We show that the upper order of the size of the torsion subgroup of a CM
elliptic curve over a degree d number field is d log log d.Comment: 6 pages. This is a minor revision incorporating expository
  suggestions made by the refere

Aoki

Bateman

Bourdon

Breuer

Clark

Cohen

Davenport

Goldfeld

Haneke

Hardy

Hindry

Paul Pollack

Pete L. Clark

Pintz

Rosen

Silverberg

Silverman

English

arXiv

Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 353 (2015) 683–688Contents lists available at ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comNumber theoryThe truth about torsion in the CM caseLa vérité sur la torsion dans le cas CMPete L. Clark, Paul PollackDepartment of Mathematics, University of Georgia, Athens, GA, 30602, USAa r t i c l e i n f o a b s t r a c tArticle history:Received 9 March 2015Accepted after revision 22 May 2015Available online 2 July 2015Presented by Jean-Pierre SerreLet TCM(d) be the maximum size of the torsion subgroup of an elliptic curve with complex multiplication over a degree d number field. We show there is an absolute, effective constant C such that TCM(d) ≤ Cd log log d for all d ≥ 3.© 2015 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éSoit TCM(d) la taille maximale du sous-groupe de torsion d’une courbe elliptique à multiplications complexes, définie sur un corps de nombres de degré d. Nous montrons qu’il existe C une constante absolue, effective, telle que TCM(d) ≤ Cd log log(d) pour tout d ≥ 3.© 2015 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.For a commutative group G , we denote by G[tors] the torsion subgroup of G .1. IntroductionThe aim of this note is to prove the following result.Theorem 1. There is an absolute, effective constant C such that for all number fields F of degree d ≥ 3 and all elliptic curves E/F with complex multiplication,#E(F )[tors] ≤ Cd log log d.It is natural to compare this result with the following one.Theorem 2. (See Hindry–Silverman [9].) For all number fields F of degree d ≥ 2 and all elliptic curves E/F with j-invariant j(E) ∈OF , we have#E(F )[tors] ≤ 1 977 408d log d.E-mail addresses: plclark@gmail.com (P.L. Clark), pollack@uga.edu (P. Pollack).http://dx.doi.org/10.1016/j.crma.2015.05.0041631-073X/© 2015 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.684 P.L. Clark, P. Pollack / C. R. Acad. Sci. Paris, Ser. I 353 (2015) 683–688Every CM elliptic curve E/F has j(E) ∈OF , and only finitely many j ∈OF are j-invariants of CM elliptic curves E/F . But the improvement of log log d over log d is interesting in view of the following result.Theorem 3. (See Breuer [4].) Let E/F be an elliptic curve over a number field. There exists a constant c(E, F ) > 0, integers 3 ≤ d1 <d2 < . . . < dn < . . . and number fields Fn ⊃ F with [Fn : F ] = dn such that for all n ∈ Z+ we have#E(Fn)[tors] ≥{c(E, F )dn log log dn if E has CM,c(E, F )√dn log log dn otherwise.Let TCM(d) be the maximum size of the torsion subgroup of a CM elliptic curve over a degree d number field. Theorems 1and 3 tell us that TCM(d) has upper order d log log d:0 < lim supd→∞TCM(d)d log log d< ∞.To our knowledge, this is the first instance of an upper order result for torsion points on a class of abelian varieties over number fields of varying degree.Define T (d) as for TCM(d) but replacing “CM elliptic curve” with “elliptic curve”, and define T¬CM(d) as for TCM(d)but replacing “CM elliptic curve” with “elliptic curve without CM”. Hindry and Silverman ask whether T¬CM(d) has upper order√d log log d. If so, the upper order of T (d) would be d log log d [5, Conjecture 1].2. Proof of the Main Theorem2.1. Torsion points and ray class containmentLet K be a number field. Let OK be the ring of integers of K , K the discriminant of K , w K the number of roots of unity in K and hK the class number of K . By an “ideal of OK ” we shall always mean a nonzero ideal. For an ideal a of OK , we write K (a) for the a-ray class field of K . We also put |a| = #OK /a andϕK (a) = #(OK /a)× = |a|∏p|a(1 − 1|p|).An elliptic curve E defined over a field of characteristic 0 has complex multiplication (CM) if End E  Z; then End E is an order in an imaginary quadratic field. We say E has O-CM if End E ∼=O and K -CM if End E is an order in K .Lemma 4. Let K be an imaginary quadratic field and a an ideal of OK . ThenhK ϕK (a)6≤ hK ϕK (a)w K≤ [K (a) : K ] ≤ hK ϕK (a).Proof. This follows from [6, Corollary 3.2.4]. Theorem 5. Let K be an imaginary quadratic field, F ⊃ K a number field, E/F a K -CM elliptic curve and N ∈ Z+ . If (Z/NZ)2 ↪→ E(F ), then F ⊃ K (NOK ) .Proof. The result is part of classical CM theory when End E = OK is the maximal order in K [15, II.5.6]. We shall reduce to that case. There is an OK -CM elliptic curve E ′/F and a canonical F -rational isogeny ι: E → E ′ [5, Prop. 25]. There is a field embedding F ↪→ C such that the base change of ι to C is, up to isomorphisms on the source and target, given by C/O →C/OK . If we putP = 1/N +O ∈ E[N], P ′ = 1/N +OK ∈ E ′[N],then ι(P ) = P ′ and P ′ generates E ′[N] as an OK -module. By assumption P ∈ E(F ), so ι(P ) = P ′ ∈ E ′(F ). It follows that (Z/NZ)2 ↪→ E ′(F )[tors]. Remark 6. In fact one can show — e.g., using adelic methods — that for any K -CM elliptic curve E defined over C, the field obtained by adjoining to K ( j(E)) the values of the Weber function at the N-torsion points of E contains K (NOK ) .P.L. Clark, P. Pollack / C. R. Acad. Sci. Paris, Ser. I 353 (2015) 683–688 6852.2. Squaring the torsion subgroup of a CM elliptic curveTheorem 7. Let K be an imaginary quadratic field, let F ⊃ K a field extension, and let E/F be a K -CM elliptic curve. Suppose that for positive integers a and b we have an injection Z/aZ ×Z/abZ ↪→ E(F ). Then [F (E[ab]) : F ] ≤ b.Proof. Step 1: Let O = End E . For N ∈ Z+ , let CN = (O/NO)× . Let E[N] = E[N](F ). As an O/NO-module, E[N] is free of rank 1. Let gF = Aut(F/F ), and let ρN : gF −→ GL2(Z/NZ) be the mod N Galois representation associated with E/F . Because E has O-CM and F ⊃ K , we haveρN :gF −→ AutO E[N] ∼= GL1(O/NO) ∼= (O/NO)× = CN .Let  be the discriminant of O. Then e1 = 1, e2 = +√2 is a Z-basis for O. The induced ring embedding O ↪→ M2(Z) is given by αe1 + βe2 →[α β−β24β α+β]. SoCN ={[α β−β24β α + β]| α,β ∈ Z/NZ, and α2 + αβ +(2 − 4)β2 ∈ (Z/NZ)×}.From this we easily deduce the following useful facts:(i) CN contains the homotheties {[α 00 α]| α ∈ (Z/NZ)×}.(ii) For all primes p and all A, B ≥ 1, the natural reduction map C p A+B → C p A is surjective and its kernel has size p2B .Step 2: Primary decomposition reduces us to the case a = p A , b = pB with A ≥ 0 and B ≥ 1. By induction it suffices to treat the case B = 1: i.e., we assume E(F ) contains full p A -torsion and a point of order p A+1 and show [F (E[p A+1]) : F ] ≤ p.Case A = 0:• If (p)= 1, then C p is conjugate to {[α 00 β]| α,β ∈ F×p}. If α = 1 (resp. β = 1) the only fixed points (x, y) ∈ F2p of [α 00 β]have x = 0 (resp. y = 0). Because E(F ) contains a point of order p we must either have α = 1 for all [α 00 β]∈ ρp(gF ) or β = 1 for all [α 00 β]∈ ρp(gF ). Either way, #ρp(gF ) | p − 1.• If (p)= −1, then C p acts simply transitively on E[p] \ {0}, so if we have one F -rational point of order p then E[p] ⊂E(F ), so #ρp(gF ) = 1.• If (p)= 0, then C p is conjugate to {[α β0 α]| α ∈ F×p , β ∈ Fp}[3, §4.2]. Since E(F ) has a point of order p, every element of ρp(gF ) has 1 as an eigenvalue and thus ρp(gF ) ⊂{[1 β0 1]| β ∈ Fp}, so has order dividing p.Case A ≥ 1: By (ii), K = ker C p A+1 → C p A has size p2. Since (Z/p AZ)2 ↪→ E(F ), we have ρp A+1(gF ) ⊂ K. Since E(F ) has a point of order p A+1, by (i) the homothety [1+p A 00 1+p A]lies in K\ρp A+1 (gF ). Therefore ρp A+1(gF ) K, so #ρp A+1(gF ) | p. 2.3. Uniform bound for Euler’s function in imaginary quadratic fieldsLet a be an ideal in an imaginary quadratic field K . To apply the results of Section 2.1, we require a lower bound on ϕK (a)|a| . For fixed K , it is straightforward to adapt a classical argument of Landau (see the proof of [10, Theorem 328, p. 352]). Replacing Landau’s use of Mertens’ Theorem with Rosen’s number field analogue [13], one obtains the following result: let γ denote the Euler–Mascheroni constant, and let χ(·) = (K· ) be the quadratic Dirichlet character associated with K . Thenlim inf|a|→∞ϕK (a)|a|/ log log |a| = e−γ · L(1,χ)−1.Alas, this result is not sufficient for our purposes. There are two sources of difficulty. First, the right-hand side depends on K , and can in fact be arbitrarily small (see [2, (4′)]). Second, it only addresses limiting behavior as |a| → ∞. However, looking back at Lemma 4 we see that a lower bound on hK ϕK (a)|a| would suffice. The factor of hK allows us to prove a totally uniform lower bound.Theorem 8. There is a positive, effective absolute constant C such that, for all imaginary quadratic fields K and all nonzero ideals aof OK with |a| ≥ 3, we have686 P.L. Clark, P. Pollack / C. R. Acad. Sci. Paris, Ser. I 353 (2015) 683–688ϕK (a) ≥ ChK· |a|log log |a| .Lemma 9. For a fundamental quadratic discriminant  < 0, let K = Q(√), and let χ(·) = (· ). There is an effective constant C > 0such that for all x ≥ 2,∏p≤x(1 − χ(p)p)≥ ChK. (1)Proof. By the quadratic class number formula, hK  L(1, χ)√|| [7, eq. (15), p. 49]. Writing L(1, χ) = ∏p(1 − χ(p)/p)−1and rearranging, we see (1) holds iff∏p>x(1 − χ(p)p) √||, (2)with an effective and absolute implied constant. By Mertens’ Theorem [10, Theorem 429, p. 466], the factors on the left-hand side of (2) indexed by p ≤ exp(√||) make a contribution of O (√||). Put y = max{x, exp(√||)}; it suffices to show that ∏p>y (1 − χ(p)/p)  1. Taking logarithms, this will follow if we prove that ∑p>y χ(p)/p = O (1). For t ≥ exp(√||), the explicit formula gives S(t) := ∑p≤t χ(p) log p = −tβ/β + O (t/ log t), where the main term is present only if L(s, χ) has a Siegel zero β . (Cf. [7, eq. (8), p. 123].) We will assume the Siegel zero exists; otherwise the argument is similar but simpler. By partial summation,∑p>yχ(p)p= − S(y)y log y+∞∫yS(t)t2(log t)2(1 + log t)dt 1 +∞∫ytβt2 log tdt.Haneke, Goldfeld–Schinzel, and Pintz have each shown that β ≤ 1 − c/√||, where the constant c > 0 is absolute and effective [8,11,12]. Using this to bound tβ , and keeping in mind that y ≥ exp(√||), we see that the final integral is at most∞∫exp(√||)exp(−c log t/√||)t log tdt.A change of variables transforms the integral into ∫ ∞1 exp(−cu)u−1 du, which converges. Assembling our estimates com-pletes the proof. Proof of Theorem 8. Write ϕK (a) = |a| ∏p|a(1 − 1/|p|), and notice that the factors are increasing in |p|. So if z ≥ 2 is such that ∏|p|≤z |p| ≥ |a|, thenϕK (a)|a| ≥∏|p|≤z(1 − 1|p|). (3)We first establish a lower bound on the right-hand side, as a function of z, and then we prove the theorem by making a convenient choice of z. We partition the prime ideals with |p| ≤ z according to the splitting behavior of the rational prime plying below p. Noting that p ≤ |p|, Mertens’ Theorem and Lemma 9 yield∏|p|≤z(1 − 1|p|)≥∏p≤z(1 − 1p)(1 − (p )p) (log z)−1∏p≤z(1 − (p )p) (log z)−1 · h−1K . (4)With C ′ a large absolute constant to be described momentarily, we setz = (C ′ log |a|)2. (5)We must check that ∏|p|≤z |p| ≥ |a|. The Prime Number Theorem impliesP.L. Clark, P. Pollack / C. R. Acad. Sci. Paris, Ser. I 353 (2015) 683–688 687∏|p|≤z|p| ≥∏p≤z1/2p ≥∏p≤C ′ log |a|p ≥ |a|,provided that C ′ was chosen appropriately. Combining (3), (4), and (5) givesϕK (a)  |a| · (log z)−1 · h−1K  h−1K · |a| · log(log(|a|))−1. 2.4. Proof of Theorem 1Let F be a number field of degree d ≥ 3, and let E/F be a K -CM elliptic curve. We may assume #E(F )[tors] ≥ 3. We have E(FK)[tors] ∼= Z/aZ ×Z/abZ for positive integers a and b. Theorem 5 gives FK ⊃ K (aOK ) . Along with Lemma 4, we get2d ≥ [FK : Q] ≥ [K (aOK ) : Q] ≥ hK ϕK (aOK )3.By Theorem 7, there is an extension L/FK with (Z/abZ)2 ↪→ E(L) and [L : FK] ≤ b. Applying Theorem 5 and Lemma 4 as above we get L ⊃ K (abOK ) and[L : Q] ≥ [K (abOK ) : Q] ≥ hK ϕK (abOK )3,sod = [F : Q] ≥ [FK : Q]2= [L : Q]2[L : FK] ≥[L : Q]2b≥ hK ϕK (abOK )6b. (6)Multiplying (6) through by (ab)2 = |abOK | and rearranging, we get#E(FK)[tors] = a2b ≤ 6 dhK|abOK |ϕK (abOK ). (7)By Theorem 8 we have|abOK |ϕK (abOK ) hK log log |abOK | ≤ hK log log(a2b)2  hK log log #E(FK)[tors]. (8)Combining (7) and (8) gives#E(FK)[tors]  d log log #E(FK)[tors]and thus#E(F )[tors] ≤ #E(FK)[tors]  d log log d.3. Related workLet E be a K -CM elliptic curve defined over a number field F , and let P ∈ E(F )[tors]. Silverberg showed [14, Corollary 6.1] that if F ⊃ K then ϕ(#〈P 〉) ≤ 3[F : Q]. It follows that if F ⊃ K then ϕ(#〈P 〉) ≤ 6[F : Q]. Later Aoki showed [1, Proposi-tion 8.1] that if F ⊃ K then ϕ(#〈P 〉) ≤ 2[F : Q]. Silverberg’s and Aoki’s bounds are the real truth: there are points of order 6when F =Q and of order 7 when F = K = Q(√−3).These results give an O (d log log d) bound on the exponent of E(F )[tors] and thus imply #E(F )[tors] = O ((d log log d)2), which was later superseded by Theorem 2. If F ⊃ K , then E(F )[tors] has a cyclic subgroup of index at most 2. Thus the work of Silverberg and Aoki yields Theorem 1 when F ⊃ K , in fact in the more explicit form#E(F )[tors] ≤ (4 eγ + o(1))d log log d, as d → ∞.AcknowledgmentsWe thank John Voight for suggesting that the proof of Theorem 1 ought to be in reach, Alice Silverberg for providing pointers to the literature, and the referee for a rapid, careful reading accompanied by useful comments. We are extremely grateful to Abbey Bourdon for pointing out an error in a previous draft. The second author is supported by NSF award DMS-1402268.688 P.L. Clark, P. Pollack / C. R. Acad. Sci. Paris, Ser. I 353 (2015) 683–688References[1] N. Aoki, Torsion points on abelian varieties with complex multiplication, in: Algebraic Cycles and Related Topics, Kitasakado, 1994, World Sci. Publ., River Edge, NJ, USA, 1995, pp. 1–22.[2] P.T. Bateman, S. Chowla, P. Erdős, Remarks on the size of L(1, χ), Publ. Math. (Debr.) 1 (1950) 165–182.[3] A. Bourdon, P.L. Clark, J. Stankewicz, Torsion points on CM elliptic curves over real number fields, submitted for publication, http://arxiv.org/abs/1411.2742.[4] F. Breuer, Torsion bounds for elliptic curves and Drinfeld modules, J. Number Theory 130 (2010) 1241–1250.[5] P.L. Clark, B. Cook, J. Stankewicz, Torsion points on elliptic curves with complex multiplication (with an appendix by Alex Rice), Int. J. Number Theory 9 (2013) 447–479.[6] H. Cohen, Advanced Topics in Computational Number Theory, Graduate Texts in Mathematics, vol. 193, Springer-Verlag, 2000.[7] H. Davenport, Multiplicative Number Theory, third edition, Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York, 2000.[8] W. Haneke, Über die reellen Nullstellen der Dirichletschen L-Reihen, Acta Arith. 22 (1973) 391–421;W. Haneke, Corrigendum, Acta Arith. 31 (1976) 99–100.[9] M. Hindry, J. Silverman, Sur le nombre de points de torsion rationnels sur une courbe elliptique, C. R. Acad. Sci. Paris, Ser. I 329 (2) (1999) 97–100.[10] G.H. Hardy, E.M. Wright, An Introduction to the Theory of Numbers, sixth edition, Oxford University Press, Oxford, UK, 2008.[11] D.M. Goldfeld, A. Schinzel, On Siegel’s zero, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4) 2 (1975) 571–583.[12] J. Pintz, Elementary methods in the theory of L-functions. II. On the greatest real zero of a real L-function, Acta Arith. 31 (1976) 273–289.[13] M. Rosen, A generalization of Mertens’ theorem, J. Ramanujan Math. Soc. 14 (1999) 1–19.[14] A. Silverberg, Points of finite order on abelian varieties, in: p-Adic Methods in Number Theory and Algebraic Geometry, in: Contemp. Math., vol. 133, Amer. Math. Soc., Providence, RI, USA, 1992, pp. 175–193.[15] J. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, vol. 151, Springer-Verlag, 1994.

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The truth about torsion in the CM case

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The truth about torsion in the CM case

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