THE TRUTH ABOUT TORSION IN THE CM CASE

PETE L. CLARK AND PAUL POLLACK

Abstract. 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.

1. Introduction The 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 (Hindry–Silverman [HS99]). 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] ≤ 1977408d log d.

Every CM elliptic curve E/F has j(E) ∈ OF , but only finitely many j ∈ OF are j-invariants of CM elliptic curves E/F . Thus Theorem 1 has a significantly stronger hypothesis and a slightly stronger conclusion than Theorem 2. But the improvement of log log d over log d is interesting in view of the following result.

Theorem 3 (Breuer [Br10]). 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 ® c(E,F )dn log log dn if E has CM, #E(Fn)[tors] ≥ √ c(E,F ) dn log log dn otherwise.

Thus if TCM(d) is the largest size of a torsion subgroup of a CM elliptic curve defined over a number field of degree d, its upper order is d log log d: T (d) lim sup CM ∈ ]0, ∞[. 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. √ Hindry and Silverman ask whether d log log d is the upper order of the size of the torsion subgroup of an elliptic curve without complex multiplication over a degree d number field. If so, the upper order of the size of the torsion subgroup of an elliptic curve over a degree d number field would be d log log d [CCRS13, Conj. 1]. 1 2 PETE L. CLARK AND PAUL POLLACK

2. Proof of the Main Theorem 2.1. Preliminary Results

Let K be a number field. Let OK be the ring of integers of K, ∆K the discriminant of K, wK 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 (a) K for the a-ray class field of K. We also put |a| = #OK /a and

Y Å 1 ã ϕ (a) = #(O /a)× = |a| 1 − . K K |p| p|a An elliptic curve E defined over a field of characteristic 0 has complex multiplica- tion (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 . Then

hK ϕK (a) hK ϕK (a) (a) ≤ ≤ [K : K] ≤ hK ϕK (a). 6 wK

Proof. This follows from [Co00, 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 [Si94, II.5.6]. We shall reduce to that case. There is an OK -CM elliptic 0 0 curve E/F and a canonical F -rational isogeny ι: E → E [CCRS13, 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 put

0 0 P = 1/N + O ∈ E[s],P = 1/N + OK ∈ E [N],

0 0 0 then ι(P ) = P and P generates E [N] as an OK -module. By assumption P ∈ E(F ), 0 0 2 0 so ι(P ) = P ∈ E (F ). It follows that (Z/NZ) ,→ E (F )[tors]. 

2.2. Squaring the Torsion Subgroup of a CM Elliptic Curve Theorem 6. Let O be an order in the imaginary quadratic field K. Let F be a field ∼ extension of K, and let E/F be an O-CM elliptic curve: i.e., End E = O. Suppose that for positive integers a | b we have an injection Z/aZ × Z/bZ ,→ E(F ). Then b 2 there is a field extension L/F with [L : F ] ≤ a and an embedding (Z/bZ) ,→ E(L).

+ × Proof. Step 1: For N ∈ Z , let CN = (O/NO) . Let E[N] = E[N](F ); it is isomorphic as an O-module to O/NO. Let gF = Aut(F /F ), and let ρN : gF → GL2(Z/NZ) be the mod N Galois representation associated to E/F . Because E has O-CM and F ⊃ K, we have ρ (g ) ⊂ Aut E[N] ∼ C . N F O = N √ ∆+ ∆ Let ∆ be the discriminant of O. Then e1 = 1, e2 = 2 is a basis for O as a Z-module. This induces a Z-algebra embedding O ,→ M2(Z) with image THE TRUTH ABOUT TORSION IN THE CM CASE 3

2 nh α β∆−β∆ i o 4 | α, β ∈ . It follows that β α+β∆ Z

ñ 2 ô ß α β∆−β∆ C = 4 | α, β ∈ /N , and N β α + β∆ Z Z Å∆2 − ∆ã ™ α2 + ∆αβ + β2 ∈ ( /N )× . 4 Z Z From this we easily deduce the following useful facts: α 0 × (i) CN contains the homotheties {s(α) = [ 0 α ] | α ∈ (Z/NZ) }. (ii) For all primes p and all 1 ≤ A ≤ B, the natural reduction map CpB → CpA is surjective and its kernel has size p2(B−A). Step 2: Primary decomposition reduces us to the case a = pA, b = pB with 0 ≤ A < B. By induction it suffices to treat the case A = B − 1. Case A = 0: We may assume there is no injection (Z/pZ)2 ,→ E(F ). Ä ∆ ä   α 0  × • If p = 1, then Cp is conjugate to d(α, β) = 0 β | α, β ∈ Fp . If α =6 1 2 (resp. β 6= 1) the only fixed points (x, y) ∈ Fp of d(α, β) have x = 0 (resp. y = 0). Because E(F ) contains a point of order p we must either have α = 1 for all d(α, β) ∈ ρp(gF ) or β = 1 for all d(α, β) ∈ ρp(gF ). Either way, #ρp(gF ) | p − 1. Ä ∆ ä • If p = −1 then Cp acts simply transitively on E[p] \{0}, so as soon as we have one F -rational point of order p we have E[p] ⊂ E(F ). So this case cannot occur. Ä ∆ ä   α β  × • If p = 0, then Cp is conjugate to m(α, β) = 0 α | α ∈ Fp , β ∈ Fp [BCS15, §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 ) ⊂ {m(1, β) | β ∈ Fp} so has order dividing p. 2 A−1 2 Case A ≥ 1: By (ii), K = ker CpA → CpA−1 has size p . Since (Z/p Z) ,→ E(F ), A we have ρpA (gF ) ⊂ K. Since E(F ) has a point of order p , by (i) the homothety A−1 s(1 + p ) lies in K\ ρpA (gF ). Therefore ρpA (gF ) ( K, so #ρpA (gF ) | p.  2.3. Minimal Order of Euler’s Function in Imaginary Quadratic Fields Theorem 7. There is a positive, effective constant C such that for all imaginary quadratic fields K and all nonzero ideals a of OK with |a| ≥ 3, we have C |a| ϕK (a) ≥ · . hK log log |a| √ Lemma 8. For a fundamental quadratic discriminant ∆ < 0 let K = Q( ∆), and ∆
let χ(·) = · . There is an effective constant C > 0 such that for all x ≥ 2, Y Å χ(p)ã C (1) 1 − ≥ . p hK p≤x p Proof. By the quadratic class number formula, hK  L(1, χ) |∆| [Da00, eq. (15), Q −1 p. 49]. Writing L(1, χ) = p(1 − χ(p)/p) and rearranging, we see (1) holds iff Y Å χ(p)ã » (2) 1 −  |∆|, p p>x 4 PETE L. CLARK AND PAUL POLLACK with an effective and absolute implied constant. By Mertens’ Theorem [HW08, Thm. 429], the factors on the left-hand side of (2) indexed by p ≤ exp(p|∆|) make a contribution of O(p|∆|). Put y = max{x, exp(p|∆|)}; it suffices to show Q that p>y (1 − χ(p)/p)  1. Taking logarithms, this will follow if we prove that P p p>y χ(p)/p = O(1). For t ≥ exp( |∆|), the explicit formula gives S(t) := P β p≤t χ(p) log p = −t /β +O(t/ log t), where the main term is present only if L(s, χ) has a Siegel zero β. (C.f. [Da00, eq. (8), p. 123].) We will assume the Siegel zero exists; otherwise the argument is similar but simpler. By partial summation, X χ(p) S(y) Z ∞ S(t) = − + (1 + log t) dt p y log y t2(log t)2 p>y y Z ∞ tβ  1 + 2 dt. y t log t Haneke, Goldfeld–Schinzel, and Pintz have each shown that β ≤ 1 − c/p|∆|, where the constant c > 0 is absolute and effective [Ha73, GS75, Pi76]. Using this to bound tβ, and keeping in mind that y ≥ exp(p|∆|), we see that the final integral is at most Z ∞ exp(−c log t/p|∆|) √ dt. exp( |∆|) t log t R ∞ −1 A change of variables transforms the integral into 1 exp(−cu)u du, which con- verges. Assembling our estimates completes the proof.  Q Proof of Theorem 7. Write ϕK (a) = |a| p|a(1 − 1/|p|), and notice that the factors Q are increasing in |p|. So if z ≥ 2 is such that |p|≤z |p| ≥ |a|, then Å ã ϕK (a) Y 1 (3) ≥ 1 − . |a| |p| |p|≤z 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 p lying below p. Noting that p ≤ |p|, Mertens’ Theorem and Lemma 8 yield Å ã Å ãÅ ∆
ã Y 1 Y 1 p 1 − ≥ 1 − 1 − |p| p p |p|≤z p≤z Å ∆
ã Y p (4)  (log z)−1 1 −  (log z)−1 · h−1. p K p≤z With C0 a large absolute constant to be described momentarily, we set (5) z = (C0 log |a|)2. Q We must check that |p|≤z |p| ≥ |a|. The Prime Number Theorem implies Y Y Y |p| ≥ p ≥ p ≥ |a|, |p|≤z p≤z1/2 p≤C0 log |a| provided that C0 was chosen appropriately. Combining (3), (4), and (5) gives −1 −1 −1 −1 ϕK (a)  |a| · (log z) · hK  hK · |a| · log(log(|a|)) .  THE TRUTH ABOUT TORSION IN THE CM CASE 5

2.4. Proof of Theorem 1

Let F be a number field of degree d ≥ 3 and let E/F be a K-CM elliptic curve. We may assume that #E(F )[tors] ≥ 3 and also, by replacing F by FK, that F ⊃ K. We have E(F )[tors] =∼ Z/aZ × Z/bZ for positive integers a | b. By Theorem 6, there 2 b is an extension L/F with (Z/bZ) ,→ F and [L : F ] ≤ a . By Theorem 5 we have

hK ϕK (aOK ) d ≥ [K(aOK ) : ] ≥ Q 3 and similarly h ϕ (bO ) [L : ] ≥ K K K , Q 3 so [L : ] h ϕ (bO ) a d ≥ Q ≥ K K K . b/a 3 b 2 Multiplying through by b = |bOK |, we get d |bO | (6) #E(F )[tors] = ab ≤ 3 K . hK ϕK (bOK ) By Theorem 7 we have

|bOK | 2 (7)  hK log log |bOK | ≤ hK log log(ab)  hK log log #E(F )[tors]. ϕK (bOK ) Combining (6) and (7) gives #E(F )[tors]  d log log #E(F )[tors] and thus #E(F )[tors]  d log log d. Acknowledgments. We thank John Voight for suggesting that the proof of Theo- rem 1 ought to be in reach. We are deeply indebted to Abbey Bourdon for pointing out an error in a previous draft.

References [BCS15] A. Bourdon, P.L. Clark and J. Stankewicz, Torsion points on CM elliptic curves over real number fields, submitted for publication. [Br10] F. Breuer, Torsion bounds for elliptic curves and Drinfeld modules. J. Number Theory 130 (2010), 1241–1250. [CCRS13] P.L. Clark, B. Cook and J. Stankewicz, Torsion points on elliptic curves with complex multiplication (with an appendix by Alex Rice). International Journal of Number Theory 9 (2013), 447–479. [Co00] H. Cohen, Advanced Topics in Computational Number Theory. Graduate Texts in Mathematics 193, Springer-Verlag, 2000. [Da00] H. Davenport, Multiplicative Number Theory. Third edition. Graduate Texts in Mathe- matics, 74. Springer-Verlag, New York, 2000. [Ha73] W. Haneke, Uber¨ die reellen Nullstellen der Dirichletschen L-Reihen. Acta Arith. 22 (1973), 391–421. Corrigendum in 31 (1976), 99–100. [HS99] M. Hindry and J. Silverman, Sur le nombre de points de torsion rationnels sur une courbe elliptique. C. R. Acad. Sci. Paris S´er.I Math. 329 (1999), no. 2, 97–100. [HW08] G.H. Hardy and E.M. Wright, An introduction to the theory of numbers. Sixth edition. Oxford University Press, Oxford, 2008. [GS75] D.M. Goldfeld and A. Schinzel, On Siegel’s zero. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2 (1975), 571–583. [Pi76] 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. 6 PETE L. CLARK AND PAUL POLLACK

[Si94] J. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Graduate Texts in Mathematics 151, Springer-Verlag, 1994.