains the CMfield K = Q ( √ D ), so that the action of O ( D ) is defined and rational over F . Then,in additional to its Z /N Z [Gal F ]-module structure, E [ N ] also has the structure ofa O -module. Morever, the F -rationality of the endomorphisms means preciselythat for all σ ∈ Gal F and ϕ ∈ O ( D ), we have σϕ = ϕσ , i.e., the two actionscommute with each other. In fact, since N = 0 in E [ N ], E [ N ] is naturally a O ( D ) ⊗ Z /N Z = O ( D ) /N O ( D )-module. This can be expressed more concisely as the fact that E [ N ] is a ( Z /N Z [Gal F ] , O ( D ))-bimodule,but for our purposes there is no particular advantage to using this terminology. ORSION POINTS ON ELLIPTIC CURVES WITH COMPLEX MULTIPLICATION 13
Lemma 13. ( [Pari89, Lemma 1] ) The N -torsion group E [ N ] is free of rank asa (right) O ( D ) ⊗ Z /N Z -module, i.e., isomorphic to O ( D ) ⊗ Z /N Z itself. In particular, the natural Z /N Z -linear action of O ( D ) ⊗ Z /N Z on E [ N ] is faithful,so we have an embedding of Z /N Z -algebras ι : O ( D ) ⊗ Z /N Z ֒ → End( E [ N ]) ∼ = M ( Z /N Z ) . Let us denote the image of ι by C N . Now, for any σ ∈ Gal F , the matrix ρ N ( σ ) givesan invertible O ( D ) ⊗ Z /N Z -linear map of E [ N ]. Since the O ( D ) ⊗ Z /N Z -linearendomorphisms of the free one-dimensional module E [ N ] are precisely multiplica-tion by an element of O ( D ) ⊗ Z /N Z and the invertible ones are elements of theunit group of this ring, we conclude ρ N (Gal F ) ⊂ C × N . This shows that the CM case is much different, because the Galois extension F ( E [ N ]) /F is in this case abelian and has size at most C × N , or approximately N , whereas Serre’s theorem asserts that in the non-CM case ρ N (Gal F ) has, forsufficiently large prime N , size ( Z /N Z ) = ( N − N − N ) ∼ N .To give more precise results, we must consider separately whether N splits, staysinert or ramifies in O ( D ).Case 1 (split case): ( DN ) = 1. Then one sees (e.g. by direct computation) that C N , as a F N -algebra, is isomorphic to F N ⊕ F N ; therefore the unit group C × N isisomorphic to ( Z /N Z ) × ⊕ ( Z /N Z ) × . Thus there are precisely two one-dimensionalsubspaces V , V of E [ N ] which are simultaneous eigenspaces for C N . By takinggenerators e of V and e of V as basis, we get C N ∼ = { (cid:20) a b (cid:21) a, b ∈ F N } . The same considerations show that there is, up to conjugacy, a unique subalgebraof M ( F N ) isomorphic to F N ⊕ F N ; such an algebra is called a split Cartan sub-algebra and its unit group a split Cartan subgroup .Case 2 (inert case): ( DN ) = −
1. Then one sees that C N ∼ = F N , a finite fieldof order N , so that C × N is cyclic of order N −
1. Again ones sees that F N isunique up to conjugacy as a subalgebra of M ( F N ) (e.g. the result is a special caseof the Skolem-Noether theorem on simple subalgebras of central simple algebras;or just do a direct computation). Such an algebra is called a nonsplit Cartansubalgebra and the unit group is called a nonsplit Cartan subgroup .Case 3 (ramified case): N divides D . Then C N ∼ = F N [ t ] / ( t ), i.e., is generatedover the center (the scalar matrices) by a single nilpotent matrix g . Since theeigenvalues of g are F N -rational, we can put g in Jordan canonical form, and thisgives a choice of basis such that C N ∼ = { (cid:20) a b a (cid:21) a, b ∈ F N } . Again C N is unique up to conjugacy; for lack of a better name, we shall call it a pseudo-Cartan subalgebra . Evidently C N ∼ = Z N − ⊕ Z N ∼ = Z N − N . We now introduce a third operator on E [ N ]: by Fact 1 above, we can choosean embedding of K into C which carries Q ( j D ) into the real numbers. With thisunderstanding, complex conjugation c induces an F N -linear automorphism of E [ N ]. Lemma 14.
Let N be an odd positive integer. The characteristic polynomial ofcomplex conjugation acting on the free -dimensional Z /N Z -module E [ N ] is t − . Proof: Clearly c satisfies the polynomial t −
1, so what we must show is that c = ±
1. If c = 1 then c acts trivially on each N -torsion point and we would havedim Z /N Z E [ N ]( R ) = 2. If c = − N is odd), c acts nontrivially on each N -torsion point, and we would have dim E [ N ]( R ) = 0. But it is easy to see thatthe correct answer is dim E [ N ]( R ) = 1: indeed, a little thought shows that the one-dimensional compact real Lie group E ( R ) is isomorphic either to S (if a definingWeierstrass cubic has one real root) or to S × Z / Z (if all 2-torsion points (if adefining Weierstrass cubic has three real roots), and either way E [ N ]( R ) ∼ = Z /N Z . Lemma 15. ( [S67] , [S66] ) Let E/ Q ( j D ) be an O ( D ) -CM elliptic curve, and let σ be the nonidentity element of Aut( Q ( j D , √ D ) / Q ( j D )) .a) As operators on E [ N ] , we have σ = c .b) Therefore Q ( j D , E [ N ]) contains Q ( √ D ) . There is also a natural nontrivial action of complex conjugation on O ( D ), and thehomomorphism ι : O ( D ) → End( E [ N ]) is c -equivariant: ι ◦ c = c ◦ ι . This, togetherwith the nontriviality of the c -action on O ( D ), is equivalent to the fact that conju-gation by c stabilizes C N and induces a nontrivial involution on it.In the split case we find that, with respect to the chosen basis e , e of C N -eigenspaces, c is equal to either permutation matrix (cid:20) (cid:21) or its negative. Eitherway, the effect of conjugation by c is (cid:20) a b (cid:21) (cid:20) b a (cid:21) . Explicit computationshows that the Cartan subgroup C × N has index 2 in its normalizer N ( C × N ).In the inert case, conjugation by c stablizes C N ∼ = F N and induces the uniquenontrivial Galois automorphism, the Frobenius map: Frob N : x x N . The el-ements of N ( C × N ) \ C × N correspond to Frob N -semilinear automorphisms of the 1-dimensional F N -vector space V = E [ N ], i.e., maps σ : V → V such that for v, w ∈ V , σ ( vw ) = Frob N ( v ) σ ( w ). Such a map is uniquely specified by σ (1), sothat N ( C × N ) \ C × N = N −
1, i.e., [ N ( C × N ) : C × N ] = 2.In the ramified case, complex conjugation induces a nontrivial involution of the(non-semisimple) F N -algebra C N ∼ = F N [ t ] / ( t ). The automorphism group Aut( C N / F N )is isomorphic to ∼ = F × N − so has a unique element of order 2, t
7→ − t . Thereforeconjugation by c has the effect (cid:20) a b a (cid:21) (cid:20) a − b a (cid:21) . Note that this case isdifferent from the previous two in that the normalizer of C × N is the entire Borelsubgroup { (cid:20) a b c (cid:21) | a, b, c ∈ F N , ac = 0 } .Given all this information, one readily deduces the following result: ORSION POINTS ON ELLIPTIC CURVES WITH COMPLEX MULTIPLICATION 15
Theorem 16.
Let F be a number field, and E /F an elliptic curve with O ( D ) -CM. Let M = F ( E [ N ]) be the field extension of F obtained by adjoining x and y coordinates of all the N -torsion points of E .a) The CM field K = Q ( √− D ) is contained in M , so we get a short exact sequence (8) 1 → Gal(
M/KF ) → Gal(
M/F ) → Gal(
KF/F ) → . b) Under the natural embedding ρ N : Gal( M/F ) ֒ → GL ( F N ) , the subgroup Gal(
M/KF ) embeds in the unit group C × N .c) The sequence (8) splits, with a splitting given by a choice of an involution c ∈ N ( C × ) \ C × . This result gives upper bounds on the the degree [ F ( E [ N ]) : F ] which improveupon the obvious bound of ( F N ): Corollary 17. a) If ( DN ) = 1 , then [ F ( E [ N ]) : F ] | N − .b) If ( DN ) = − , then [ F ( E [ N ]) : F ] | N − .c) If ( DN ) = 0 , then [ F ( E [ N ]) : F ] | N − N ) . Proof: Using the exact sequence (8) we see that
M/F ) =
M/KF ) · K/F ) | C N ) × · . And we know that C × N has order ( N − , ( N −
1) or N − N according to whether N splits, is inert, or is ramified in O ( D ).The slogan here is that the image of the Galois representation ρ N should be “aslarge as possible”, up to a factor which is uniformly bounded as N varies, but in theCM case GL ( F N ) is impossibly large. The correct answer is again due to Serre: Theorem 18. (Open Image Theorem, CM case [S66] ): Let F be a number fieldand E /F be an elliptic curve with O -CM. Then for all sufficiently large primes N ,we have: • ρ N (Gal F )) = N ( C N ) , if K = Q ( √− D ) is not contained in F , • ρ N (Gal F ) = C × N , if K ⊂ F . Since Serre’s theorem only holds for sufficiently large primes N , the case of N | D can be completely ignored. Nevertheless Theorem 18 tells us to “expect” that the N -torsion fields will be as large as possible. In the next section we use elementarygroup theory to deduce consequence for the least degree of an N -torsion point.2.4. Orbits under C × N and applications. We maintain the notation of the previous section: E /F is an elliptic curve with O ( D )-CM; N is an odd prime number; C N = ι ( O ⊗ Z /N Z ) ⊂ End( E [ N ]); C × N isthe unit group of C N ; N ( C × N ) is the normalizer. Lemma 19. a) The orbits of C × N on E [ N ] \ { } are as follows:(i) If ( DN ) = 1 , the two one-dimensional eigenspaces for C N give two orbits of size N − ; all the remaining points lie in a single orbit of size ( N − .(ii) If ( DN ) = − , E [ N ] \ { } forms a single C × N -orbit.(iii) If ( DN ) = 0 , the unique one-dimensional eigenspace for C N gives an orbit ofsize N − ; the remaining points form a single orbit of size N − N .b) If ( DN ) = 1 , the two orbits of size N − for C × N form a single orbit for N ( C × N ) . Proof: A pleasant elementary computation that we leave to the reader.In the statement of the following result we employ the following convention: if p and q are nonzero rational numbers, we say p | q if qp ∈ Z . Corollary 20.
Let E /F be an O ( D ) -CM elliptic curve defined over a number field F . Suppose that the image ρ N (Gal KF ) of the mod N Galois representation hasindex I in C × N . Let P ∈ E ( C ) be any point of exact order N , and let F ( P ) be theextension of F obtained by adjoining the coordinates of P .(i) If ( DN ) = 1 and √ D ∈ F , then I ( N − | [ F ( P ) : F ] (ii) If ( DN ) = 1 and √ D is not in F , then I | [ F ( P ) : F ] .(iii) If ( DN ) = − , then I ( N − | [ F ( P ) : F ] .(iv) If ( DN ) = 0 , then I ( N − | [ F ( P ) : F ] . Proof: Consider of field extensions F ⊂ F ( P ) ⊂ F ( E [ N ]). Then F ( E [ N ]) /F ( P )is Galois, with Galois group canonically isomorphic to ρ N (Gal F ) ∩ G ( P ), where G ( P ) ⊂ GL ( F N ) is the stabilizer of the point P . By the orbit-stabilizer theorem,[ F ( P ) : F ] is equal to the orbit of P under the action of Gal F .In case (i) we have √ D ∈ F , so that the image of Galois lies in the split Cartansubgroup C × N ∼ = F × N ⊕ F × N . By Lemma 19 the full C × N -orbits have sizes N − N − . Since we are assuming that [ C × N : ρ N (Gal F )] | I , it follows that every ρ N (Gal F )-orbit has size a multiple of N − I . Case (ii) is similar except in this casereplace the gcd of all sizes of C × N orbits with the gcd of all sizes of N ( C × N )-orbits,which according to Lemma 19 is 2( N − √ D lies in the ground field F : in case (iii) thisis because the orbit size for C × N is already as large as possible; in case (iv) this isbecause the minimal C × N -orbit is stable under complex conjugation.3. Proof of Theorem 1
As in Remark 1.2, Theorem 1a) is precisely the D = − D = − w ( D ) = 6, and an odd prime splits completely in Q ( √−
3) iff N ≡ O ( D )-CM point on X ( N ) of degree D . If D = − N is greater than or equal to some absolute con-stant N , we have d ≥ N − if N ≡ d ≥ N − if N ≡ − D = −
4, so w ( D ) = 4, and then Theorem 3 says that for N greater than or equal to another absolute constant N , we have d ≥ N − if N ≡ d ≥ N − if N ≡ − D , so w ( D ) = 2 and then by Theorem 1.3, d ≥ N − .Altogether we see that if N ≥ max(5 , N , N ) then d ≥ N − in all cases, equalitycan be met iff N ≡ O ( − j -invariant 0), and the next smallest possible degree is N − , for an O ( − j -invariant 1728. This completes the proof of Theorem 1. ORSION POINTS ON ELLIPTIC CURVES WITH COMPLEX MULTIPLICATION 17 Proof of Theorem 2
Let N be an odd prime number; let K = Q ( √ D ) be an imaginary quadraticfield; and let E /F be an O K -CM elliptic curve. Suppose that there exists a point P ∈ E ( F ) of order N . Let M be the compositum of the CM field K with the N -torsion field F ( E [ N ]). We know that K ( j ( E )) = K (1) is the Hilbert class fieldof K and M/K ( N ) is abelian of degree divisible by N − w ( K ) · ( N − (cid:0) D N (cid:1) ) by Corollary 9.Split case: ( D N ) = 1. We know that Gal( M/K (1)) is contained in a split Car-tan subgroup C ( N ) ∼ = ( Z /N − Z ) of GL ( Z /N Z ) with index dividing w ( K ). Ifwe had equality – i.e., [ M : K (1)] = ( N − – then by the work of the previoussection, for any N -torsion point P ∈ E ( C ) we must have N − | [ K (1)( P ) : K ].Moreover, as we saw above, passing to a subgroup of index i cuts down this degreeby at most a factor of i , so N − w ( K ) | [ K (1)( P ) : K (1)] , and therefore h ( K ) w ( K ) · ( N − | [ K (1)( P ) : K ] | [ KF : K ] . Since δ ( F, K ) · [ KF : K ] = [ F : Q ] , Theorem 2a) follows.Ramified case: ( D N ) = 0. In this case Gal( M/K (1)) is contained in a pseudo-Cartan subgroup C × N ∼ = Z /N Z × Z /N − ( Z /N Z ) with index dividing w ( K ).As above, the smallest orbit of C × N on the N -torsion has size N −
1, leading to thebound h ( K ) w ( K ) · ( N − | [ F : Q ] . In this case, Gal(
KF/K ) acts trivially on the unique N -torsion subgroup stabilizedby C × N . From this, one sees that we gain an extra factor of 2 iff K does contain F ,giving the divisibility relation as in Theorem 2b).Inert case: ( D N ) = −
1. In this case Gal(
M/K (1)) is contained in a nonsplitCartan subgroup C × N ∼ = ( Z / ( N − Z ) of GL ( Z /N Z ) with index dividing w ( K ).As above, the N -torsion points form a single orbit under C × N , so arguing as in thesplit case we get h ( K ) w ( K ) · ( N − | [ F : Q ] . This completes the proof of Theorem 2.5.
Proof of Theorem 3
A technical lemma.
Let w be a positive even integer, and let ζ = ζ w = e πi/w be a primitive w throot of unity. Let G = h s | s w = 1 i be a cyclic group of order w . Let M be anabelian group endowed with the following additional structures: • a Z -linear action of G , and • A ring homomorphism Z [ ζ ] → End( M ).We require first that ζ w · x = − x for all x ∈ M . We also require that thesetwo actions commute with each other: for all x ∈ M , ζσx = σζx .For i ∈ Z /w Z , we define M i = { x ∈ M | σx = ζ i x } , and M = M i ∈ Z /w Z M i . Consider the Z -module homomorphism Φ : M → M given ( x i ) P i x i . Let˜Φ = Φ ⊗ Z Z [ w ] : M ′ = M ⊗ Z [ w ] → M ′ = M ⊗ Z [ w ]. Lemma 21.
Both ker(Φ) and coker(Φ) are w -torsion Z -modules. It follows that:a) The map ˜Φ is an isomorphism of Z [ w ] -modules.b) We have dim Q ( M ⊗ Q ) = dim Q ( M ⊗ Q ) , and for any prime p not dividing w , Φ induces an isomorphism from the p -primary torsion subgroup M [ p ∞ ] of M to the p -primary torsion subgroup M [ p ∞ ] of M . Proof: It is enough to show that the kernel and cokernel of Φ are w -torsion; for ifso, tensoring the short exact sequences0 → ker(Φ) → M Φ → Φ( M ) → → M / ker(Φ) Φ → M → coker(Φ) → Z -modules with the flat Z -module Z [ w ] shows that ˜Φ is an isomorphism.Step 1: We show ker(Φ) = ker(Φ)[ w ]. Let P = ( P , . . . , P w − ) be an elementof ker Φ, so that P + · · · + P w − = 0 . Applying σ , we obtain P + ζP + · · · + ζ w − P w − = 0 . Applying σ w − AP = 0, where A = . . . ζ . . . ζ w − ... ... . . . ...1 ζ w − . . . ζ ( w − w − . It is therefore also a solution to A P = 0, where A = w . . . . . .
00 0 . . . w ... ... . .. w w . . . . ORSION POINTS ON ELLIPTIC CURVES WITH COMPLEX MULTIPLICATION 19
Thus wP = wP w − = · · · = wP = 0, i.e., wP = 0.Step 2: We show coker(Φ) = coker(Φ)[ w ]. Let P ∈ M . Define a w × w matrix B = P σ ( P ) σ ( P ) . . . σ w − ( P ) P ζ − σ ( P ) ζ − σ ( P ) . . . ζ − ( w − σ w − ( P )... ... ... . . . ... P ζ − ( w − σ ( P ) ζ − w − σ ( P ) . . . ζ − ( w − w − σ w − ( P ) . Notice that the sum of all the entries of B is wP : indeed, this is the sum of theentries in the first column, and since for any j = 0 (mod w ) we have P w − i =1 ζ − ji = 0,each of the other columns sums to 0. Now for 1 ≤ i ≤ w , put P i − = w − X k =0 ζ − ki σ k ( P ) . Then σ ( P i ) = w − X k =0 ζ − ki σ k +1 ( P ) = ζ i w − X k =0 ζ − ( k +1) i σ k +1 ( P ) = ζ i P, so P i − ∈ M i . Therefore wP = Φ(( P , . . . , P w − )) ∈ Φ( M ) . This completes the proof of the lemma.5.2.
Application to the proof of Theorem 3.
Now let O be an imaginary quadratic order of discriminant D , K = Q ( √ D ), andlet N > w = w ( O ) be a prime which splits in K . Let K D = K ( j D ), and let E/K D be an O -CM elliptic curve. By the work of § .
3, we know that there exists an ex-tension K D ( P ) /K D , which is cyclic of degree dividing N −
1, such that over K D ( P ) E has a point P of exact order N . Let us first assume that [ K D ( P ) : K D ] = N − N − N ≡ w ). Therefore, by Galois theory, thereexists a unique subextension K D ⊂ L ⊂ K D ( P ) with G = Gal( K D ( P ) /L ) cyclic oforder w . Now we are in the setup of the previous section: take M = E ( K D ( P ));the G -action is the restriction of the natural Gal( K D /K D )-action on E ( K D ( P )),the Z [ ζ ]-action comes from the fact that O = End( E ) contains the w th roots ofunity, and the compatibility of these two actions is a consequence of the rationalityof the endomorphisms over K D (hence also over L ). Since E ( K D ( P )) contains apoint whose order is a prime N not divisible by w , by Lemma 21 there exists some i ∈ Z /w Z such that M i contains an element of order N .Using the theory of twisting in the Galois cohomology of elliptic curves, we mayinterpret M i as the group of L -rational points on a K D ( P ) /L -twisted form of theelliptic curve E . Specifically, the set of such twisted forms are parameterized by H (Gal( K D ( P ) /L ) , Aut( E )) = Hom( G, Z /w Z ) ∼ = Z /w Z , the last isomorphism being given by( ϕ : G → Z /w Z ) ζ i ϕ = ϕ ( σ ) . Corresponding to ζ i = ζ i ϕ ∈ Z /w Z we build a twisted Gal( K D ( P ) /L )-action on E ( K D ( P )): σ · i x := ζ − i σx. This is exactly the relation defining M i . In other words, the abstract decompositionof the Z [ w ]-module M ′ ∼ → M ′ corresponds to a decomposition of the Mordell-Weilgroup – up to w -torsion – of E ( K D ( P )) into a direct sum of the Mordell-Weil groupsof the w different twists of E /L via the cyclic extension K D ( P ) /L and the automor-phism group of E . (When w = 2, this result – decomposition of the Mordell-Weilgroup under a quadratic extension – is very well known.) Thus we have producedan O -CM elliptic curve over a field of degree N − w ( O ) with a rational N -torsion point,giving the statement of Theorem 3a).It remains to deal with the case in which d = [ K D ( P ) : K ] strictly divides N − w | d , we can run through the above argument verbatim, getting in fact an O -CMelliptic curve with a rational N -torsion point over a field of degree dw , which is apriori stronger than what we are trying to prove. This necessarily is the case if w = 2. If w = 4 and d is a multiple of 2 but not a multiple of 4, we run through theabove argument using quadratic twists instead of quartic twists. If w = 6 and d isa multiple of 2 but not of 6, then we run through the above using quadratic twistsinstead of sextic twists. One sees easily that we get exactly the same bounds. Thiscompletes the proof of Theorem 3a).Proof of Theorem 3b): Suppose first that N is an odd prime with ( DN ) = 1. Let F D = Q ( j ( E )) = Q ( j D ) be the number field generated by the j-invariant of the qua-dratic order O ( D ), and let E /F D be any O ( D )-CM elliptic curve. Serre’s Theorem18 says that there exists N = N ( D ) such that if N ≥ N , the image ρ N (Gal F D )in GL ( F N ) will be N ( C × N ), the normalizer of a split Cartan subgroup, and thenCorollary 20 applies to show that the least degree [ F D ( P ) : F D ] is a multiple of2( N − F , E ′ /F an O ( D )-CM elliptic curvewith an F -rational point of prime order N ≥ N . The theory of twisting – to-gether with the Kummer isomorphism H (Gal F , µ d ) ∼ = F × /F × d – implies firstthat F ⊃ F D , and second that there exists an extension L of F , of degree w ( O )such that E /L ∼ = E ′ /L . Therefore, since E ′ has an F -rational torsion point of order N , E has an L -rational torsion point of order N , so2( N − | [ F D : Q ] | [ L : F D ][ F D : Q ] = [ L : Q ] = [ L : F ][ F : Q ] = w ( O )[ F : Q ] , and hence 2( N − w ( O ) | [ F : Q ] . The argument in the case ( DN ) = − N such that N ≥ N implies that, for our fixed E /F D as above we have [ F D ( P ) : F D ] = N − P in all of GL ( F N ), hence thelargest possible order, so there is no further contribution coming from the action ofcomplex conjugation) and arguing as before we get N − w ( O ) | [ F : Q ] . ORSION POINTS ON ELLIPTIC CURVES WITH COMPLEX MULTIPLICATION 21
Since we are taking N arbitrarily large compared to D , we do not have to worryabout the ramified case. 6. Proof of Theorem 4
For a negative quadratic discriminant D , write d D ( N ) for the least degree of an O ( D )-CM point on X ( N ), and d CM ( N ) for the least degree of a CM point on X ( N ), so d CM ( N ) = min D d D ( N ).We will need the following two estimates: Lemma 22.
Suppose D is a positive integer and N a prime, with ( − DN ) = 1 . Thenthere exists a CM point on X ( N ) of degree dividing N − h ( Q ( √− D )) . Proof of Lemma 22: this is an immediate consequence of the theory of Galoisrepresentations on CM elliptic curves as recalled in § . Lemma 23. As D tends to −∞ through quadratic discriminants (i.e., D ≡ , ), the class number h ( D ) of the imaginary order of discriminant D is O ( √ D log D ) . Proof: A consequence of Dirichlet’s class number formula; see e.g. [Coh07, § N ≡ d CM ( N ) ≤ N − .This is stronger than the bounds we are claiming for arbitrary N , so we may assumethat N ≡ − N , let D be a negative quadratic discriminant not divisible by N . Then1 = (cid:18) DN (cid:19) ⇐⇒ (cid:18) | D | N (cid:19) = − , so we are interested in the least positive integer M which is first, a quadratic non-residue modulo N and second, is congruent to 0 or − − M is animaginary quadratic discriminant.In fact this latter condition is nothing to worry about: let M be the least posi-tive quadratic nonresidue modulo N . Then certainly M is squarefree, so M is not0 (mod 4). If M ≡ − D = − M is the discriminant of Q ( √− M ).If M ≡ , − M but − M which is the discriminant of Q ( √− M ). But if M is a quadratic nonresidue modulo the odd prime N , so is 4 M ,and if we know that M = O ( f ( N )) for some function f , then of course the sameholds for 4 M .So what is the order of the least quadratic nonresidue modulo N ? This is a fa-mous classical problem. The trivial bound – taking into account only that there arein all N − quadratic nonresidues – is N , but a bit of thought and experimentationsuggests that M should be considerably smaller than this. Long ago Vinogradovconjectured that M = O ǫ ( N ǫ ), i.e., that M grows more slowly than any power of N ,but we are still far away from an unconditional proof of this. In 1952 N.C. Ankenyshowed that, conditionally on GRH, M = O ((log N ) ) [Ank52]. In his review ofthis paper [Erd52], P. Erd¨os remarks that it is known that M is not O (log N ), sothat Ankeny’s bound seems to get admirably close to the truth. Vinogradov himselfwas able to show unconditionally that M = o ( N ); for more than fifty years, thebest unconditional bound has been due to D.A. Burgess: M = O ǫ ( N c + ǫ ), where c = e − / = 0 . . . . is “Burgess’ constant” [Bur57].So, for a large prime N , let M be the least quadratic nonresidue modulo N and D = − M if M ≡ − D = − M otherwise. Applying Lemma 22 andthen Lemma 23, we get d CM ( N ) = O ( N h ( D )) = O ( N p | D | log | D | ) . Substituting in the unconditional Burgess bound for D , we get d CM ( N ) = O ǫ ( N c/ ǫ/ log( N c + ǫ )) . That this bound hold for all ǫ > d CM ( N ) = O ǫ ( N c/ ǫ ) . Applying instead Ankeny’s bound, we get, conditionally on GRH, d CM ( N ) = O ( N p (log N ) log(log N ) = O ( N log N log log N ) . Proof of Theorem 5
Although not necessary from a logical point of view, we believe it will make foreasier reading if we discuss first the special case in which the endomorphism ring isthe maximal order and second the (less) special case in which the conductor of theorder is prime to N before discussing the general case. Case 1: fundamental discriminants
Suppose that there exists some positivenumber C such that for every odd prime N , there exists a point on X ( N ) withCM by the full ring of integers of some imaginary quadratic field, and of degree atmost CN . We will derive a contradiction. H := 6 C + 1. Recall that the set of negative quadratic discriminants D suchthat h ( D ) ≤ H is finite [Deu33], [Hei34], [Sie35]. Let us write out this set as { D , . . . , D n } .Let P be the set of primes which are 1 (mod 4) and divide D k for some1 ≤ k ≤ n . Put R = P . Similarly, let P be the set of primes which are 3(mod 4) and divide some D k . Put S = P . Lemma 24.
The set P H of odd primes N such that { ( DN ) = − ∀ D | h ( D ) ≤ H } is infinite; indeed it has density at least ( ) R + S +2 . Proof: Let N be any prime number satisfying:(i) N ≡ Np ) = 1 for all p ∈ P .(iii) ( Nq ) = − q ∈ P .By the Cebotarev density theorem (or even the quantitative version of Dirichlet’stheorem on primes in arithmetic progressions), the set of such primes N has density( ) R + S +2 . We claim that all such primes lie in P H . Indeed, we may write D k = ( − · a +2 b p · · · p r q · · · q s = ( − s +1 a +2 b r Y i =1 p i s Y j =1 ( − q j ) , ORSION POINTS ON ELLIPTIC CURVES WITH COMPLEX MULTIPLICATION 23 where a, b ∈ { , } , p i ∈ P and q j ∈ P . Then (cid:18) D k N (cid:19) = (cid:18) − N (cid:19) s +1 (cid:18) N (cid:19) a +2 b r Y i =1 (cid:16) p i N (cid:17) s Y j =1 (cid:18) − q j N (cid:19) =( − s +1 · · r Y i =1 (cid:18) Np i (cid:19) · s Y j =1 (cid:18) Nq j (cid:19) = ( − s +1 ( − s = − . Let
N > H be a prime in P H , and let D be any negative quadratic discriminant. If( DN ) = −
1, then by Theorem 2 we have d D ( N ) ≥ N − , which for sufficiently large N , is greater than CN . Otherwise ( DN ) = −
1, and by Theorem 2 we have d D ( N ) ≥ h ( D )6 ( N − > H N − > CN, since N > H . Case 2: Orders of conductor prime to N : Suppose that O ( D ) is an orderof conductor f in the imaginary quadratic field K = Q ( √ D ); let F be a numberfield and E /F be a O -CM elliptic curve. Proposition 25. there exists an F -rational isogeny ι : E → E ′ , where E ′ /F is anelliptic curve with O K -CM. Moreover ι is cyclic of degree f . This is “well known”, but lacking a convenient reference we shall sketch the proof.Over the complex numbers we may view E as C / O , and then the map is just thenatural map C / O → C / O K . The rationality of the map over F follows easily fromthe fact that O is the unique subring of O K of index f .The isogeny ι induces a homomorphism of Mordell-Weil groups ι ( F ) : E ( F ) → E ′ ( F ). According to the Proposition, the kernel of ι ( F ) is f -torsion. Moreover,using the existence of a dual isogeny ι ∨ : E ′ → E such that ι ∨ ◦ ι = [ f ], ι ◦ ι ∨ = [ f ],one sees that also the cokernel of ι ( F ) is f -torsion. In particular, if N is an oddprime with ( N, f ) = 1, then ι ( F ) : E ( F )[ N ] ∼ → E ′ ( F )[ N ] . In particular, if E has an F -rational torsion point of order N , so does E ′ . Fromthis it follows that – still for N prime to f – the least degree of an O ( f D )-CMpoint on X ( N ) is at least as large as that of an O ( D )-CM point on X ( N ). Thatis, we have succeeded in reducing Case 2 to Case 1. Case 3: General Case : Finally suppose we have D = f D with N | f , andconsider an O ( D )-CM elliptic curve E defined over a number field F , with an F -rational N -torsion point. To simplify the analysis, we assume F contains theCM-field K (this extra factor of 2 will not effect the asymptotic analysis).The above geometric description of the isogeny ι shows that dim F N ker( ι ) ∩ E [ N ] =1, i.e., there exists a single point P ∈ E [ N ]( C ) such that h P i = ker( ι ) ∩ E [ N ].Consider first any N -torsion point P which is not in h P i . Then ι ( P ) is an F -rational point on the O ( D )-CM elliptic curve, i.e., as in Case 2, we immediatelyreduce to Case 1. So it suffices to assume that the point P is F -rational and derivelower bounds on [ F : K ]. As in § .
3, Case 3, the mod N Galois representation ρ N : Gal F → GL ( Z /N Z ) iscontained in a “pseudo-Cartan subgroup”; taking an ordered basis with P as thefirst vector, we have ρ (Gal F ) ⊂ C × N ∼ = { (cid:20) a b a (cid:21) a ∈ F × N , b ∈ F N } . So our assumption that P is F -rational means precisely that ρ (Gal F ) ⊂ { (cid:20) b (cid:21) b ∈ F N } . Thus det( ρ (Gal F )) = 1, so by Corollary 11 we deduce F ⊃ K ( ζ N ). Now K ( ζ N )and the ring class field K ( j ( E )) are extensions of K of degrees at least N − and N − respectively. Moreover, Fact 1e) implies that, loosely speaking, these twoextensions are close to being disjoint over K , so that K ( ζ N , j ( E )) has degree atleast a universal constant times ( N − .Let us now see this in more detail: let E ′′ be an elliptic curve with O ( N D )-CM, i.e., with the same CM field but conductor N instead of its multiple f . Byclass field theory K ( j ( E )) ⊂ K ( j ( E ′′ )). But K ( j ( E ′′ )), being the ring class fieldof conductor N , is contained in the N -ray class field K ( N ), whereas explicit classfield theory shows Gal( K ( N ) /K ) is a finite abelian group with either 1 or twogenerators. Therefore the degree of the maximal exponent 2 abelian subextensionof K ( j ( E )) /K is at most 4. Combining all estimates, we get[ F : K ] ≥ [ K ( j ( E ) , ζ N ) : K ] ≥ ( N − . This is obviously not O ( N ), so the proof is complete.8. Proof of Theorem 6
Here, briefly, is the idea: Start with E/ Q of j -invariant 0. Enumerate the oddprimes p n which are 1 mod 3 (hence split in Q ( √− K n be the least fieldover which E acquires a point of order N n := p · · · p n . The degree of this field isat most 2 n Y i =1 ( p i −
1) = 2 ϕ ( N n ) , and it is known that N n ϕ ( N n ) ≫ log log N n .Proof: Let K = Q ( √− E /K an O ( − y = x + 1).Let p < p < . . . be the primes congruent to 1 (mod 3), i.e., the primes whichsplit in K . It follows from the material reviewed in § . i there is apoint P i on E of order i , such that [ K ( P i ) : K ] | ( p i − n , the field L n := K ( { P i } ni =1 ) has a point of order N n = p · · · p n (namely P + . . . + P n ) and d n := [ L n : K ] ≤ n Y i =1 ( p i −
1) = 2 ϕ ( N n ) . Then | E ( L n )[tors] | d n ≥ N n ϕ ( N n ) , ORSION POINTS ON ELLIPTIC CURVES WITH COMPLEX MULTIPLICATION 25 and to complete the proof it is sufficient to establish the followingClaim: There exists
C > n , N n ϕ ( N n ) ≥ C p log(log( d n )) . The proof of the claim rests on an asymptotic formula due to Mertens, namely Y p ≤ x − p − ∼ e − γ log( x ) , where the product is taken over all primes less than or equal to x , and γ is Euler’sconstant [BD04, Cor. 6.19]. From the Prime Number Theorem for ArithmeticProgressions [BD04, Thm. 9.12], it follows that Y p ≤ x,p ≡ − p − ) ∼ e − γ/ p log( x ) . Let us now write N n ϕ ( N n ) = n Y i =1 p i p i − Y p ≤ x ( n ) ,p ≡ − p − . Then we have N n ϕ ( N n ) ∼ e − γ/ p log( x ( n )) . Again applying the Prime Number Theorem for Arithmetic Progressions, it followsthat log( x ( n )) ∼ log( n ), and also thatlog( N n ) = n X i =1 log( p i ) ∼ n X i =1 i log( i ) ∼ n n X i =1 i = n ( n + 1) log( n ) . This implies that log(log( N n )) ∼ log( n ) ∼ log( x ( n )). Thus N n ϕ ( N n ) ∼ e − γ/ p log(log( N n )) ≥ e − γ/ p log(log( ϕ ( N n ))) ≥ e − γ/ p log(log( d n / , which is sufficient to give the result.Remark 8.1: The reader may be wondering whether we could have done betterby applying Theorem 1, which says that we can get an O ( − p i − . However, the factor of 6 that we gained in the proof of this result was viaour ability to make a single cyclic twist to get more torsion. However we cannotindependently make cyclic twists for each prime p i . Thus we could improve d n to ϕ ( p ··· p n )3 but not to n ϕ ( p · · · p n ). In fact Serre’s Theorem (Theorem 18) im-plies that among constructions working with a fixed elliptic curve, or even a fixed j -invariant, our lower bound is asymptotically optimal.9. Proof of Theorem 7
Theorem 26. (Abramovich, [Abr96] ) Let Γ ⊂ P SL ( Z ) be a congruence subgroup,and X Γ = Γ \H the corresponding modular curve. The gonality of X Γ is at least [ P SL ( Z ) : Γ] . Remark 9.1: This result uses results of differential geometry and spectral theory,including an upper bound on the leading nontrivial eigenvalue for the Laplacian onthe Riemannian manifold X Γ : Abramovich’s theorem uses the bound λ ≤ , dueto Luo, Rudnick and Sarnak. Selberg has conjectured that λ ≤ , which wouldallow replacement of by . Theorem 27. (Faltings, Frey [Fre77] ) Let X be a curve defined over a numberfield K with at least one K -rational point. If, for any positive integer d , X /K hasinfinitely many points of degree d , then Gon K ( X ) ≤ d . Remark 9.2: The hypothesis is satisfied for all classical modular curves X Γ uni-formized by congruence subgroups of P SL ( Z ) since such curves always have acusp rational over their “reflex field” K ( K = Q for the curves X ( N )).When N is prime, the index of Γ ( N ) in P SL ( Z ) is N − . Thus we getGon C ( X ( N )) ≥ N − unconditionally , and Gon C ( X ( N )) ≥ N − conditionally on Selberg’s eigenvalue conjecture.Therefore we get12 Gon Q ( X ( N )) ≥
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Department of Mathematics, Boyd Graduate Studies Research Center, Universityof Georgia, Athens, GA 30602-7403, USA
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