ON INVARIANTS OF ELLIPTIC CURVES ON AVERAGE

Transcription

1 ON INVARIANTS OF ELLIPTIC CURVES ON AVERAGE AMIR AKBARY AND ADAM TYLER FELIX Abstract. We rove several results regarding some invariants of ellitic curves on average over the family of all ellitic curves inside a bo of sides A and B. As an eamle, let E be an ellitic curve defined over Q and be a rime of good reduction for E. Let e E be the eonent of the grou of rational oints of the reduction modulo of E over the finite field F. Let C be the family of ellitic curves E a,b : y = 3 + a + b, where a A and b B. We rove that, for any c > and k N, e k E = c 0li k+ + O E C k+ log c as, as long as A, B > e c log / and AB > log 4+c, where c is a suitable ositive constant. Here c 0 is an elicit constant which deends only on k, and li = dt/ log t. We rove several similar results as corollaries to a general theorem. The method of the roof is caable of imroving some of the known results for a bo of size A, B > ɛ and AB > log δ to a thinner bo of size A, B > e c log / and AB > log δ.,. INTRODUCTION AND RESULTS Let E be an ellitic curve defined over Q of conductor N. For a rime of good reduction i.e. N, let E be the reduction mod of E. It is known that E F, the grou of rational oints of E over the finite field F is the roduct of at most two cyclic grous, namely E F Z/i E Z Z/e E Z, where i E divides e E. Thus, e E is the eonent of E F and i E is the inde of the largest cyclic subgrou of E F. In recent years there has been a lot of interest in studying the distribution of the invariants i E and e E. Borosh, Moreno, and Porta [8] were the first to comutationally studied i E and conjectured that, for some ellitic curves, i E = occurs often. We note that i E = if and only if E F is cyclic. Let N E = #{ ; N and E F is cyclic}.. Then Serre [6], under the assumtion of the generalized Riemann hyothesis GRH for division fields QE[k], roved that N E c E li as, where c E > 0 if and only if QE[] Q. Here li = dt/log t. For the curves with comle multilication CM, Murty [5] removed the assumtion of the GRH and showed that under GRH one can obtain the estimate O log log /log for the error term in the asymtotic formula for N E. The value of the error term is imroved to O 5/6 log /3 in [0]. Also following the method of [5] in the CM case, the error term O/log A for any A > is established in [3]. Another roblem closely related to cyclicity is finding the average value of the number of divisors of i E as varies over rime. Let τn denote the number of divisors of n. In [], Akbary and Ghioca roved that τi E = c E li + O 5/6 log /3 Date: March 6, Mathematics Subject Classification. G05, G0. Key words and hrases. reduction mod of ellitic curves, invariants of ellitic curves, average results. Research of the first author is artially suorted by NSERC. Research of the second author is suorted by a PIMS ostdoctoral fellowshi.

2 if GRH holds, and τi E = c E li + O log A for A >, if E has CM. In the above asymtotic formulas c E is a ositive constant which deends only on E. A more challenging roblem is studying the average value of i E. In [3], Kowalski roosed this roblem and roved unconditionally that the lower bound log log holds for / log i E if E has CM. He also showed that for a non-cm curve the above quantity is bounded from the below. A more aroachable roblem is finding the average value of e E. Freiberg and Kurlberg [6] were the first to consider this roblem and established conditional unconditional in CM case asymtotic formulas for e E. The best result to date is due to Feli and Murty [4] who roved more generally that for k a fied ositive integer the following asymtotic formula holds: e k E = c E,kli k+ + O k E, where /log A E = 5/6 log, if E has CM if GRH holds and c E,k is a ositive constant deending on E and k. Feli and Murty derived their result as a consequence of a more general theorem on asymtotic distribution of i s. Their general theorem also imly the best known results on the cyclicity, the Titchmarsh divisor roblem, and several other similar roblems. To state their result, let gn be an arithmetic function such that gn +β log γ,. where β and γ are arbitrary, and let n Then the following is roved in [4, Theorem.c]. f n = gd..3 d n Theorem. Feli and Murty. Under the assumtion of GRH and bound. for β < / and arbitrary γ, we have f i E = c E f li + O 5+β 6 log β+γ 3, where c E f is a constant deending only on E and f. They also roved an unconditional version of the above theorem for CM ellitic curves see [4, Theorem.a]. Our goal in this aer is to rove that Theorem. holds unconditionally on average over the family of all ellitic curves in a bo. More recisely, we consider the family C of ellitic curves E a,b : y = 3 + a + b, where a A and b B. It is not that difficult to rove a version of Theorem. on average over a large bo. However it is a challenging roblem to establish the same over a thin bo. By a thin bo we mean, as a function of, either A or B can be as small as ɛ for any ɛ > 0. Here we rove a stronger result in which one of A and B can be as small as ec log / for a suitably chosen constant c > 0. Before stating our main theorem, we note that, at the eense of relacing β and γ by larger non-negative values, we can assume that β and γ are non-negative.

3 Theorem.. Let c > be a ositive constant and let f be the summatory function.3 of a function g that satisfies. for certain values of β and γ. Assume that AB > log 4+c if 0 β < / and AB > /+β log γ+6+c log log if / β <. Then there is a ositive constant c > 0 such that if A, B > e c log /, we have where E a,b C f i Ea,b = c 0 f li + O gd c 0 f := dψdϕd. d log c The imlied constant deends on g, β, γ, and c. Here ϕn = n d n / and ψn = d n + /. This theorem is comarable to Stehens s average result on Artin s rimitive root conjecture. Let a be a non-zero integer other than or a erfect square and let A a be the number of rimes not eceeding, for which a is a rimitive root. The following result has been roved in [8] and [9]. Theorem.3 Stehens. There eist a constant c > 0 such that, if N > e c log /, then A a = A li + O N log c, a N where A = /ll and c is an arbitrary constant greater than. l rime The line of research on Artin rimitive root conjecture on average started with the work of Goldfeld [9] that used multilicative character sums and the large sieve inequality to establish a weaker version of Theorem.3. The etension of the method of character sums to the average questions on a two arameters family, in the case of ellitic curves inside a bo, was ioneered by Fouvry and Murty in [5] on the average Lang-Trotter conjecture for suersingular rimes. Their work was etended to the general Lang-Trotter conjecture by David and Paalardi [3]. The best result on the size of the bo a A and b B is due to Baier [4] who established the Lang-Trotter conjecture on average under the condition A, B > /+ɛ and AB > 3/+ɛ,.4 where ɛ > 0. The suersingular case of this result is due to Fouvry and Murty [5, Theorem 6]. Baier [5] has also established an average result for the Lang-Trotter conjecture on the range A, B > log 60+ɛ and 3/ log 0+ɛ < AB < e /8 ɛ,.5 where ɛ > 0. Note that.5 is suerior to.4 if A and B are not very large. There are also average results for other distribution roblems for ellitic curves. Banks and Sharlinski [7] considered such average roblems in a very general setting by emloying multilicative characters and consequently roved average results for the cyclicity roblem, the Sato-Tate conjecture, and the divisibility roblem on a bo a A, b B satisfying the conditions A, B ɛ and AB +ɛ,.6 where ɛ > 0. Another notable result is related to Koblitz conjecture. Let A conjecture of Koblitz redicts that π twin E := #{ ; #E F is rime}. π twin E c E log, as, where c E is a secifically defined constant deending on E. The following result on Koblitz conjecture on the average over the family C is known. 3,

4 Theorem.4 Balog, Cojocaru, and David. Let A, B > ɛ and AB > log 0. Then, as, π twin E = l l l 3 l + log + O log 3. E C rime l See [6, Theorem ]. The error term in the above theorem is estimated by a careful analysis of some multilicative character sums. We rove our Theorem. by a generalization of a modified version of [6, Lemma 6] see Lemma 3.. We have used some results of Stehens [9] to sharen the estimates given in [6, Lemma 6], and thus we could establish our results, for β < /, on a bo of size A, B > ec log / and AB > log δ,.7 for aroriate ositive constants c and δ. As far as we know this is the thinnest bo used so far for an ellitic curve average roblem. Our Theorem. has many alications. Here we mention some direct consequence of it on the cyclicity roblem, the Titchmarsh divisor roblem, and comutation of the k-th ower moment of the eonent e E. Corollary.5. Let c > and AB > log 4+c. There is c > such that if A, B > e c log / then, as, the following statements hold. i ii µd N E = dψdϕd li + O log c E C d where N E is the cyclicity counting function and µd is the Möbius function. iii For k N we have τi E = dψdϕd li + O log c E C E C e k E = d d d k+ ψdϕd lik+ + O,. k+ log c Part i of the above corollary gives a strengthening of a result of Bank and Sharlinski [7, Theorem 8] where asymtotic formula in i was roved in the weaker range.6. Parts ii and iii establish unconditional average versions of some results given in [] and [4]. Remarks.6. i As corollaries of Theorem. we can also establish unconditional average results for f i E where f n is one of the functions log n α, ωn k, Ωn k, kωn, or τ k n r. Here α is an arbitrary ositive real number and k and r are fied non-negative integers. See [4,. 76] for conditional results for these functions in the case of a single ellitic curve. ii Under the conditions of Theorem. one can also obtain average results for f n = n β and f n = σ β n = m n m β as long as β <. More recisely, for A and B satisfying the conditions of Theorem. we have, for c >, E C i β E = d where g is the unique arithmetical function satisfying n β = gm. gd dψdϕd li + O log c m n This stos short of roviding an answer on average to a roblem roosed by Kowalski [3, Problem 3.] that asks about asymtotic behavior of i E. iii Following the roof of Theorem., one can imrove the condition A, B > ɛ in Theorem.4 to A, B > e c log /, for some suitably chosen constant c. 4,.

5 iv Lemma 3. is the difficult art of the roof of Theorem.. The roof of Lemma 3. follows the method used in the roof of [6, Lemma 6] which itself is based on [7] and combines it with some devices from [9]. A new ingredient in the roof of Lemma 3. is an asymtotic estimate due to Howe see Lemma. for the number of ellitic curves over F whose d-torsion subgrou over F is isomorhic to two coies of Z/dZ, where d is a divisor of. Another new feature is a successful alication of Burgess s bound see Lemma.6 in handling terms obtained from the error term of Howe s estimate. v One other novel feature of the roof of Theorem. is shar estimates of the error terms arising from the curves of j-invariant 0 or 78, which are estimated using some results from the theory of CM curves see Lemma.3. A trivial estimation of these terms will result in unsatisfactory uer bounds on admissible values of A and B in Theorem.. Following the ideas of the roof of Theorem. and by a careful analysis of some character sums one can show that c 0 f li closely aroimates f i E for almost all curves E C. Here we rove the following more general theorem. Theorem.7. Let f n be an arithmetic function satisfying f n n β log n γ,.8 where 0 β < / and γ 0. Suose AB > log 6 if 0 β < /4 and AB > 3 +β log 4γ+4 log log 4 if /4 β < 3/4. Then there is a ositive constant c > 0 such that if A, B > e c log /, we have f i E c 0 f li = O log. E C The following is a direct consequence of Theorem.7. Corollary.8. Let h be a ositive real function such that lim h = 0. Under the assumtions of Theorem.7, for any > we have f i E c 0 f li h log,.9 for almost all E C. More recisely.9 holds ecet ossibly for O h of curves in C. Remarks.9. It is ossible to establish a version of Theorem.7 using the bound. instead of.8. However we found that.8 will make the resentation of the roof more convenient. Note that if f n = gd n β log n γ then, by the Möbius inversion formula, we have gn +β log γ+. n d n The structure of the aer is as follows. In Section we summarize results that will be used in the roof of our two theorems. Section 3 is dedicated to a detailed roof of Theorem. and Corollary.5. In Section 4 we briefly summarize the roof of a technical lemma which is a two-dimensional version of Lemma 3.. The roof is tedious and divides to several subcases. We treat some cases and briefly comment on the remaining ones. Finally in Section 5 we rove Theorem.7. Notation.0. Throughout the aer and q denote rimes for simlicity in most cases we assume that, q, 3, ϕn is the Euler function, ωn is the number of distinct rime divisors of n, Ωn is the total number of rime divisors of n, τd is the total number of divisors of n, n is the largest rime factor of n, τ k n is the number of reresentations of n as a roduct of k natural numbers, µn is the Möbius function, ψn = n d n + /d, and π; d, a is the number of rime not eceeding that are congruent to a modulo d. Moreover, K is a quadratic imaginary number field of class number, Na is the norm of an ideal a of K, Nα is the norm of an element α in K, always denotes a degree 5

6 rime ideal of K with N =, and d s is the largest divisor of integer d comosed of rimes that slit comletely in K. We denote the finite field of elements by F and its multilicative grou by F. For two functions f and g 0, we use the notation f = Og, or alternatively f g, if f /g is bounded as.. LEMMAS Let E s,t denote an ellitic curve over F given by the equation y = 3 + s + t; s, t F, where at least one of s or t is non-zero. Let E s,t [d]f denote the set of d-torsions of E s,t with coordinates in F. The following lemma essentially is due to Howe see [,. 45]. Lemma.. i For d N and a fied rime, let S d := { s, t F F ; E s,t [d]f Z/dZ Z/dZ }. For d, we have #S d = dψdϕd + O3/. Moreover, if d or d > +, then #S d = 0. ii The assertions in i hold if we relace S d with S d, where S d = { s, t F F ; E s,t [d]f Z/dZ Z/dZ }. Proof. i We know that ellitic curves isomorhic over F to E s,t are in the form E su 4,tu6, where u F. Let Aut F E s,t be the grou of automorhisms over F of the ellitic curve E s,t. So the number of ellitic curves isomorhic to E s,t over F is / Aut F E s,t. Let [E s,t ] denote the class of all ellitic curves over F that are isomorhic over F to E s,t. We have [E s,t ] S d #S d = [E s,t ] S d Aut F E s,t. Now the result follows since by [,. 45], we have, for d, Aut F E s,t = dψdϕd + O/.. Moreover, by [7, Corollary III.8..], if d then Z/dZ E s,t F [d], and so #S d = 0. Also if d > + and Z/dZ E s,t F [d] E s,t F, then + + < d #E s,t F. On the other hand #E s,t F + +, by Hasse s theorem. This is a clear contradiction. ii We can deduce this by following the roof of art i and observing that there are O isomorhism classes over F containing a curve of the form E 0,t or E s,0. Remarks.. i For any rime, we know that Aut F E s,t = O. In fact, for, 3, from [7, Theorem III.0.], we know that 6 if s = 0 and mod 6 Aut F E s,t = 4 if t = 0 and mod 4. otherwise ii We note that, using Howe s notation [, Page 45], we have Aut F E s,t = dψdϕd + O ψd/d ωd, [E s,t ] S d where ωd is the number of distinct rime divisors of d and the imlied constant is absolute. However, the term ωd is a bound for j gcdd, µ j. In our case, gcdd, d =, since d. d Thus, the term ωd can be removed. Also, ψd/d =, and thus. is correct. 6

7 Let K be a quadratic imaginary number field of class number. Let be a degree rime ideal of K with N =. Let π be the unique generator of. Note that if is unramified, then π is unique u to units, and if it is ramified, then π is unique u to units and comle conjugate. We have N = Nπ =. Lemma.3. Suose that d s is the largest divisor of d comosed of rimes that slit comletely in K. i For ositive integer d with d / log we have ii For ositive integer d, we have N d π π ωd s τd s ϕd N d π π τd s. d log/d. iii Let E s,t : y = 3 + s + t be an ellitic curve over F with st = 0. We have #E s,t F = + or #E s,t F = π π and Nπ =, where π Z[ + i 3/] or Z[i]. iv Let gd be an arithmetic function satisfying. with β <. Then we have s,t F st=0 d E s,t F [d] Z/dZ gd log. Proof. The roofs of i and ii are identical to the roofs of Proositions. and.3 of []. iii See [, Chater 8, Theorems 4 and 5]. iv We observe that the condition E s,t F [d] Z/dZ imlies that d and d #E s,t F. By art iii we know the ossibilities for #E s,t F. Now if #E s,t F = +, then we conclude that d = since d and d +. On the other hand if #E s,t F = π π where π Z[+i 3/] or Z[i], we let 0 < ɛ < β. So by emloying i and ii, the sum in iv is bounded by mod 4 + d + gd N d π π log + log gd d ɛ d /5 We net recall a version of the large sieve inequality for multilicative characters. + gd d ɛ log. d> /5 Lemma.4 Gallagher. Let M and N be ositive integers and a n n=m+ M+N be a sequence of comle numbers. Then q M+N M+N ϕq a n χn N + Q a n, q Q χq n=m+ n=m+ where Q is any ositive real number, and χq denotes a sum over all rimitive Dirichlet characters χ modulo q. Proof. See [8,. 6]. To state the net lemma, we need to describe some notation. Let We also set τ k,b n := # { a, a,..., a k [, B] k N k ; n = a a a k }. ΨX, Y := n X n Y where m is the largest rime factor of m. Note that we define 0 = ± =. 7,

8 Lemma.5 Stehens. i For k N, if B k 8 then b B k τ k,b n < B k ΨB, 9 log k. ii For a sufficiently large constant c > 0 there eists c > 0 such that if e c log / < B 8 then /k ΨB, 9 log / e c log / / log log, where k = [ log / log B ] +. iii For a sufficiently large constant c > 0 there eists c 3 > 0 such that if e c log / < B 4 then /k ΨB, 9 log / e c 3 log / / log log, where Proof. See [9, Lemmas 8, 9, and 0]. Lemma.6 Burgess. k = [ 4 log / log B ] +. i For any rime, non-rincile character χ, r N, and B, we have χb B r+ r 4r log, b B where the imlied constant is absolute. ii Let ɛ > 0, n >, χ be a non-rincial character, r N, and B. Then, if n is cube-free or r =, we have χb B r+ r n 4r +ɛ, b B where the imlied constant may deend on ɛ and r. Proof. See [9, Theorems and ]. Lemma.7. i Friedlander and Iwaniec Let Q and N be ositive integers. Then we have 4 χn N Q log 6 Q, χmod Q n N where denotes a sum over all rimitive Dirichlet characters modulo Q. ii Suose that Q is the roduct of two distinct rimes. Then we have 4 χn N Q log 6 Q. χmod Q χ χ 0 n N Proof. i This is [7, Lemma 3]. ii Let Q = q with q. To see that the result is true if the summation is over all non-rincial characters, we need only establish the inequality for imrimitive characters. The only non-rincial imrimitive characters modulo q are of the form χ χ 0 or χ 0 χ, where χ 0 and χ 0 are the rincial characters modulo and q, resectively, and χ and χ are rimitive characters modulo and q, resectively. Then, artition the summation over all characters into a summation over rimitive characters modulo q, rimitive characters modulo and rimitive characters modulo q. Hence, the assertion can be obtained by using the triangle inequality and the result for rimitive characters in art i. We summarize several elementary estimations that are used in the roofs of net sections. Lemma.8. i Brun-Titchmarsh inequality Let ɛ > 0. Then for d ɛ, we have π; d, a ϕd log. 8

9 ii Let θ < and ɛ > 0. Then for d ɛ, we have iii For 3 and d we have iv We have mod d mod d ϕd θ θ ϕd log. log log + log d. ϕd log log d. d v Under the assumtion of bound., for any real θ we have gd d θ + y +β θ log y γ+. d y Proof. i See [, Theorem 7.3.]. ii This is a consequence of artial summation and art i. iii See [, Section 3., Eercise 9]. iv See [0,. 67, Theorem 38]. v This comes by straightforward alications of artial summation and bound.. 3. PROOFS OF THEOREM. AND COROLLARY Basic set u. Let C be the family of ellitic curves E a,b : y = 3 + a + b, where a A, b B, and at least one of a or b is non-zero. Note that Let for all n N. We have E a,b C f i Ea,b = = 4AB + OA + B. f n = gd d n Aut E F s,t f i Es,t s,t F a A, b B: u< a su 4 mod b tu 6 mod Net by alying Remark. i in the above identity recall that, 3, we have where E a,b C f i Ea,b = Error Term = f i Es,t s,t F st=0 9 a A, b B, u< a su 4 mod b tu 6 mod f i Es,t a A, b B ab 0 mod. + Error Term,. 3.

10 Now by considering a A, b B, u< a su 4 mod b tu 6 mod = AB + a A, b B, u< a su 4 mod b tu 6 mod AB and alying it in the revious identity we arrive at f i Ea,b = The Main Term + Error Term + Error Term, where and E a,b C Error Term = 3.. The Main Term. We have The Main Term = 4AB Let G = The Main Term = 4AB d d + f i Es,t gd dψdϕd f i Es,t f i Es,t a A, b B, u< a su 4 mod b tu 6 mod = 4AB = 4AB and G = d d + By using these notations and emloying Lemma. we have The Main Term = 4AB G G + O = 4AB S + OS. AB. gd d i E s,t gd# S d. d gd Estimation of S. Let α R >0 be fied. The Siegel-Walfisz Theorem imlies π; d, = li ϕd + O log C for any d log α and any C > 0. Then, by the Brun-Titchmarsh inequality Lemma.8 i, the fact that ψd d, and., we have gdπ; d, gdπ; d, S = + dψdϕd dψdϕd d log α = li d log α <d + gd dψdϕd + O gd log C dψdϕd + O log d 0 d>log α gd dψdϕd.

11 Note that, for any ε > 0, we have gd dψdϕd gd Thus, for β <, d>y d d>y d 3 ε gd c 0 := dψdϕd y β ε. is a constant and S = c 0 li + O, log C where C := C C, α, β, ε is an aroriate ositive constant. Since α is arbitrary, we can choose α so that C is any constant bigger than. So S = c 0 li + O log c, 3. where c can be chosen as any number bigger than Estimation of S. We first emloy the Brun-Titchmarsh inequality Lemma.8 i and. to deduce + β log γ log log if β 0 G = gd π; d, β log γ log log if β = 0 d + By artial summation and 3.3, we have G S = +β log γ log log. 3.4 In conclusion, since β < The Main Term = 4AB where c can be taken as any number bigger than. c 0 li + O log c 3.3. Error Term. Recall the eression 3. for Error Term. We have Error Term f i Es,t AB + A + B s,t F st=0 s,t F st=0 d E s,t F [d] Z/dZ gd + A + B An alication of art iv of Lemma.3 in the latter sum yields Error Term gd + d d + A + B, 3.5 s,t F st=0 d E s,t F [d] Z/dZ gd. log. 3.6 By emloying Lemma.8 iii and iv and usual estimates, the first of these summations is bounded as follows. gd gd = gd log log log d. 3.7 d d + d + mod d From alying art v of Lemma.8 in 3.7 we have Error Term β log γ+ log log + d + A + B log. 3.8

12 3.4. Error Term. We summarize the main result of this section in the following lemma, which can be considered as a generalization and an imrovement of Lemma 6 of [6]. Lemma 3.. Let r N, 0 β < 3/, γ R 0, and g : N C be a function such that gd +β log γ. d Then there are ositive constants c and c such that if A, B > ec log / we have gd d s,t< E s,t F [d] Z/dZ u< a su 4 mod b tu 6 mod AB β log γ+ log log + log γ log log + log + e c / log log A + B log + +β + A /r + B /r +β + r+ 4r log γ+ log log log γ log log + AB 3 log + + β log γ+3 log log β 4 log γ+3 log log Proof. Throughout, χ, with or without subscrit, will denote a character modulo. As usual, χ 0 will be the rincial character modulo. Let be a fied rime, and let s, t F be fied. By [6, Equation ], we have where We use the identity and note that = χ,χ χ 4 χ6 =χ 0 u< a su 4 mod b tu 6 mod = χ,χ χ 4 χ6 =χ 0 χ sχ taχ Bχ, Aχ := χa and Bχ := χb. a A χ sχ taχ Bχ χ 0sχ 0 taχ 0 Bχ χ χ 0 χ 4 =χ 0 χ 0sχ 0 taχ 0 Bχ 0 = χ sχ 0 taχ Bχ 0 + a A χ 0 χ χ 6 =χ 0 b B χ 0 sχ taχ 0 Bχ χ 0 a b B χ χ 0 χ 4 χ6 =χ 0 χ 0 b = AB. χ sχ taχ Bχ AB + O + A + B.

13 Therefore, gd d = gd d + s,t< E s,t F [d] Z/dZ =: Σ + Σ + Σ 3 + Σ 4. s,t< E s,t F [d] Z/dZ χ χ 0 χ 4 =χ 0 u< a su 4 mod b tu 6 mod O χ sχ 0 taχ Bχ 0 + We will evaluate each summation searately. AB AB + A + B + χ χ 0 χ 4 χ6 =χ 0 χ 6 =χ 0 χ 0 sχ taχ 0 Bχ χ sχ taχ Bχ Estimation of Σ. We have Σ := gd d AB AB 3 + A + B 3 d d + s,t< E s,t F [d] Z/dZ gd d AB O + A + B s,t< E s,t F [d] Z/dZ gd dψdϕd + O3/ A + B + d d + gd dψdϕd + O3/. We denote the first summation by Σ, and the second by Σ,. By artial summation and 3.3, we have as β < 3/. By Equations 3. and 3.4, we have Σ, Σ, β log γ+ log log + log γ log log 3.9 A + B A + B d d + log + +β gd dψdϕd + Therefore, Σ is bounded by the error terms in the lemma. / d d + gd log γ log log

14 3.4.. Estimations of Σ and Σ 3. For Σ, we have Σ := gd A By Lemma., we have Now, Σ B B d d + d d + gd d d + d d + d d + =: Σ, + Σ.. gd s,t< E s,t F [d] Z/dZ s,t< E s,t F [d] Z/dZ gd gd dψdϕd Σ, = B χ 6 =χ 0 d + χ 6 =χ 0 χ 6 =χ 0 Bχ χ 6 =χ 0 χ 6 =χ 0 Bχ s,t< E s,t F [d] Z/dZ χ 0 sχ taχ 0 Bχ A a A a. Bχ dψdϕd + O 3/ Bχ + B gd dψdϕd Let k = [ log / log B] +. By Hölder s inequality, we have mod d χ 6 =χ 0 Bχ mod d χ 6 =χ 0 / mod d χ 6 =χ 0 k d d + gd χ 6 =χ 0 Bχ Bχ. 3. b B mod d χ 6 =χ 0 χ b π; d, k τ k,b bχ b b B k where τ k,b n := # { a, a,..., a k [, B] k N k : n = a a a k }. By Lemma.4, we have τ k,b bχb χ χ 0 b B k Suose k =. That is, B >. Then, we obtain τ B bχ b b B k Therefore from 3. we have mod d χ 6 =χ 0 k k k, 3. + B k τ k,b b. 3.3 b B k B. / Bχ B ϕd / log / 4

15 after using Lemma.8 i. Substituting this into Equation 3., we obtain Σ, / gd / log / dψdϕd 3/ log /, d as β < 3/ and the summation above was reviously determined to be a constant. Now suose k = [ log / log B] + >. Then B and < B k B 4. Then, by Lemma.5 i and ii, 3., 3.3, and the trivial bound for π; d,, we have k Bχ + B k B k ΨB, 9 log k k d mod d χ 6 =χ 0 B d 3/4 k / ΨB, 9 log log c / B e d3/4 d log log, 3.4 where c > 0 if c is sufficiently large. Substituting 3.4 into 3., we obtain log Σ, e c / gd c log log d 7/4 ψdϕd e log /, log log as β < 3/. For Σ,, by Lemma.6 i,., and Lemma.8 i, ii, and v, we have Σ, = B B r / d + d d + gd gd mod d χ 6 =χ 0 +β + r+ 4r log γ+ log log. B r Bχ b B r +r+ 4r log χ 6 =χ 0 d + gd mod d + r+ 4r log log B r / d + The roof of the bound for Σ gives us the same bound for Σ 3, mutatis mutandis Estimation of Σ 4. For Σ 4, we have Σ 4 = gd where = = d d + d + d + gd gd s,t< E s,t F [d] Z/dZ mod d mod d W,d χ, χ := χ χ 0 χ 4 χ6 =χ 0 χ χ 0 χ 4 χ6 =χ 0 χ χ 0 χ 4 χ6 =χ 0 Aχ Bχ s,t< E s,t F [d] Z/dZ 5 χ 6 =χ 0 gd d χ sχ taχ Bχ s,t< E s,t F [d] Z/dZ Aχ Bχ W,d χ, χ, χ sχ t. χ b b B χ sχ t

16 Alying the Cauchy-Schwarz inequality twice, we obtain 4 Aχ Bχ W,d χ, χ W,d χ, χ χ χ 0 χ χ 0 χ 4 χ6 =χ 0 By Lemma.7, we have Hence, Similarly, Also, χ χ 0 χ 4 χ6 =χ 0 Aχ 4 = W,d χ, χ = χ,χ = χ χ 0 χ χ 0 χ 4 χ6 =χ 0 χ χ 0 χ 4 χ6 =χ 0 χ a a A χ χ 0 χ 4 χ6 =χ 0 4 χ a a A χ a a A χ,χ s,t< E s,t F [d] Z/dZ 4 A log χ χ 0 χ χ 0 χ 4 χ6 =χ 0 χ a a A A log 6. Bχ 4 B log 6. χ sχ t s,t< s,t < E s,t F [d] Z/dZ E s,t F [d] Z/dZ = s,t< E s,t F [d] Z/dZ Aχ 4 4 s,t < E s,t F [d] Z/dZ χ 4 χ6 =χ 0 χ χ 0 χ 4 χ6 =χ 0 χ s χ t χ χ χ sχ s χ tχ t Bχ 4 4 dψdϕd + 7/ 3.5 by Lemma.. Putting all this information together, we obtain 4 χ χ 0 χ 4 χ6 =χ 0 Hence, Aχ Bχ W,d χ, χ Σ 4 d + gd mod d AB 0 log d ψd ϕd + AB 9/ log + AB 9 log. dψdϕd ABlog 3 / d / ψd / ϕd / + 3/8 d /4 ψd /4 + /4 ϕd /4 AB 3 log + + β log γ+3 log log β 4 log γ+3 log log,. as β < 3/. This comletes the roof. 6

17 3.5. Proof of Theorem.. Proof. By combining 3.5, 3.8, and Lemma 3., we have gd f i Ea,b = dψdϕd li + E, E a,b C where E log c + A + B log + +β log γ+ d + A /r + B /r +β + r+ 4r log γ+ log log + AB 3 log + + β log γ+3 log log 5/4 + 5+β 4 log γ+3 log log for given c > and A, B > e c log /. Now we choose r large enough such that +β + r+ <. 4r Note that we can do this if β <. So we arrive at the following uer bound for E. We have E log c + e c r log / + 3 log + 5+β 4 log γ+3 log log. AB Now the result follows by choosing AB log 4+c if β < / and AB /+β log γ+6+c log log if / β < Proof of Corollary.5. Proof. Parts i and ii hold, since the characteristic function of {} can be written as µd and the divisor function can be written as d n τn =. Thus, gd = µd and gd = both satisfy. with β = 0 and γ =. For iii, let f n = /n k, where k N. Then, writing f n = gd, gives us that Therefore, by Theorem., we have d n d n gn = n µ f d d τn. E C d n d n i E k = c kli + O log c. 3.6 Let a E be defined by #E F = + a E. Hasse s Theorem says that a E. Note that k + e k ae k k k k j = = i E C E C E i E C E k + a E j j i j= E k k k k k j = i E C E k + O j k j i E C j= E k k = i E C E k + O k k i E C E k k = i E k + O k k+ log. E C 7,

18 For the first art in the above, by 3.6, we have k i E C E k = k i E C E k k t k i E C t E k dt = c k k k+ li + O log c c k k t k t k lit dt + O k log t c dt = c k k li c k k t k k+ lit dt + O log c. Then, the result holds since that there eists a constant C such that li k+ + C = li k k t k lit dt. 4. A TECHNICAL LEMMA Lemma 4.. Let r N and ε > 0 be fied. Let g : N C be a function such that gd +β log γ, d where 0 β < 3/4 and γ R 0. Then there are ositive constants c and c 3 such that if A, B > ec log / we have 4 q s,t F q d i E s,t d i Es,t q log γ log log + + A /r + B /r where c 3 is a ositive constant. gdgd u<, u <q a su 4 mod,a s u 4 mod q b tu 6 mod,b t u 6 mod q log + e AB q log c / 3 A + B log log 3+β + r+ r +ε log γ log log + 3 log + +β 4 log γ+3 log log, AB Proof. Throughout, a rime suerscrit will denote that underlying object is related to the rime q. Note that, for, q rime, s, t F and s, t F q fied, by orthogonality relations, we have = 4 χ su 4 χ a χ tu 6 χ b u<, u <q a su 4 mod,a s u 4 mod q b tu 6 mod,b t u 6 mod q where = u< u <q q 4 q a A b B χ mod q χ χ,χ mod χ 4 χ6 =χ 0 χ mod s u 4 χ a q χ,χ mod q χ 4 χ 6 =χ 0 Aχ := χa and Bχ := χb. a A 8 χ mod q χ χ mod t u 6 χ b χ sχ tχ s χ t Aχ χ Bχ χ, b B

19 Thus, u<, u <q a su 4 mod,a s u 4 mod q b tu 6 mod,b t u 6 mod q = 6 j= S j, q, s, t, s, t, where S j corresonds to one of the cases arising from choices of each of the following conditions: χ = χ 0, χ = χ 0 χ = χ 0, χ = χ 0 χ = χ 0, χ χ 0 : χ 6 = χ 0 χ = χ 0, χ χ 0 : χ 6 = χ 0 χ χ 0, χ = χ 0 : χ 4 = χ 0 χ χ χ 0, χ χ 0 : χ 4 χ6 = χ χ 0, χ = χ 0 : χ 4 = χ 0 0 χ χ 0, χ χ 0 : χ 4 χ 6 = χ 0 From these 6 cases, there are essentially five different cases to handle. Case : all four of χ, χ, χ, χ are rincial. Let this corresond to j =. Then, for q, we have S, q, s, t, s, t = AB q + O AB + O q AB q + O A + B q. Thus, we have 4 q = 4 s,t F q d i E s,t d i Es,t q q gdgd s,t F q d i E s,t d i Es,t q u<, u <q a su 4 mod,a s u 4 mod q b tu 6 mod,b t u 6 mod q 6 gdgd j= AB q S, q, s, t, s, t + O AB q + AB q + A + B q. The sums corresonding to j =, 3,..., 6 are dealt with in Cases, 3, 4, and 5. Here, we will bound the sums corresonding to the error terms above. We have 4 q 3 s,t F q d i E s,t d i Es,t q gd d i E s,t q gdgd AB q q s,t F q d i Es,t q gd. The first summation can be bounded as we bound Σ, in 3.4., and the second summation can be bounded as we bound Σ, in That is, by 3.9, 3.0, and β < 3/4, we have 4 q s,t F q d i E s,t d i Es,t q 9 gdgd AB q log γ log log.

20 The same bound holds for the term coming from OAB/q. For the last error term, by 3.0, we have 4 gdgd A + B q q s,t F q A B + A + B log. d i E s,t d i Es,t q gd d i E s,t q q s,t F q d i Es,t q gd Case : Eactly two of χ, χ, χ, χ are rincial. We have two subcases to consider. Subcase : Eactly one of χ or χ is rincial and eactly one of χ or χ is rincial. We will bound the summation when χ = χ 0 and χ = χ 0. The bound for when χ = χ 0 and χ = χ 0 is similar. The estimation is analogous to estimations of Σ and Σ 3 in subsection We note that χ 0 χ 0 is the rincial character modulo q since q. Hence, Aχ 0 χ 0 A. Thus, 4 gdgd χ tχ q 4 q t Aχ 0 χ 0 Bχ χ B B d + s,t F q q gd d i E s,t d i Es,t q d d + d q d q+ d + gd gd gd dψdϕd + O3/ = σ + σ + σ 3 + σ 4, E s,t F [d] Z/dZ s,t F E s,t F q [d ] Z/d Z mod d q q mod d qq d ψd ϕd + Oq3/, χ χ 0 χ 6 =χ 0, χ 6 =χ 0, χ χ 0 χ 6 =χ 0, χ 6 =χ 0 q, χ χ 0 χ 6 =χ 0, χ 6 =χ 0 Bχ χ Bχ χ where σ is the sum corresonding to the roduct of the main terms in 4., σ 4 corresonds to the roduct of error terms in 4., and σ and σ 3 corresond to the mied terms. We will evaluate each of these summations searately. For the first summation we have σ = B d + gd dψdϕd d + gd d ψd ϕd Let k = [4 log / log B] +. By Hölder s inequality, we have, χ mod d χ 0 q mod d χ 6 =χ 0, χ 6 =χ 0 Bχ χ, χ mod d χ 0 q mod d χ 6 =χ 0, χ 6 =χ 0 mod d q mod d k π; d, π; d, k 0, χ χ 0 χ 6 =χ 0, χ 6 =χ 0, χ mod d χ 0 q mod d χ 6 =χ 0, χ 6 =χ 0,χ χ 0 4. Bχ χ. 4. k χ χ b b B k τ k,b bχ χ b, b B k 4.3 k

21 where τ k,b n := # { a, a,..., a k [, B] k N k : n = a a a k }. By Lemma.4, we have τ k,b bχb χ χ 0 b B k 4 + B k τ k,b b. 4.4 b B k Suose k =. That is, B > 4. Then, we obtain τ,b bχ χ b b B k χ χ 0 B. Therefore by emloying Lemma.8 i in 4.3, we have, χ mod d χ 0 q mod d χ 6 =χ 0, χ 6 =χ 0 Bχ χ B ϕd / ϕd / log. Substituting this into Equation 4., we obtain σ log d gd dψdϕd 3/ d gd d ψd ϕd 3/ log, as β < 3/4. The latter summations were reviously determined to be constants. Now suose k = [4 log / log B] + >. Then B 4 and 4 < B k B 4 8. Then, by Lemma.5 i and iii, 4.3, 4.4, and the trivial bounds for π; d, and π; d,, we have, χ mod d χ 0 q mod d χ 6 =χ 0, χ 6 =χ 0 Bχ χ k 4 dd + B k B k ΨB, 9 log k k B dd 3/4 k ΨB, 9 log / log B dd e c / 3/4 3, 4.5 log log where c 3 > 0 if c is a suitable large constant. Substituting 4.5 into 4., we obtain σ log e c / 3 log log d gd d 7/4 ψdϕd d gd d 7/4 ψd ϕd e c 3 log / log log, as β < 3/4.

22 By Lemma.6 ii, for any r N and ε > 0, we have that our second summation σ is bounded by gd gd B dψdϕd q / χ χ b r,ε B d + d + + r+ 4r +ε B /r log gd dψdϕd d r+ r +ε log log B /r log d + d + gd dψdϕd d + gd d + gd d q mod d q mod d q mod d q mod d gd q / r q +r+ 4r +ε q q mod d, χ χ 0 χ 6 =χ 0, χ 6 =χ 0, χ χ 0 χ 6 =χ 0, χ 6 =χ 0 b B B r q r+ 4r +ε 3+β + r+ r B/r +ε log γ log log. In the above estimations we emloyed Lemma.8 v and the fact that β < 3/4. We obtain a similar bound for σ 3. Finally, by Lemma.6 ii and Lemma.8 v, for any r N and ε > 0, we have that our fourth summation σ 4 is bounded by gd gd B / q / χ χ 0 b B /r d + d + gd d + + r+ r +ε log log B /r log B /r +β+ r+ d + d + gd mod d gd d r +ε log γ log log. mod d d + q q mod d q q q mod d gd d, χ χ 0 χ 6 =χ 0, χ 6 =χ 0 r +r+ 4r +ε Adding the above bounds for σ, σ, σ 3, and σ 4 concludes Subcase of Case. Subcase : Either both χ and χ are rincial or both χ and χ are rincial. Without loss of generality we assume that χ = χ 0 and χ = χ 0. We have 4 gdgd χ sχ taχ χ q 4 q 0 Bχ χ 0 where = d + gd s,t F q d + d i E s,t d i Es,t q gd W,q χ, χ := mod d q mod d χ χ 0 χ 4 χ6 =χ 0 q χ χ 0 χ 4 χ6 =χ 0 b B Aχ χ 0 Bχ χ 0 W,qχ, χ, χ sχ t. 4.6 E s,t F [d] Z/dZ s,t F q E s,t F q [d ] Z/d Z

23 By alying the Cauchy-Schwarz inequality twice, we obtain 4 Aχ χ 0 Bχ χ 0 W,qχ, χ W,q χ, χ χ χ 0 χ χ 0 χ 4 χ6 =χ 0 From Lemma.7 we have and χ χ 0 χ 4 χ6 =χ 0 χ χ 0 χ 4 χ6 =χ 0 χ 4 χ6 =χ 0 Aχ χ 0 4 A qlog q 6 Bχ χ 0 4 B qlog q 6. We have W,q χ, χ W,q χ, χ W,q χ, χ χ,χ χ χ 0 χ 4 χ6 =χ 0 = = = χ,χ s,t F q E s,t F [d] Z/dZ E s,t F q [d ] Z/d Z χ sχ t u,v F s,t F q u,v F q E s,t F [d] Z/dZ E u,v F [d] Z/dZ E s,t F q [d ] Z/d Z E u,v F q [d ] Z/d Z s,t,u,v F q E s,t F [d] Z/dZ E s,t F q [d ] Z/d Z E u,v F q [d ] Z/d Z χ χ 0 χ 4 χ6 =χ 0 u,v F u,v F q E u,v F [d] Z/dZ E u,v F q [d ] Z/d Z Aχ χ 0 4 χ u χ v χ sχ u χ tχ v χ q q χ dψdϕd + 3/ χ χ 0 χ 4 χ6 =χ 0 Bχ χ 0 4 q 4 d ψd ϕd + q3 3 q 5 dd ψdψd ϕdϕd + 3 q 4 dψdϕd + 5/ q 5 d ψd ϕd + 5/ q 4, which imlies Aχ χ 0 Bχ χ 0 W,qχ, χ χ χ 0 χ 4 χ6 =χ 0 ABlog q 3 q 3 dψdϕd / d ψd ϕd + q 5/ dψdϕd / + 7/4 q 3 d ψd ϕd + 7/4 q 5/. In the above inequalities, we have used the facts that a+b+c+d a +b +c +d and a+b+c+d /4 a /4 + b /4 + c /4 + d /4, where the imlied constants are absolute

24 Substituting the first term in 4.7 into the original summation in 4.6, we obtain a bound of AB d + AB 3 log gd d / ψd / ϕd / d + d + gd d / ψd / ϕd 3/ gd d ψd ϕd d + mod d q mod d qlog q 3 gd d ψd ϕd AB 3 log, 4.8 as β < 3/4. Similarly by substituting the second, the third, and the fourth terms in 4.7 into the original summation in 4.6, we obtain a bound of AB 5+β/ log γ+ log log + +β/4 log γ+ log log + 9+4β/4 log γ+3 log log. 4.9 Adding 4.8 to 4.9 concludes Subcase of Case. Case 3: Eactly three of χ, χ, χ, and χ are rincial. In this case by following the method of Subcase of Case we can conclude that the sum in question is bounded by the same bound in Subcase of Case. Case 4: Eactly one of χ, χ, χ, χ is rincial. In this case by following the method of Subcase of Case we can conclude that the sum in question is bounded by the same bound in Subcase of Case. Case 5: All four of χ, χ, χ, χ are non-rincial. In this case by following the method of Subcase of Case we can conclude that the sum in question is bounded by the same bound in Subcase of Case. Proof. We will evaluate the following summation: 5. PROOF OF THEOREM.7 f i E c 0 li E C = E C f i E f i E q + f i E c 0 li 4 f i E + c 0 li. 5.

25 For the first summation in 5. we have f i E f i E q E C = q s,t F q d i E s,t d i Es,t q Aut F E s,t Aut Fq E s,t q Aut F E s,t Aut Fq E s,t q Aut F E s,t Aut Fq E s,t q gdgd u<, u <q a su 4 mod,a s u 4 mod q b tu 6 mod,b t u 6 mod q s,t F d i E s,t st=0 s,t F d i Es,t q q d i E s,t s,t F q d i Es s t,t q =0 s,t F st=0 s,t F q s t =0 d i E s,t d i Es,t q gdgd gdgd gdgd u<, u <q a su 4 mod,a s u 4 mod q b tu 6 mod,b t u 6 mod q u<, u <q a su 4 mod,a s u 4 mod q b tu 6 mod,b t u 6 mod q u<, u <q a su 4 mod,a s u 4 mod q b tu 6 mod,b t u 6 mod q = S + S + S 3 + S Let S be the corresonding bound in Lemma 4. to a function gn satisfying gn +β log γ+. We have S = OS + 4AB = OS + 4AB n qq s,t F q f i Es,t f i Es,t f i Es,t q From the calculation of the main term in Section 3. we have f i Es,t = c 0 li + O log c f i Es,t. 5.3 for any c >. Since i Es,t + and f n n β log n γ, we have f i Es,t +β log γ. 5.5 As β < 3/4, alying 5.3 and 5.4 in 5.5 yields S = c 0 li + OS + O log c for any c >

26 We will net bound S a similar argument will bound S 3. We have S Aut F E s,t Aut Fq E s,t q Aut F E s,t Aut Fq E s,t q s,t F d i E s,t st=0 s,t F d i Es,t q q s,t F d i E s,t st=0 s,t F d i Es,t q q gd gd gd gd u<, u <q a su 4 mod,a s u 4 mod q b tu 6 mod,b t u 6 mod q u <q a s u 4 mod q b t u 6 mod q s,t F st=0 gd d i E s,t q s,t F q Aut Fq E s,t q d i Es,t q gd u <q a s u 4 mod q b t u 6 mod q. 5.7 By Lemma.3 iv, the first term in the above roduct is bounded by / log. The second term in the above roduct can be bounded by q q s,t F q d i Es,t q gd u <q a s u 4 mod q b t u 6 mod q AB q + q q s,t F q d i Es,t q gd AB q. Following the comutations in Section 3. we can conclude that q q s,t F q d i Es,t q gd AB q AB log. This together with Lemma 3. imly that, under the assumtions of Theorem.7, the second term of the roduct in 5.7 is also bounded by / log. Thus, we have For S 4, we have Note that S 4 a A, b B ab 0 mod q q s,t F st=0 s,t F q s t =0 S log. 5.8 d i E s,t d i Es,t q gd gd a A, b B ab 0 mod ab 0 mod q AB q + O A + B + B q + B + B AB + OA + B. q q 6.

27 Thus, S 4 q s,t F st=0 s,t F q s t =0 d i E s,t d i Es,t q The summation in 5.9 corresonding to AB/q can be bounded by gd log log log gd d d + mod d AB gd gd q + A + B. 5.9 d + β log log log γ+4. By emloying Lemma.3 iv, the summation in 5.9 corresonding to A + B can be bounded by A B + gd d i E s,t A + B log. In conclusion we have s,t F st=0 S 4 β log log log γ+4 + A + B log. 5.0 Thus, under the assumtions of Theorem.7, by alying 5.6, 5.8, and 5.0 in 5., we have f i E f i E q = c 0 li + OS + O E C Net we bound f i E. Let G : N C be defined by f n = Gd. Then, we have Gn n d f d n d n d n d f d Thus, alying the roof of Theorem. for G and f yields f i E log + A + B log + +β log γ+3 + E C Therefore d log +β log γ A /r + B /r +β + r+ 4r log γ+ log log + AB 3 log + +β log γ+4 log log 5/ β 4 log γ+4 log log f i E = OS. 5. E C Now by alying 5. and 5. to 5. we conclude that, under the assumtions of Theorem.7, we have f i E c 0 li = OS + O log. E C Since S = O /log the result follows. 7.

28 References [] A. Akbary and D. Ghioca, A geometric variant of Titchmarsh divisor roblem, Int. J. Number Theory 8 0, [] A. Akbary and V. K. Murty, Reduction mod of subgrous of the Mordell-Weil grou of an ellitic curve, Int. J. of Number Theory 5 009, [3] A. Akbary and V. K. Murty, An analogue of the Siegel-Walfisz theorem for the cyclicity of CM ellitic curves mod, Indian J. Pure Al. Math. 4 00, [4] S. Baier The Lang-Trotter conjecture on average, J. Ramanujan Math. Soc , [5] S. Baier A remark on the Lang-Trotter conjecture, New directions in value-distribution theory of zeta and L-functions, Ber. Math, Shaker-Verlag, Aachen, 009, 8. [6] A. Balog, A.-C. Cojocaru, and C. David, Average twin rime conjecture for ellitic curves, American J. of Math. 33 0, [7] W. D. Banks and I. E. Sharlinski, Sato-Tate, cyclicity, and divisibility statistics on average for ellitic curves of small height, Israel Journal of Mathematics , [8] I. Borosh, C. J. Moreno, and H. Porta, Ellitic curves over finite fields. II, Math. Com , [9] D. A. Burgess, On character sums and L-series, II, Proc. London Math. Soc , [0] A. C. Cojocaru and M. R. Murty, Cyclicity of ellitic curves modulo and ellitic curve analogues of Linnik s roblem, Math. Ann , [] A. C. Cojocaru and M. R. Murty, An Introduction to Sieve Methods and their Alications, Cambridge University Press, 006. [] H. Davenort, Multilicative Number Theory, third edition, Sringer, 000. [3] C. David and F. Paalardi, Average Frobenius distribution of ellitic curves, Int. Math. Res. Not. 999, [4] A. Feli and M. R. Murty, On the asymtotics for invariants of ellitic curves modulo, J. Ramanujan Math. Soc. 8 No. 3, [5] E. Fouvry and M. R. Murty, On the distribution of suersingular rimes, Canad. J. Math , [6] T. Freiberg and P. Kurlberg, On the average eonent of ellitic curves modulo, Int. Math. Res. Not. 03, 9. doi:0.093/imrn/rns80 [7] J. B. Friedlander and H. Iwaniec, The divisor roblem for arithmetic rogressions, Acta Arith , [8] P. X. Gallagher, The large sieve, Mathematika 4 967, 4 0. [9] M. Goldfeld, Artin s conjecture on average, Mathematika 5 968, 3 6. [0] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth Edition, Oford Science Publication, 979. [] E. W. Howe, On the grou orders of ellitic curves over finite fields, Comositio Mathematica , [] K. Ireland and M. Rosen, A classical introduction to modern number theory, Sringer-Verlag, New York, 990. [3] E. Kowalski, Analytic roblems for ellitic curves, J. Ramanujan Math. Soc. 006, 9 4. [4] H. L. Montgomery, Toics in Multilicative Number Theory, Lecture Notes in Mathematics 7, Sringer-Verlag, Berlin- Heidelberg, 97. [5] M. R. Murty, On Artin s conjecture, J. Number Theory 6 983, [6] J.-P. Serre, Résumé des cours de , Ann. Collège de France 978, [7] J. H. Silverman, The Arithmetic of Ellitic Curves, Sringer-Verlag, New York, 986. [8] P. J. Stehens, An average result for Artin s conjecture, J. Number Theory 8 976, [9] P. J. Stehens, Prime divisors of second-order linear recurrences. II, J. Number Theory 8 976, University of Lethbridge, Deartment of Mathematics and Comuter Science, 440 University Drive, Lethbridge, AB, TK 3M4, Canada address: amir.akbary@uleth.ca University of Lethbridge, Deartment of Mathematics and Comuter Science, 440 University Drive, Lethbridge, AB, TK 3M4, Canada address: adam.feli@uleth.ca 8