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QUOTIENTS of ELLIPTIC CURVES OVER FINITE FIELDS 1. Introduction by Tate's Isogeny Theorem [14, P. 139], Two Elliptic Curves Ov

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QUOTIENTS of ELLIPTIC CURVES OVER FINITE FIELDS 1. Introduction by Tate's Isogeny Theorem [14, P. 139], Two Elliptic Curves Ov QUOTIENTS OF ELLIPTIC CURVES OVER FINITE FIELDS JEFFREY D. ACHTER AND SIMAN WONG Abstract. Fix a prime `, and let Fq be a finite field with q ≡ 1 (mod `) elements. If 2 2 ` > 2 and q ` 1, we show that asymptotically (` − 1) =2` of the elliptic curves E=Fq with complete rational `-torsion are such that E=hP i does not have complete rational `-torsion for any point P 2 E(Fq) of order `. For ` = 2 the asymptotic density is 0 or 1=4, depending whether q ≡ 1 (mod 4) or 3 (mod 4). We also show that for any `, if E=Fq has an Fq-rational point R of order `2, then E=h`Ri always has complete rational `-torsion. 1. Introduction By Tate's isogeny theorem [14, p. 139], two elliptic curves over a finite field k are k- isogenous if and only if they have the same number of k-rational points. However, the isogeny need not preserve the group structure of the k-rational points of the respective 2 curves. For example, for E : y = x(x − 1)(x + 1) we have E(F5) ' Z=2 × Z=2. By explicit computations (cf. [15], [13, p. 70]), we find that E=h(0; 0)i is F5-isomorphic to E as elliptic curves (not via the quotient map, of course), but that the group of F5-rational points of both E=h(1; 0)i and E=h(−1; 0)i are cyclic. In this paper we investigate the problem of how often the quotient map by a k-rational torsion point preserves the group structure. Note that the group homomorphism induced by an isogeny φ is an isomorphism on the prime-to-deg(φ) part of the group of rational points, so it suffices to focus on the k-rational deg(φ)-torsion. Let ` be a prime, and let E=K be an elliptic curve over a field. Denote by [`]: E!E the multiplication-by-` isogeny on E, and denote by E[`](K) the K-rational points of the kernel of this isogeny. For the rest of this paper, we take K to be Fq, a finite field of size q with q ≡ 1 (mod `). This condition is necessary for the set S(Fq; `) := the set of Fq-isomorphism classes of elliptic curves E with E[`](Fq) = E[`](Fq) to be non-empty; this follows from properties of the Weil pairing ([13, p. 96]; see also 2 Thm 2.1(c)). If ` = 2 then q is odd, so y = x(x − 1)(x + 1) belongs to S(Fq; 2); in particular 4 S(Fq; 2) is not empty. In general, S(Fq; `) is not empty for q ≥ ` (see Lemma 2.2). We also define S0(Fq; `) = fE 2 S(Fq; `): E=hP i 62 S(Fq; `) for every non-zero point P 2 E[`] g; S∗(Fq; `) = fE 2 S(Fq; `): E=hP i 2 S(Fq; `) for every non-zero point P 2 E[`] g: and ratios #S0(Fq; `) #S∗(Fq; `) R0(Fq; `) = and R∗(Fq; `) = : #S(Fq; `) #S(Fq; `) 1991 Mathematics Subject Classification. Primary 11G20; Secondary 11G25, 14H10. Key words and phrases. Elliptic curves, finite fields, isogeny, moduli, monodromy, quotients. Jeffrey Achter's work is supported in part by a grant from the Simons foundation (204164). Siman Wong's work is supported in part by NSF grant DMS-0901506. 1 Theorem 1.1. (a) Let ` be an odd prime. For all q ≡ 1 (mod `), we have 2 7 7 (` − 1) ` 1 ` R0(Fq; `) − < ; R∗(Fq; `) − < : 2`2 q1=2 `3 q1=2 (b) Suppose ` = 2. For 1 ≤ j ≤ 3, define n E=hP i 2 S( ; 2) for at least j o S ( ; 2) = E 2 S( ; 2) : Fq : j Fq Fq distinct points P 2 E of order 2 Then the ratio Rj(Fq; 2) = #Sj(Fq; 2)=#S(Fq; 2) is given by q (mod 4) R0(Fq; 2) R1(Fq; 2) R2(Fq; 2) R3(Fq; 2) 1 (q) 1 (q) 1 0 1 + 1 + 1 4 q 4 q 1 (q) 3 (q) 3 (q) 3 − 2 + 2 + 2 0 4 q 4 q 4 q where each ji(q)j ≤ 23=12 and depends on q (mod 24) only. If a curve in S(Fq; `) belongs to S0(Fq; `), then some quotient of this curve by an `-torsion 2 point must have an Fq-rational point of order ` . Apply the dual isogeny and we are led to 2 ask how often an elliptic curve E=Fq with an Fq-rational point R of order ` is such that E=h[`]Ri is in S(Fq; `). The answer is surprisingly simple and uniform. Theorem 1.2. Let ` be a prime, and let Fq be a finite field of cardinality q ≡ 1 (mod `). Let 2 E be an elliptic curve over Fq with an Fq-rational point R of order ` . Then E=h[`]Ri has complete Fq-rational `-torsion. Theorem 1.2 implies that if S(Fq; `) is empty then no elliptic curve over Fq has an Fq-point of order `2. This already follows essentially from Deuring's work on complex multiplication [2]; see Theorem 2.1 below. In fact, from these classical works we can derive closed form expressions (Theorem 5.3) for S(Fq; `) and S∗(Fq; `). While it is not clear (to us) how to use these expressions to prove Theorem 1.1, we can use them to test how sharp the estimates are in the Theorem. See Section 5 for more details. In Section 2, we collect useful, classical facts about elliptic curves over finite fields. In Section 3, we prove Theorem 1.2; combined with an equidistribution argument, this allows us to prove the asymptotic result Theorem 1.1(a). Section 4 takes up the analysis of 2- torsion, and in particular proves Theorem 1.1(b). In Section 5, we follow the suggestion (indeed, detailed sketch) of the referee to derive and interpret formulae for S(Fq; `), S0(Fq; `) and S∗(Fq; `) in terms of class numbers. Finally, many of our results, especially those of Section 3, generalize readily to abelian varieties of arbitrary dimension. This is explained in greater detail in Section 6. 2. Existence of curves with split torsion In this section we prove Lemma 2.2 which is stated in the introduction. It is in fact an immediate consequence of the work of Schoof [9], which in turn is based on Deuring's work on complex multiplication [2]. We now summary the relevant parts of Schoof's work, concentrating on the case of ordinary curves in view of the discussion in Section 5. First, we introduce some notation. 2 Given a binary quadratic form ax2 + bxy + cy2, its discriminant is defined to be b2 − 4ac, and we say that it is primitive if gcd(a; b; c) = 1. Fix a negative integer ∆ ≡ 0 or 1 (mod 4). 1 Denote by H(∆) (resp. h(∆)) the number of SL2(Z)-equivalent classes of binary quadratic forms (resp. primitive binary quadratic forms) of discriminant ∆. We have the classical relation (cf. [9, Prop. 2.2]) X (2.1) H(∆) = h(∆=d2); d where d run through all positive integers such that d2j∆ and that ∆=d2 ≡ 0 or 1 (mod 4). In particular, h(∆) is the usual class number of O(∆), the unique complex quadratic order of discriminant ∆. 2 For any integer t ≤ 4q, denote by I(Fq; t) the set of Fq-isomorphism classes of elliptic curves with q +1−t Fq-rational points. Either each elliptic curve in the isogeny class I(Fq; t) is supersingular, or each is ordinary; the isogeny class is ordinary if and only if gcd(t; q) = 1. For a prime number `, let S(Fq; `; t) = S(Fq; `) \ I(Fq; t), and define S∗(Fq; `; t) and S0(Fq; `; t) analogously. The following results are taken verbatim from [9]. Theorem 2.1 (Schoof). Let q be a prime power and let t be an integer such that t2 ≤ 4q and gcd(t; q) = 1. 2 (a) [9, Thm. 4.3(i)] We have E 2 I(Fq; t) if and only if O(t − 4q) ⊂ EndFq (E). (b) [9, Thm. 4.3(i) and Thm. 4.5(i)] Let O be a complex quadratic order that contains 2 O(t − 4q). Then there are h(O) isomorphism classes of elliptic curves E 2 I(Fq; t) ∼ such that EndFq (E) = O. (c) [9, Prop. 3.7] Let ` be a prime number such that gcd(`; q) = 1. We have E 2 S(Fq; `; t) 2 t2−4q if and only if q + 1 − t ≡ 0 mod ` and O( `2 ) ⊂ EndFq (E). Lemma 2.2. Let n be an odd integer with q ≡ 1 (mod n). Then there exists an ordinary 4 elliptic curve E 2 S(Fq; n) if q ≥ n . Proof. We can find an integer t 2 [0; n2] such that t ≡ q + 1 (mod n2). Replace t by t + n2 if necessary, we can further require that gcd(t; q) = 1 and t2 ≤ (t + n2) ≤ 4n2 ≤ 4q. Apply Theorem 2.1 and we are done. Remark 2.3. For odd `, the curves in S(Fq; `) are parameterized by the Fq-points of the fine moduli Y (`). The modular curve X(`) has genus 1 + (`2 − 1)(` − 6)=24, so the Riemann 6 hypothesis implies that X(`)(Fq) is not empty for q ` , which is weaker than Lemma 2.2. 3. The case of ` odd 3.1. Pointwise results. Throughout this section we consider an elliptic curve E over a field K in which the odd, rational prime ` is invertible.
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