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On the group orders of elliptic curves over finite fields. (English) Zbl 0793.14023
Suppose that a finite field \(\mathbb{F}_ q\) of \(q\) elements and a positive integer \(N\) are given. Denote by \(V(\mathbb{F}_ q;N):=\{E/ \mathbb{F}_ q; N | \# E(\mathbb{F}_ q)\}/ \cong_{\mathbb{F}_ q}\) the set of \(\mathbb{F}_ q\)- isomorphism classes of elliptic curves \(E\) over \(\mathbb{F}_ q\) such that the order of the group \(E(\mathbb{F}_ q)\) of \(\mathbb{F}_ q\)-rational points is divisible by \(N\). The main result of the paper is to give an estimate for the weighted cardinality of the set \(V(\mathbb{F}_ q;N)\). (The weighted cardinality of a set \(S\) of \(\mathbb{F}_ q\)-isomorphism classes of elliptic curves over \(\mathbb{F}_ q\) is defined to be \(\#'S=\sum_{[E] \in S} {1 \over \# \operatorname{Aut}_{\mathbb{F}_ q} (E)}\), where \([E]\) stands for the \(\mathbb{F}_ q\)-isomorphism classes of elliptic curve \(E.)\) It is easily seen that the weighted cardinality of the set of all \(\mathbb{F}_ q\)-isomorphism classes of elliptic curves over \(\mathbb{F}_ q\) is equal to \(q\). The paper gives an estimate for the ratio \(\#' V(\mathbb{F}_ q:N)/q\). First fix some notation: \(\lfloor x \rfloor\) (resp. \(\lceil x \rceil)\) denotes the greatest integer less (resp. least integer greater) than or equal to \(x \in \mathbb{R}\).
Theorem. There is a constant \(C \leq 1/12+5 \sqrt 2/6 \approx 1.262\) such that the following statement is true: Given a prime power \(q\), let \(r\) be the multiplicative arithmetic function such that for all primes \(\ell\) and positive integer a holds \(r(\ell^ a)= 1/ (\ell^{a-1} (\ell-1))\) if \(q \not\equiv 1 \pmod {\ell^ c}\); \(r(\ell^ a)=(\ell^{b+1}+\ell^ b-1)/ \ell^{a+b-1} (\ell^ 2-1))\) if \(q \equiv 1 \pmod {\ell^ c}\) where \(b=\lfloor a/2 \rfloor\) and \(c=\lceil a/2 \rceil\). Then for all positive integer \(N\), one has \[ \left | {\#' V(\mathbb{F}_ q;N) \over q}-r(N) \right | \leq {CN \rho (N)2^{\nu (N)} \over \sqrt q}, \] where \(\rho (N)=\prod_{p | N} {p+1 \over p- 1}\) and \(\nu (N)\) is the number of prime divisors of \(N\).
This extends the estimate of H. W. Lenstra jun. given in Ann. Math., II. Ser. 126, 649-673 (1987; Zbl 0629.10006), for the special case when \(N\) and \(q\) are distinct primes with \(q>3\). – The strategy of proof is the same as that of Lenstra, namely, to make use of certain modular curves and estimate the number in question on them. The modular curves used here are the quotients of some familiar modular curves: e.g., for every pair of positive integers \((m,n)\) with \(m | \text{gcd} (n,q- 1)\), the author constructs a modular curve over \(\mathbb{F}_ q\) that parametrizes elliptic curves over \(\mathbb{F}_ q\) having \(\mathbb{F}_ q\)- rational points \(P\) and \(Q\) of order \(m\) and \(n\), respectively, with \(P\) and \((n/m)Q\) having a given Weil pairing. Let \(W(\mathbb{F}_ q;m,n)=\{E/ \mathbb{F}_ q; E[n] (\mathbb{F}_ q) \cong (\mathbb{Z}/m \mathbb{Z}) \times (\mathbb{Z}/n \mathbb{Z})\}/ \cong_{\mathbb{F}_ q}\). First it is shown that \(V(\mathbb{F}_ q;N)=\coprod W(\mathbb{F}_ q;d,N/ \text{gcd} (d,t(N)))\) where the coproduct runs over \(d\) such that \(d | \text{gcd} (u(N),q-1)\). (Here \(t\) and \(u\) are multiplicative arithmetic function defined on prime powers \(\ell^ a\) by \(t(\ell^ a)=\ell^{\lfloor a/2 \rfloor}\) and \(u(\ell^ a)=\ell^{\lceil a/2 \rceil}\).) Then the inequality in the theorem follows from giving an estimate for the quantity in the right hand side of the following identity: \(\#' V(\mathbb{F}_ q;N)=\sum \#' W(\mathbb{F}_ q;d,N/ \text{gcd} (d,t(N)))\) where the sum is taken over all \(d\) such that \(d | \text{gcd} (u(N),q-1)\).

MSC:
14H52 Elliptic curves
14G15 Finite ground fields in algebraic geometry
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
11G20 Curves over finite and local fields
14G05 Rational points
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References:
[1] Enrico Bombieri : Counting points on curves over finite fields (d’après S. A. Stepanov) , Séminaire Bourbaki 430 (1972/73), pp. 234-241 in Lecture Notes in Mathematics 383, Springer-Verlag, New York, 1974. · Zbl 0307.14011 · numdam:SB_1972-1973__15__234_0 · eudml:109832
[2] P. Deligne and M. Rapoport : Les schémas de modules des courbes elliptiques , pp. 143-316 in Modular Functions of One Variable II , Lecture Notes in Mathematics 349, Springer-Verlag, New York, 1973. · Zbl 0281.14010
[3] Robin Hartshorne : Algebraic Geometry , Graduate Texts in Mathematics 52, Springer-Verlag, New York, 1977. · Zbl 0367.14001
[4] Nicholas M. Katz and Barry Mazur: Arithmetic Moduli of Elliptic Curves , Annals of Mathematics Studies 108, Princeton University Press, Princeton, New Jersey, 1985. · Zbl 0576.14026 · doi:10.1515/9781400881710
[5] Serge Lang : Algebra , Addison-Wesley Publishing Company, Menlo Park, California, 1984. · Zbl 0712.00001
[6] H.W. Lenstra, Jr. : Factoring integers with elliptic curves , Ann. of Math. 126 (1987) 649-673. · Zbl 0629.10006 · doi:10.2307/1971363
[7] Jean-Pierre Serre : Algebraic Groups and Class Fields , Graduate Texts in Mathematics 117, Springer-Verlag, New York, 1988. · Zbl 0703.14001
[8] Jean-Pierre Serre : Cohomologie Galoisienne , Lecture Notes in Mathematics 5, Springer-Verlag, New York, 1964. · Zbl 0259.12011
[9] Joseph H. Silverman: The Arithmetic of Elliptic Curves , Graduate Texts in Mathematics 106, Springer-Verlag, New York, 1986. · Zbl 0585.14026
[10] André Weil : The field of definition of a variety , Amer. J. of Math. 78 (1956) 509-524. · Zbl 0072.16001 · doi:10.2307/2372670
[11] André Weil : Sur les Courbes Algébriques et les Variétés qui s’en Déduisent , Hermann, Paris, 1948. · Zbl 0036.16001
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