GROUP STRUCTURES OF ELLIPTIC CURVES OVER FINITE FIELDS

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1 GROUP STRUCTURES OF ELLIPTIC CURVES OVER FINITE FIELDS VORRAPAN CHANDEE, CHANTAL DAVID, DIMITRIS OUOULOPOULOS, AND ETHAN SMITH Abstract. It is we-known that if E is an eiptic curve over the finite fied F p, then EF p ) Z/mZ Z/mkZ for some positive integers m, k. Let SM, ) denote the set of pairs m, k) with m M and k such that there exists an eiptic curve over some prime finite fied whose group of points is isomorphic to Z/mZ Z/mkZ. Banks, Pappaardi and Shparinski recenty conjectured that if og M) 2 ɛ, then a density zero proportion of the groups in question actuay arise as the group of points on some eiptic curve over some prime finite fied. On the other hand, if og M) 2+ɛ, they conjectured that a density one proportion of the groups in question arise as the group of points on some eiptic curve over some prime finite fied. We prove that the first part of their conjecture hods in the fu range og M) 2 ɛ, and we prove that the second part of their conjecture hods in the imited range M 4+ɛ. In the wider range M 2, we show that a positive density of the groups in question actuay occur.. Introduction Let E be an eiptic curve over F p, and denote with EF p ) its set of points over F p. It is we-known that EF p ) admits the structure of an abeian group. It is then natura to ask for a description of the groups that arise this way as p runs through a primes and E through a curves over F p. This question was first addressed by Banks, Pappaardi and Shparinski in [2]. Beow we reproduce part of the discussion from [2]. The first reevant property is that the size of EF p ) can never be very far from p +. Indeed, if #EF p ) = p + a p, then Hasse proved that a p 2 p. Setting x := x + 2 x = x ) 2 and x + := x x = x + ) 2. for each x, this is equivaent to saying that #EF p ) p, p + ). It foows from the work of Deuring [7] that for any integer N satisfying p < N < p +, there exists an eiptic curve E/F p with #EF p ) = N. Soving the inequaities for p aows us to concude that, given a positive integer N, there is a finite fied F p and an eiptic curve E/F p with #EF p ) = N if and ony if there is a prime p N, N + ). However, this resut does not take into account the actua group structure of EF p ). The second reevant property is that, as an abstract abeian group, EF p ) has at most two invariant factors. In other words, we may write that EF p ) G m,k := Z/mZ Z/mkZ for some unique positive integers m, k. Refining the ideas aready present in the work of Deuring, one can argue that there is an eiptic curve E/F p with EF p ) G m,k if and ony if N = m 2 k p, p + ) and p mod m). Arguing as before aows us to concude that, given a group G m,k of order N = m 2 k, there is a finite fied F p and an eiptic curve E/F p with EF p ) G m,k if and ony if there is a prime p mod m) in the interva

2 N, N + ). The atter condition is equivaent to the assertion that there is a prime of the form p = km 2 + jm + with j < 2 k. See Lemma 2.2 beow. The above characterization gives some interesting consequences. Note that when k is very sma it is unikey that there is a finite fied F p and a curve E/F p such that EF p ) G m,k simpy because the interva N, N + ) is too short. For exampe, there is no curve over F p such that EF p ) Z/Z Z/Z, since none of the three integers 22, 22, 22 + is prime. Other exampes of groups not occurring are given by Banks, Pappaardi, and Shparinski in [2]. In order to study the question of which groups G m,k occur as group structures of eiptic curves over F p from an average point of view, the authors of [2] defined SM, ) := {m M, k : there is a prime p and a curve E/F p such that EF p ) G m,k }. They proved the foowing resut for the cardinaity of SM, ). Theorem. Banks, Pappaardi, and Shparinski [2]). Let M 2 and. Then for every fixed, we have M #SM, ) og M. If M 43/94 ɛ, then #SM, ) M og. Finay, if M /2 ɛ, then #SM, ) M og ). 2 Moreover, the authors of [2] conjectured the foowing. Conjecture.2 Banks, Pappaardi, Shparinski [2]). { om) if og M) #SM, ) = 2 ɛ, M + o if og M) 2+ɛ. The motivation behind the above conjecture can be expained by a simpe heuristic. An integer n is prime with probabiity about. For G og n m,k to be the group of a curve E over some finite fied, we need at east one of the integers n = km 2 + jm + with j < 2 k to be prime. If we assume that these events occur independenty of each other, the probabiity that none of the integers n = km 2 + jm +, j < 2 k, is prime is about ogm 2 k) ) 4 k. This quantity becomes ess than one as soon as k ogm 2 k). In particuar, if k og m) 2+ɛ, then we expect with probabiity that km 2 + jm + is prime for some j 2 k, 2 k). One can make the even boder guess that if k is arge enough, then there is aways some j 2 k, 2 k) for which km 2 + jm + is prime. This question is competey out of reach with the current technoogy, as we do not even know whether there are primes in every interva of the form x, x + x ) with x arge enough. The best resut known, due to Baker, Harman and Pintz [], is that x, x + x ) contains primes for every sufficienty arge x. 2

3 In this paper we improve upon Theorem.. Our first resut is that the first part of Conjecture.2 hods for M, in the predicted range. Theorem.3. Let M 2 and. Then we have that #SM, ) M3/2 og M. In particuar, if og M) 2 ɛ for some fixed ɛ > 0, then #SM, ) = o ɛ M) as M. We aso prove that the second part of Conjecture.2 hods for a restricted range of M and. Theorem.4. Fix A and ɛ > 0. If M /4 ɛ, then ) M #SM, ) = M + O ɛ,a. og ) A Finay, we show that a ower bound of the correct order of magnitude aso hods in some arger range. Theorem.5. For M /2, we have that #SM, ) M. Notation. Given an integer n, we et P + n) and P n) denote its argest and smaest primes factors, respectivey, with the notationa conventions that P + ) = and P ) =. As usuay, τ, µ, φ and Λ denote the divisor, the Möbius, the totient and the von Mangodt function, respectivey. Furthermore, we et πx; q, a) be the number of primes up to x that are congruent to a mod q) and ψx; q, a) := Λn). n a mod q) The etters p and aways denote prime numbers. Finay, we write f a,b,... g if there is a constant c, depending at most on a, b,..., such that f cg, and we write f a,b,... g if f a,b,... g and g a,b,... f. 2. Preiminaries and Cohen-Lenstra heuristics In this section we expain how the existence of an eiptic curve over a prime finite fied with a given group structure is equivaent to the existence of a prime in a certain interva with a given congruence condition. Some of the resuts and arguments of this section are very simiar to Section 3 of [2], but we reproduce them here for the sake of competeness. The first emma is a resut of Rück [2], who used the work of Deuring, Waterhouse, and Tate-Honda to characterize those groups which actuay occur as the group of points on eiptic curves over finite fieds. Lemma 2. Rück). Let N = h be a possibe order #EFp ) for an eiptic curve E/F p, i.e., N p, p + ). Then a the possibe groups EF p ) with #EF p ) = N are Z/p hp Z Z/ b Z Z/ h b Z ) p 3

4 where b are arbitrary integers satisfying 0 b min v p ), h 2 ). As a coroary of the above emma, we have the foowing resut, which is Lemma 3.5 in [2]. Coroary 2.2. Let m and k be integers. There is a prime p and a curve E over F p such that EF p ) G m,k if and ony if there is a prime p mod m) in the interva I m 2 k := km 2 2m k +, km 2 + 2m ) k + or, equivaenty, if and ony if there is a prime p = km 2 + jm + with j < 2 k. Proof. Suppose that there exists an eiptic curve E over F p such that EF p ) G m,k. As mentioned in the introduction, we must have that N = m 2 k = #EF p ) p, p + ). Soving for p as in the introduction gives that p N, N + ) = I mk 2. Since the m-torsion points are contained in EF p ) and since the Wei pairing is surjective, F p must contain the m-th roots of unity, which is equivaent to saying that p mod m). Conversey, suppose that there is a prime p I m 2 k such that p mod m), and et N = km 2. It is easy to check that p + N 2 p, that is to say that N is an admissibe order. Writing N = km 2 = h, we ceary have that v m) h /2. Furthermore, since p mod m), we aso have that v p ) v m) for each m. Thus, we may take b = v m) in Lemma 2. for a m. So, in particuar, h b = v m) + v k), and we concude that G m,k = Z/ v m) Z Z/ v m)+v k) Z ) is an admissibe group. This competes the proof of the emma. The fact that the groups G m,k are more ikey to occur when m is sma can be seen using the Cohen-Lenstra heuristics which predict that random abeian groups naturay occur with probabiity inversey proportiona to the size of their automorphism groups. In particuar, those groups which are neary cycic are the most ikey to occur. In order to see that the probabiity of occurrence of the groups G m,k is reay in correspondence with the weights suggested by the Cohen-Lenstra heuristics, one shoud count the number of times a given group G m,k occurs as EF p ), and not ony if it occurs. More precisey, given a group G of order N and a prime p, et The quantity in question then is the sum M p G) := # {E/F p : EF p ) G}. MG) := N <p<n + M p G). Using the proper generaization of Deuring s work, MG) can be reated to a certain average of ronecker cass numbers. See [3]. It is shown in [6] that, under a suitabe hypothesis for the number of primes in short arithmetic progressions, MG m,k ) 4 N/ og N #G m,k A G m,k ) #AutG m,k ) N 3/2 N = m 2 k, m og k) A, k ), 2.) where G m,k ) is non-zero and uniformy bounded for a integers m and k. So we see that the average frequency of occurrence of groups of eiptic curves over finite fieds is compatibe with the Cohen-Lenstra heuristics. 4

5 As we mentioned above, the resuts of [6] are conditiona under some hypothesis for the number of primes in short arithmetic progressions because the intervas N, N + ) are so short that even the Riemann hypothesis does not guarantee the existence of a prime. Nevertheess, it is possibe to obtain unconditiona resuts dispaying the Cohen-Lenstra phenomenon, by showing that the asymptotic in 2.) is an upper bound for a groups G, and a ower bound for most of the groups G moduo constants). This work is in progress [4]. The proof of the ower bound for most of the groups G has simiarities with the proof of Theorem.4 of the present paper and, in particuar, it requires the generaization of Seberg s theorem about primes in short arithmetic progressions due to the third author [9], but it invoves more technica difficuties, as one needs to combine this with the arguments of [6]. 3. Auxiiary resuts In this section, we coect some technica resuts that wi be needed to prove the theorems. First, we state the fundamenta emma of the combinatoria sieve see, for exampe, [6, Theorem 3, p. 60]), which wi be used in the proof of Theorem.3. Given a finite set of integers A and a number y, we set SA, y) := #{a A : P a) > y}. As is customary, we assume that there is a mutipicative function ρ and a number X such that for every integer d #{a A : a 0 mod d)} = X ρd) d + R d for some rea number R d, which we think of as an error term. Then we have the foowing resut. Lemma 3.. Let A, ρ, X and {R d : d N} be as above. If ρp) min{2, p } for a primes p, then we have that SA, y) = X ρ) ) { + Ou u/2 ) } + O µ 2 d) R d, y uniformy for a y and u. The next emma wi be used in the proof of Theorem.3. d y u, P + d) y Lemma 3.2. Fix ɛ > 0 and et χ be a non-principa character mod q. For every y, we have that χ) ) ɛ q /2+ɛ. y Proof. Mertens s estimate impies that χ) ) q /2+ɛ y exp{q /2+ɛ }< y χ) ). Moreover, by the discussion in [5, p. 23], we have that x Λn)χn) ɛ og x x exp{q/2+ɛ }), 3.) n x 5

6 using the trivia bound β < c/q /2 og q) for the Siege zero provided by the cass number formua. Partia summation then impies that og χ) ) = Λn)χn) n og n exp{q /2+ɛ }< y which competes the proof of the emma. = n exp{q /2+ɛ }< y exp{q /2+ɛ }<n y Λn)χn) n og n + O), The next emma, which is essentiay due to Eiott, aows us to bound the vaue of L, χ) by a very short product for most quadratic characters χ. Lemma 3.3. Let δ 0, ] and Q 3. There is a set E δ Q) Z [, Q] of size Q δ such that if χ is a non-principa, quadratic Dirichet character moduo some q Q and of conductor not in E δ Q), then y< z χ) ) δ z y og Q). Proof. We borrow from the proof of Proposition 2.2 in [8], which is essentiay due to Eiott. Without oss of generaity, we may assume that Q is arge enough. By Theorem in [0], for every σ 0 [4/5, ], Q 2 and T, there are Q 2 T ) 2 σ 0)/σ 0 og Q) 4 primitive characters of conductor beow Q whose L-function has a zero in the region {s = σ + it C : σ σ 0, t T }. Let E δ Q) be the set of conductors corresponding to these exceptiona characters with σ 0 = δ/2 /2 and T = Q 3. If χ is a Dirichet character mod q [, Q] of conductor not in E δ Q), then Ls, χ) has no zeroes in {s = σ + it C : σ δ/2, t Q 3 }. So by [5, eqn. 7), p. 20] appied with T = min{q 3, x}, we find that n x Λn)χn) x δ/2 og 2 x + x og2 x Q 3 + og 2 Q δ x og x + og2 Q 2 x e Q ). The above estimate aso hods for x e Q by 3.). Together with partia summation, this impies that { og χ) ) } = Λn)χn) n og n = Λn)χn) n og n + O) δ 3.2) y< z n y< z y<n z for z y og 2 Q, that is to say, the emma does hod in this range of y and z. Finay, if og Q y < og 2 Q, then setting w = min{z, og 2 Q}, we have that χ) ) = χ) ) χ) ) δ y< z y< w w< z by 3.2) and Mertens s estimate, and the emma foows. Next, we state the Bombieri-Vinogradov theorem [3, 4, 5], which wi be used to prove Theorem.5. 6

7 Lemma 3.4 Bombieri-Vinogradov). Let A > 0 be fixed. Then there exists a B = BA) > 0, depending on A, such that max y x q x /2 /og x) B a,q)= iy) πy; q, a) φq) x og x) A. Finay, in order to prove Theorem.4, we need the foowing short interva version of the Bombieri-Vinogradov theorem, due to the third author [9]. Lemma 3.5. Fix ɛ > 0 and A. For x h 2 and Q 2 h/x /6+ɛ, we have that 2x x max a,q)= q Q ψy + h; q, a) ψy; q, a) h φq) dy xh og x) A. By Coroary 2.2, we readiy have that where 4. Proof of Theorem.3 #SM, ) k j <2 k S k,j := # { m M : km 2 + jm + is prime }. S k,j, 4.) Using the combinatoria sieve to bound S k,j, one immediatey obtains as in [2], that for any fixed, #SM, ) M/ og M. By keeping track of the dependence on j, k and summing we wi prove Theorem.3. In the notation of Lemma 3., et A = {km 2 + jm + : m M}, and note that where #{a A : a 0 mod d)} = #{m M : km 2 + jm + 0 mod d)} = M ρk,jd) d + O ρ k,j d, ρ k,j d) = # { c Z/dZ : kc 2 + jc + 0 mod d) }. The Chinese remainder theorem impies that ρ j,k is a mutipicative function. Moreover, by a straightforward computation, we find that k j ) 2 2 ) if = 2, j ρ k,j ) = + 2 4k if k and 2, ) if k, j 2 7

8 for a primes. Since S k,j SA, y) + y for a y, appying Lemma 3. with y = M /2 and u = yieds the estimate S k,j M ρ ) k,j) + µ 2 d) ρ k,j d) + M /2 y d M ) /2 ) M j 2 j + 2 4k + M /2 og M k, y k, y ) M og M k φk) j 2 4k + M /2 og M. This impies that #SM, ) M og M y k, j<2 k k φk) y j 2 4k ) + M /2 3/2 og M. Observing that j 2 4k [ 4, ] for j and k as above, we fix d [, 4] and seek a bound for the sum k T d := φk) + ). First, note that + ) k, > og k, j <2 k j 2 4k= d k, >og + ) k, j <2 k j 2 4k= d exp k k, >og { } #{ k} exp, og by Mertens s estimate and the fact that k has at most og k distinct prime factors. Therefore, og 2 + ) + ) µ 2 a) = a. So T d k P + a) og µ 2 a) a k, og k, j <2 k a k, j 2 4k= d µ2 a) τa) a P + a) og ) /2 a k P + a) og µ2 a) a P + a) og ) /2 a + ), j <2 4a j 2 +d since a e π og ) for a square-free integers a with P + a) og. Consequenty, #SM, ) M d + M /2 3/2 og M og M d 4 y 8

9 Using Lemma 3.3 on truncated products of L-functions with δ = /4 and Q = 4, we find that there is a set E of O /4 ) integers in [, 4] such that if d [, 4] and the conductor of ) d is not in E, then d w 2 w og4. w < w 2 So for such a d we find that y d z d, 4.2) where z = min{y, og4)}. For the exceptiona d s, we write d = a 2 d, where d denotes the conductor of ) d and note that d a d a φa) φa) d 3/4 y by Lemma 3.2. Hence, d 4 cond d E y d #SM, ) M og M y d E d E d E d 4 z a 4/ d y a 4/ d The above reation and 4.2) then impy that a φa) d 3/4 d a 2 ) d /4 /4 /4 =. d + M3/2 og M. 4.3) In order to contro the above sum, we proceed by expanding the product to a sum and inverting the order of summation. We have that d = µa) ) d. a a d 4 z P + a) z d 4 If a =, the inner sum is 4 + O); ese, it is a. So d + µ 2 a) = + 2 πz) + 2 π og4. d 4 z P + a) z Inserting the ast estimate into 4.3), we obtain the inequaity #SM, ) M3/2 og M, 9

10 which competes the proof of Theorem.3. Define 5. Proof of Theorem.4 RM, ) := {M/2 < m M, /2 < k : there is no prime p mod m) in I m 2 k}. First, we prove an intermediate resut for the cardinaity of RM, ). Theorem 5.. Fix A and ɛ > 0. If M /4 ɛ, then #RM, ) ɛ,a M og ) A. Proof. Set h = M and Ey, h; q, a) = ψy + h; q, a) ψy; q, a) h φq), and note that if the pair m, k) RM, ), then Consequenty, Em k ) 2, h; m, ) #RM, ) M/2<m M /2<k h φm) h m. Em k ) 2, h; m, ). Next, observe that m k ) 2 J := [M 2 /0, M 2 ]. We cover the interva J by M 2 ) 2/3 subintervas J r of ength M 2 ) /3 each. If m k ) 2 J r, then for every y J r we have that Ey, h; m, ) Em k ) 2, h; m, ) 2 measj r) + #{p J r J r + h) : p mod m)} φm) by the Brun-Titchmarsh inequaity. So and consequenty, Em k ) 2, h; m, ) = #RM, ) M/2<m M measj r ) M 2 ) /3 φm) M 2 ) /3 ) M 2 ) /3 Ey, h; m, )dy + O J r φm) Ey, h; m, )dy + Jr M 2 ) /3, φm) Ey, h; m, )dy + M 2 ) Jr M ) 2 ) /3 /3 φm) J r 0 /2<k m 2 k J r.

11 For every fixed m [M/2, M] and every fixed interva J r, there are at most M 2 ) /3 /M 2 vaues of k with m k ) 2 J r. Therefore we deduce that #RM, ) Ey, h; m, )dy + M 2 ) Jr M ) 2 ) /3 M 2 ) /3 /3 φm) M 2 M 2 M/2<m M M/2<m M Since M /4 ɛ, we have that J r M 2 M 2 /20 M 2 and we can appy Lemma 3.5 to get that #RM, ) ɛ,a M 2 thus competing the proof of Theorem 5.. Ey, h; m, )dy + O 5/6 M 2/3 ). M M 2 ) /6+ɛ/2, Proof of Theorem.4. Let M /4 ɛ. Ceary, M 2 h og ) A + 5/6 M 2/3 A #{m M, k : there is no prime p mod m) in I m 2 k} M og ) A, 2 a 2M, 2 b 2 #R2 a, 2 b ). If 2 b ɛ, then we use the trivia bound #R2 a, 2 b ) 2 a+b. Otherwise, we have that 2 a /4 ɛ 2 b/4 ɛ/2), and so Theorem 5. impies that #R2 a, 2 b ) ɛ,a 2 a+b /b A. Consequenty, and Theorem.4 foows. #{m M, k : there is no prime p mod m) in I m 2 k} a+b 2 a+b A + b M A ɛ,a og ), A 2 2 a 2M 2 b ɛ 2 a 2M ɛ <2 b 2 6. Proof of Theorem.5 Note that the primes 2 and 3 are aways contained in SM, ), since I = 0, 4). So #SM, ) 2, and consequenty, we may assume without oss of generaity that is arge enough. Aso, we may assume that M is an integer, so that the interva 3M/4, M] aways contain integers. Proposition 2.2 impies that #SM, ) = Im, k), m M where Im, k) = k { if there exists a prime p I m 2 k such that p mod m), 0 otherwise.

12 Now, note that Im, k) φm) og4 k) 8m k # {p I m 2 k : p mod m)}, by the Brun-Titchmarsh inequaity. Therefore we deduce that #SM, ) og og /5<k φm) m M 2 /4<p<M 2 /3 p I m 2 k p mod m) p mod m) φm) m 6.) /5<k by switching the order of summation and restricting p in the interva M 2 /4, M 2 /3]. Fix p and m as in 6.) and note that if k is an integer for which p I m 2 k, then we necessariy have that /5 < k. So = /5<k p I m 2 k since { = # k Z : p 2 p + < k < p + 2 } p + m 2 m 2 k Z : p I m 2 k 4 p m > 4 M 2 /4 = 2 2 M 2 M 2 by our assumption that M. Consequenty, #SM, ) og M = og M = og M M 2 /4<p<M 2 /3 φm m p mod m) φm) m p m 2 πm 2 /3; m, ) πm 2 /4; m, ) ) φm) im 2 /3) im 2 /4) m φm) M, + E, 6.2) where E = φm) m πm 2 /3; m, ) πm 2 /4; m, ) im ) 2 /3) im 2 /4). φm) The sum of the main term in 6.2) is im 2 /3) im 2 /4) m 2 M 2 M ogm 2 ) M 2 og.

13 It is in this step that we need that the interva 3M/4, M] contains an integer. Furthermore, we have that E πm 2 /4; m, ) πm 2 /3; m, ) im 2 /3) im 2 /4) φm) M 2 og 2 M 2 ), by Lemma 3.4. Combining the above estimates, we find that there is an absoute constant c such that ) M #SM, ) cm + O cm, og 2 provided that is arge enough. This competes the proof of Theorem.5. Acknowedgements The authors woud ike to thank Roger Heath-Brown for suggesting some usefu references about counting primes in short intervas. The second author woud aso ike to thank Aru Shankar for enightening discussions about the Cohen-Lenstra heuristics in the context of eiptic curves and abeian varieties over finite fieds. References [] R. C. Baker, G. Harman, G. and J. Pintz, The difference between consecutive primes. II. Proc. London Math. Soc. 3) ), no. 3, [2] W. D. Banks, F. Pappaardi and I. Shparinski, On group structures reaized by eiptic curves over arbitrary finite fieds. Experimenta Mathematics 2: 202), 25. [3] E. Bombieri, On the arge sieve. Mathematika 2 965) [4] V. Chandee, C. David, D. oukouopouos and E. Smith, Eiptic curves over F p with a given group structure. In preparation. [5] H. Davenport, Mutipicative number theory. Third edition. Revised and with a preface by Hugh L. Montgomery. Graduate Texts in Mathematics, 74. Springer-Verag, New York, [6] C. David and E. Smith, A Cohen-Lenstra phenomenon for eiptic curves. Preprint, 20 pages, 202. arxiv: [7] M. Deuring, Die Typen der Mutipikatorenringe eiptischer Funktionenkorper. Abh. Math. Sem. Hansischen Univ. 4 94), [8] A. Granvie and. Soundararajan, The distribution of vaues of L, χ d ). Geom. Funct. Ana ), no. 5, [9] D. oukouopouos, Primes in short arithmetic progressions. Preprint, 202. [0] H. L. Montgomery, Zeros of L-functions. Invent. Math ), [] H. Iwaniec and E. owaski, Anaytic Number Theory, American Mathematica Society Cooquium Pubications, vo. 53, [2] H.-G. Rück, A note on eiptic curves over finite fieds. Math. Comp ), [3] R. Schoof, Nonsinguar pane cubic curves over finite fieds. J. Combin. Theory Ser. A 46:2, 987), [4] A. I. Vinogradov, The density hypothesis for Dirichet L-series. Russian) Izv. Akad. Nauk SSSR Ser. Mat ), [5], Correction to the paper of A. I. Vinogradov On the density hypothesis for the Dirichet L-series. Russian) Izv. Akad. Nauk SSSR Ser. Mat ), [6] G. Tenenbaum, Introduction to anaytic and probabiistic number theory, Cambridge University Press, Cambridge,

14 Vorrapan Chandee) Centre de recherches mathématiques, Université de Montréa, P.O. Box 628, Centre-vie Station, Montréa, Québec, H3C 3J7, Canada Chanta David) Department of Mathematics and Statistics, Concordia University, 455 de Maisonneuve West, Montréa, Québec, H3G M8, Canada Dimitris oukouopouos) Département de mathématiques et de statistique, Université de Montra, CP 628, Succ. Centre-Vie, Montréa, QC H3C 3J7 Ethan Smith) Department of Mathematics, Liberty University, 97 University Bvd, MSC Box 70052, Lynchburg, VA