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THE TRUTH ABOUT TORSION in the CM CASE 1. Introduction the Aim of This Note Is to Prove the Following Result. Theorem 1. There I

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THE TRUTH ABOUT TORSION in the CM CASE 1. Introduction the Aim of This Note Is to Prove the Following Result. Theorem 1. There I 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) 2 OF , we have #E(F )[tors] ≤ 1977408d log d: Every CM elliptic curve E=F has j(E) 2 OF , but only finitely many j 2 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 2 Z we have ¨ c(E; F )dn log log dn if E has CM; #E(Fn)[tors] ≥ p 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 2 ]0; 1[: 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. p 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 jaj = #OK =a and Y Å 1 ã ' (a) = #(O =a)× = jaj 1 − : K K jpj pja 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 'K (a) hK 'K (a) (a) ≤ ≤ [K : K] ≤ hK 'K (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 2 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 2 E[s];P = 1=N + OK 2 E [N]; 0 0 0 then ι(P ) = P and P generates E [N] as an OK -module. By assumption P 2 E(F ), 0 0 2 0 so ι(P ) = P 2 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 j 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 2 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 p ∆+ ∆ 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 j α; β 2 . It follows that β α+β∆ Z ñ 2 ô § α β∆−β∆ C = 4 j α; β 2 =N ; and N β α + β∆ Z Z Å∆2 − ∆ã ª α2 + ∆αβ + β2 2 ( =N )× : 4 Z Z From this we easily deduce the following useful facts: α 0 × (i) CN contains the homotheties fs(α) = [ 0 α ] j α 2 (Z=NZ) g. (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 j α; β 2 Fp . If α 6= 1 2 (resp. β 6= 1) the only fixed points (x; y) 2 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) 2 ρp(gF ) or β = 1 for all d(α; β) 2 ρp(gF ). Either way, #ρp(gF ) j p − 1. Ä ∆ ä • If p = −1 then Cp acts simply transitively on E[p] n f0g, 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 j α 2 Fp ; β 2 Fp [BCS15, x4:2]. Since E(F ) has a point of order p, every element of ρp(gF ) has 1 as an eigenvalue and thus ρp(gF ) ⊂ fm(1; β) j β 2 Fpg 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 n ρpA (gF ). Therefore ρpA (gF ) ( K, so #ρpA (gF ) j 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 jaj ≥ 3, we have C jaj 'K (a) ≥ · : hK log log jaj p 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) j∆j [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 È (2) 1 − j∆j; 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(pj∆j) make a contribution of O(pj∆j). Put y = maxfx; exp(pj∆j)g; 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( j∆j), 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.
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