On group structures realized by elliptic curves over arbitrary finite fields William D. Banks Department of Mathematics, University of Missouri Columbia, MO 65211 USA [email protected] Francesco Pappalardi Dipartimento di Matematica, Universit`aRoma Tre Roma, I{00146, Italy [email protected] Igor E. Shparlinski Department of Computing, Macquarie University Sydney, NSW 2109, Australia [email protected] November 2, 2018 Abstract arXiv:1003.3004v1 [math.NT] 15 Mar 2010 We study the collection of group structures that can be realized as a group of rational points on an elliptic curve over a finite field (such groups are well known to be of rank at most two). We also study various subsets of this collection which correspond to curves over prime fields or to curves with a prescribed torsion. Some of our results are rigorous and are based on recent advances in analytic number theory, some are conditional under certain widely believed conjectures, and others are purely heuristic in nature. 1 1 Introduction Let Fq denote the finite field with q elements. It is well known that the group E(Fq) of points on an elliptic curve E defined over Fq has rank at most two, and therefore, ∼ E(Fq) = Zn × Zkn (1) for some natural numbers n and k, where Zm denotes the ring of congruence classes modulo m for each natural number m; see [7, 13, 21, 23]. On the other hand, little is known about the structure of the set of groups Zn × Zkn that can be realized as the group of points on an elliptic curve defined over a finite field. For this reason, we introduce and investigate the set 2 ∼ SΠ = (n; k) 2 N : 9 prime power q and E=Fq with E(Fq) = Zn × Zkn : We are also interested in groups Zn × Zkn with a realization (1) in which q = p is a prime number, hence we study the subset Sπ ⊂ SΠ defined by 2 ∼ Sπ = (n; k) 2 N : 9 prime p and E=Fp with E(Fp) = Zn × Zkn : 2 Although one can expect Sπ and SΠ to be reasonably \dense" in N , the complementary sets also appear to be rather large. For example, here is the list of pairs (n; k) 62 SΠ with n; k 6 25: (11; 1); (11; 14); (13; 6); (13; 25); (15; 4); (19; 7); (19; 10); (19; 14); (19; 15); (19; 18); (2) (21; 18); (23; 1); (23; 5); (23; 8); (23; 19); (25; 5); (25; 14): 2 To investigate the distribution in N of the elements of Sπ and of SΠ, for natural numbers N and K we introduce the sets Sπ(N; K) = (n; k) 2 Sπ : n 6 N; k 6 K ; SΠ(N; K) = (n; k) 2 SΠ : n 6 N; k 6 K : These sets are the main objects of study in this note. For natural numbers n and k, we also put ∼ P(n; k) = primes p : 9 E=Fp for which E(Fp) = Zn × Zkn : 2 The set P(n; k) parametrizes the set of finite fields of prime cardinality over which Zn × Zkn can be realized as the group of points on an elliptic curve. For natural numbers N and K we study the double sum X X NP (N; K) = #P(n; k); n6N k6K for which we obtain an asymptotic formula in certain ranges. Finally, for natural numbers m; k we introduce and compare the sets ∼ Nm;k = n 2 N : 9 p prime and E=Fpm with E(Fpm ) = Zn × Zkn ; p m 2 Nem;k = n 2 N : 9 p prime, ` 2 Z with p = kn + `n + 1; j`j 6 2 k : We remark that the distribution of group structures generated by elliptic curves over a fixed finite field Fq has been studied in [12]. 2 Notational conventions Throughout the paper, the letter p always denotes a prime number, and q always denotes a prime power. As usual, we use π(x) to denote the number of p 6 x. For coprime integers a and m > 1, we put π(x; m; a) = # p 6 x : p ≡ a (mod m) ; Π(x; m; a) = # q 6 x : q ≡ a (mod m) : We also set X (x; m; a) = Λ(n); n6x n≡a (mod m) where Λ(n) is the von Mangoldt function. For any set A ⊆ N and real x > 0, we denote A(x) = a 2 A : a 6 x . For functions F and G > 0 the notations F = O(G), F G, and G F are all equivalent to the assertion that the inequality jF j 6 c G holds with some constant c > 0. In what follows, all constants implied by the symbols O, , and may depend (where obvious) on the small real parameter " but are absolute otherwise; we write Oρ, ρ, and ρ to indicate that the implied constant depends on a given parameter ρ. 3 3 Preliminaries Lemma 1. If q is a prime power, and E is an elliptic curve defined over Fq ∼ 2 such that E(Fq)p= Zn × Zkn, then q = kn + `n + 1 for some integer ` that satisfies j`j 6 2 k. Proof. By the Hasse bound, we can write kn2 = q + 1 − a for some integer 2 a that satisfies the bound a 6 4q. Using the Weil pairing one also sees that q ≡ 1 (mod n), hence a = `n + 2 for some integer `, and we have q = kn2 + `n + 1. Since 2 2 2 2 2 ` n + 4`n + 4 = (`n + 2) = a 6 4q = 4kn + 4`n + 4; p it follows that j`j 6 2 k as required. The following result of Waterhouse [23] (see also [22, Theorems 4.3]) is a characterization of the natural numbers N that can be realized as the cardinality of the group of Fq-rational points on an elliptic curve E defined over Fq. Lemma 2. Let q = pm be a prime power, and suppose that N = q +1−a for some integer a. Then, there is an elliptic curve E defined over such that p Fq #E(Fq) = N if and only if jaj 6 2 q and one of the following conditions is met: (i) gcd(a; p) = 1; p (ii) m even and a = ±2 q; p (iii) m is even, p 6≡ 1 (mod 3), and a = ± q; (iv) m is odd, p = 2 or 3, and a = ±p(m+1)=2; (v) m is even, p 6≡ 1 (mod 4), and a = 0; (vi) m is odd and a = 0. For every admissible cardinality N, the following result of R¨uck [13] (see also [22, Theorems 4.4]) describes the group structures that are possible for E(Fq) given that #E(Fq) = N; see also [7, 21]. 4 Lemma 3. Let q = pm be a prime power, and suppose that N is an integer such that #E(Fq) = N for some elliptic curve E defined over Fq. Write e N = p n1n2 with p - n1n2 and n1 j n2 (possibly n1 = 1). Then, there is an elliptic curve E over Fq for which ∼ e E(Fq) = Zp × Zn1 × Zn2 if and only if (i) n1 = n2 in case (ii) of Lemma 2; (ii) n1 j q − 1 in all other cases of Lemma 2. Combining Lemmas 2 and 3, we get: p Corollary 4. If p is prime and N 2 N with jp + 1 − Nj 6 2 p, then there is an elliptic curve E defined over Fp with #E(Fp) = N. In this case, if we write N = n1n3 with p - n1 and n1 j n3 (possibly n1 = 1), then n1 j p − 1 and ∼ E(Fp) = Zn1 × Zn3 . 2 Lemma 5. A prime p lies in Pp(n; k) if and only if p = kn + `n + 1 for some integer ` such that j`j 6 2 k. Proof. By definition, if p lies in P(n; k) then there is an elliptic curve E=Fp ∼ 2 such that E(Fp) = Zn × Zkn. Accordingp to Lemma 1, p = kn + `n + 1 with some integer ` such that j`j 6 2 k. p 2 2 Conversely, suppose that p = kn +`n+1 and j`j 6 2 k. Taking N = kn we have 2 2 2 2 2 jp + 1 − Nj = (`n + 2) = ` n + 4`n + 4 6 4kn + 4`n + 4 = 4p; p hence jp + 1 − Nj 6 2 p. Applying Corollary 4 with n1 = n and n3 = kn, ∼ we see that there is an elliptic curve E=Fp such that E(Fp) = Zn × Zkn, and thus p 2 P(n; k). Next, we relate NP (N; K) to the distribution of primes in short arithmetic progressions. Lemma 6. For all N; K 2 N we have X 2 2 1 2 2 2 NP (N; K) = π(Kn +`n+1; n ; `n+1)−π( 4 ` n +`n+1; n ; `n+1) : n6Np j`j62 K 5 Proof. Fix n 6 N, and let T1(n) be the collection of pairs (`; p) such that p (i) j`j 6 2 K; (ii) p is a prime congruent to `n + 1 (mod n2); 1 2 2 2 (iii) 4 ` n + `n + 1 6 p 6 Kn + `n + 1. 1 2 2 1 2 Since 4 ` n + `n + 1 = ( 2 `n + 1) cannot be prime, it is easy to see that X 2 2 1 2 2 2 #T1(n) = π(Kn + `n + 1; n ; `n + 1) − π( 4 ` n + `n + 1; n ; `n + 1) : p j`j62 K Let T2(n) be the collection of pairs (k; p) such that (iv) k 6 K; p 2 (v) p is prime and p = kn + `n + 1 for some integer ` such that j`j 6 2 k.