On the Greatest Prime Divisor of N p

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1 On the Greatest Prime Divisor of N Amir Akbary Abstract Let E be an ellitic curve defined over Q For any rime of good reduction, let E be the reduction of E mod Denote by N the cardinality of E F, where F is the finite field of elements Let P N be the greatest rime divisor of N We rove that if E has CM then for all but o/ log of rimes, P N > ϑ, where ϑ is any function of such that ϑ 0 as Moreover we show that for such E there is a ositive roortion of rimes for which P N > ϑ, where ϑ is any number less than ϑ 0 = 1 1 e 1 4 = As an alication of this result we rove the following Let Γ be a free subgrou of rank r of the grou of rational oints EQ, and Γ be the reduction of Γ mod, then for a ositive roortion of rimes, we have where ɛ > 0 Γ > ϑ 0 ɛ, Keywords: Reduction mod of ellitic curves, Ellitic curves over finite fields, Brun-Titchmarsh inequality in number fields, Bombieri-Vinogradov theorem in number fields, Abelian etensions of imaginary quadratic number fields 000 Mathematics Subject Classification Primary 11G0, Secondary 11N37 1 Introduction Let a be a fied integer and be a rime Let P + a be the greatest rime divisor of + a Many authors studied the roblem of finding good lower bounds for P + a as More recisely, for any fied integer a and real variables, ϑ, let N a, ϑ be the number of for which P + a > ϑ Then Goldfeld [G] roved that for ϑ < c 0 = 7 = 5833, 1 N a, ϑ = 1 > ηϑ log, P +a> ϑ as, where ηϑ > 0 Over the last 30 years, there has been a lot of effort to find larger values of c 0 This is a list of the major imrovements Research of the author is artially suorted by NSERC 1

2 1970 Motohashi c 0 = 6105, 197 Hooley c 0 = 6197, 1973 Hooley c 0 = 650, 198 Iwaniec c 0 = 6381, 1984 Deshouillers Iwaniec c 0 = 656, 1985 Fouvry c 0 = 6687 The best result to date is due to Baker and Harman [BH] who imroved the value of c 0 to 0677 The analogous of the above roblem have also been considered for sequences other than the sequence { + a} For eamle, Stewart [S] has roved that for all sufficiently large rimes, P 1 > 1 log 1 4 Another eamle is related to the Fourier coefficients a n of a non-cm normalized eigenform of weight k and level N with integer coefficients In [MMS], R Murty, K Murty, and Saradha showed that under the assumtion of the Generalized Riemann Hyothesis GRH, P a e log 1 ɛ for any ɛ > 0, and for a set of rimes of density 1 Moreover, they roved that unconditionally P a e og 1 ɛ for any ɛ > 0, and for a set of rimes of density one We oint out in assing that results on the greatest rime divisor of certain sequences have alications in comutational number theory and crytograhy For eamle the original roof that Primes is in P [AKS] uses a result of the form P 1 > ϑ with ϑ > 1/ In this aer we rove an ellitic analogue of such a result Let E be an ellitic curve defined over Q For any rime of good reduction, let E be the ellitic curve over the finite field F obtained by reducing E mod Let N = #E F Then we rove the following Corollary 53 Let E/Q have CM by O K ie comle multilication by the full ring of integers of an imaginary quadratic field K Let ϑ < ϑ 0 = 1 1 e 1 4 = Then for a ositive roortion of rimes, P N > ϑ The method of the roof of this theorem follows closely [G] and [M] However, in the ellitic setting, the roof involves several new ideas and modifications The new ingredients include Huley s etension of the Bombieri-Vinogradov theorem to number fields Theorem, a Brun-Titchmarsh tye inequality in number fields due to Hinz and Lodemann Theorem 1, and facts from the class field theory of the etension K KE[a] For an ideal a of O K, KE[a] is obtained by adjoining the coordinates of a-division oints of E to K We briefly describe the roof s strategy For a rime l, set π E ; l = #{, is a good rime and l N }, where a good rime means a rime of good reduction Then it is easy to show that = π E ; l, l N l + good where + = + 1 see Section 4 The estimation of the left-hand side of is straightforward On the right-hand side we emloy the number field versions of the Bombieri-Vinogradov theorem,

3 the Brun-Titchmarch inequality, and roerties of the etension K KE[a] to estimate the sum for l ϑ These estimations imly a lower bound for the sum of the right-hand side of for l > ϑ From this lower bound we deduce our result With slight modification of our arguments, we are also able to rove the following Corollary 55 Let E/Q have CM by O K Let ϑ be a function of such that ϑ 0 as Then for all but o/ log of rimes, P N > ϑ Net let Γ be a free subgrou of rank r of the grou of rational oints EQ and let Γ be the reduction of Γ mod Lang and Trotter [LT] conjectured that the density of rimes for which Γ = E F always eists This conjecture can be considered as an ellitic generalization of the celebrated Artin s rimitive root conjecture So it would be interesting to know how the size of Γ grows as In [AM] it is roved that if E has CM by O K then for all but o/ log of rimes, Γ r r+ +ɛ, where ɛ is any function of such that ɛ 0 as Here as a consequence of Corollary 53 Theorem 5, we rove the following Theorem 63 Let E/Q have CM by O K Let r, and ɛ > 0 Then for a ositive roortion of rimes, Γ > ϑ 0 ɛ We oint out that the above theorem is non-trivial if r = or 3 see Lemma 61 The structure of the aer is as follows In Section we review some basic facts regarding algebraic number fields and ellitic curves Section 3 summarizes some imortant features of the etension K KE[a] In Sections 4 and 5, we rove Theorems 5 and 54 Section 6 gives the roof of Theorem 63 Notation and Terminology We use and l to denote rational rimes We write + 1 as + K and L are number fields A rime l is called an inert res a slit, a ramified rime in an imaginary quadratic field K if l remains rime res slits comletely, ramifies in K A rime is called non-inert if it either slits or ramifies In the sums involving rimes of secial tyes for eamle ordinary, slit, inert, etc, we write the tye of the rime in the inde of the sum For eamle, ordinary is a sum over ordinary rimes and slit rimes l l slit Preliminaries The standard references for this introductory section are [N], [S1], and [S] Let K be a number field of degree n = r 1 + r with r 1 real embeddings and r comle embeddings Let O K be its ring of integers For an ideal q in O K and integers α and β O K we write α β mod q if α β q This equivalence relation defines Nq residue classes mod q Nq is called the 3

4 norm of q The residue classes relatively rime to q forms a grou under multilication We denote the order of this grou by ϕq, which is the number field analogue of the Euler function We have ϕq = Nq 1 1 N q If all real conjugates of an algebraic number α if any are ositive, we write α 0 We say that α β mod q if α β mod q, α 0, and β 0 We denote by T q the number of residue classes mod q that contain a unit If K is an imaginary quadratic field we have T q w K, where w K is the number of roots of unity in K It is known that w K =, 4, or 6 We define an equivalence relation on the set of ideals of O K as follows We say two ideals a and b are equivalent, written a b, if there eists α, β O K such that αa = βb, where α res β denotes the ideal generated by α res β This relation gives us h equivalence classes, where h is called the class number of K or O K We also say that two ideals a and b are equivalent mod q, denoted a b mod q, if they are relatively rime to q and there eists α, β O K, such that α β 1 mod q, and αa = βb Again this is an equivalence relation and we have hq classes where hq = hr 1 ϕq T q For a, q = 1, let π K ; q, a = #{ : rime ideal; N, and a mod q} Finding good estimations for π K ; q, a has fundamental imortance in the analytic theory of number fields Here we mention two imortant estimations of π K ; q, a The first one can be considered as a Brun-Titchmarsh tye inequality for number fields Theorem 1 Hinz and Lodemann If 1 Nq <, then { π K ; q, a hq log 1 + O Nq where the O-constant deends only on K } og 3 Nq log, Nq Proof See [HL], Theorem 4 The following is an etension of the Bombieri-Vinogradov theorem to K Theorem Huley For each ositive constant A, there is a ositive constant B = BA such that 1 ma a,q=1 T q π K; q, a li hq log A, Nq Q where Q = 1 log B The imlied constant deends only on A and on the field K Here li = and dy log y, Proof See [H], Theorem 1 4

5 Let E be an ellitic curve defined over Q This means that E is a non-singular curve defined by an equation y + a 1 y + a 3 y = 3 + a + a 4 + a 6 with a 1, a, a 3, a 4, a 6 Z together with a oint at infinity O given in rojective coordinates by [0, 1, 0] The discriminant of E is a olynomial in the a i which is non-zero if and only if E is nonsingular Let EQ be the set of rational oints on E together with O One can show that EQ with an aroriate addition law has a grou structure Let End Q E be the ring of endomorhisms of E defined over Q the algebraic closure of Q It is known that End Q E is either Z or is an order in an imaginary quadratic field K = Q d If End Q E Z, then E is said to have CM ie comle multilication by an order in an imaginary quadratic field K The class number of an order R in K is defined as the cardinality of the grou of rojective modules of rank 1 over R One can show that in the case R = O K, this definition of class number coincides with the definition in terms of ideal classes It is known that if E has CM, then its corresonding order has class number 1, and so it is one of the thirteen rings, Z[ d] d = 1,, 3, 7, Z[ 1+ d ] d = 3, 7, 11, 19, 43, 67, 163, Z[ 1], Z[ ] U to isomorhism over Q, there are eactly thirteen ellitic curves with CM Each class is determined by the so-called j- invariant and contains infinitely many curves For eamle, all the curves y = 3 + D have j = 178 Net we assume that E is given by an equation y + a 1 y + a 3 y = 3 + a + a 4 + a 6, a i Z, such that the valuation of the discriminant of this equation is minimal in the set of valuations of all equations for E with coefficients in Z We call such an equation a minimal Weierstrass equation Then the reduction E of E modulo rime is defined by y + ā 1 y + ā 3 y = 3 + ā + ā 4 + ā 6 where ā i F is the reduction of a, b mod If E is an ellitic curve over F ie E is non-singular, we say that E has good reduction at, and we call a good rime is a good rime if and only if, = 1 If E has good reduction at, then E F forms a grou, we let N = #E F We have the following imortant estimation for N Hasse s bound: + 1 N So if, then N + A oint Q E F is called a -division oint, if Q = O We denote the set of all -division oints of E by E [] A good rime is called suersingular if #E [] = 1, and it is called ordinary if #E [] = One can show that, for 5, a good rime is suersingular if and only if N = + 1 As art of Deuring s results regarding CM curves, we have the following theorem which gives a comlete characterization of suersingular rimes and ordinary rimes for a CM ellitic curve Theorem 3 Let E be an ellitic curve defined over Q with good reduction at Suose that E has CM by an order in an imaginary quadratic field K Then is suersingular if and only if has only one rime of K above it ie ramifies or is inert in K Proof See [L], 18, Theorem 1 So we have is ordinary, = 1, = 1 in K 1, = 1, slits comletely in K 5

6 From this observation and the Chebotarev density theorem we have #{, ordinary} 1 as This is in contrast with the non-cm case, where one can rove that #{, ordinary} log, 1 log, as Another striking difference between the CM and non-cm cases is the following In general, we know that the reduction ma r : End Q E Z Q End F E Z Q is injective Also if is an ordinary rime then End F E Z Q is an imaginary quadratic field So if E has CM by an order in an imaginary quadratic field K, the above injection imlies that End F E Z Q = K On the other hand the -th ower Frobenius morhism, y, y is an endomorhism of E that can be identified with an imaginary quadratic number π So Qπ End F E Z Q = K, which imlies K = Qπ In summary, in the CM case, for any ordinary rime there is a unique choice of an element π O K such that π reresents the -ower Frobenius morhism, = π π, π is a rime ideal of O K, and K = Qπ Moreover in this case N = Nπ 1 = + 1 π + π, where Nπ 1 denote the norm of the ideal π 1 The net statement lays an imortant role in the study of the ordinary rimes whose N is divisible by a fied rime ower Lemma 4 Let E/Q have CM by O K Let be a rime of ordinary reduction for E Then we have the following 1 Assume that l is inert in K ie l is a rime ideal of O K with Nl = l We have i If k is odd, ii If k is even, l k N π 1 mod l k+1 l k N π 1 mod l k Assume that l slits comletely in K ie l = l 1 l, l 1 l, and Nl 1 = Nl = l Then l k N π 1 mod l i 1l k i, for some 0 i k 3 Assume that l ramifies in K ie l = l and Nl = l Then l k N π 1 mod l k Proof See [C], Lemma 14 6

7 Definition For rime l and integer k 1, we define { l N l k k+1 if l is inert in K, and k is odd = l k otherwise We end this section by giving two estimations for π o E; l k =, ordinary l k N which counts the number of ordinary rimes whose N is divisible by a fied rime ower Proosition 5 Let E/Q have CM by O K Let l be rime, k 1 and 1 N l k where The imlied constant deends only on K π o E; l k K ψl k log 1,, N l k l k+1 l k 1 if l is inert in K, and k is odd, ψl k l k l k if l is inert in K, and k is even, = l k l k 1 if l slits in K, k+1 l k l k 1 if l ramifies in K Then log Proof We rove this in the case that l is inert in K, and k is odd The roof in the other cases is similar From Lemma 4, and Theorem 1 we have π o E; l k π K ; l k+1, 1 T l k+1 ϕl k+1 log Nl k+1 og O log Nl k+1 Nl k+1 The result follows, since T l k+1 6 and N l k log Proosition 6 Let E/Q have CM by O K Let l be rime, k 1, and 1 N l k + Then πe; o l k K δl k N l, k where δl k = k + 1 if l slits in K, and δl k = 1 otherwise The imlied constant deends only on K Proof We first rove that if K is an imaginary quadratic field of class number 1, then This is true since π K ; q, 1 K Nq π K ; q, 1 #{ω O K ; Nω, ω 1 mod q} #{γ O K ; Nγ + Nq } Nq K 7

8 The result follows from this observation and Lemma 4 We rove this for the case that l slits in K Proof of the other cases are similar If l slits, from art ii of Lemma 4 we have π o E; l k k i=0 K k + 1 l k π K ; l i 1l k i, 1 Finally we can also consider π s E; l k =, suersingular l k N In this case, since for 5, N = + 1, the roblem of finding uer bounds for π s E ; lk basically reduces to the classical estimations for π; l k, 1 = #{ ; 1 mod l k } Also the roblem of finding lower bounds for P N for suersingular s is essentially the same as the classical roblem of finding lower bounds for P + 1 From now on we only consider the case of ordinary rimes By emloying Bombieri-Vinogradov theorem over Q and the classical Brun-Titchmarsh inequality, one can easily write the analogous arguments for suersingular rimes 1 3 The field of a-division oints We first review some facts from class field theory Let K L be a finite Abelian etension of number fields Let be an unramified rime of K in this etension, and P be a rime above The Artin symbol L/K is the unique element of GalL/K which mas to the generator of the Galois grou of O L /P over O K / This generator is the Frobenius automorhism N Q K Let m be an ideal of K which contains all the ramified rimes in the etension L/K For rime, m = 1, we define L/K Φ {L/K, m} = This ma etended over all fractional ideals of K relatively rime to m is called the Artin ma for L/K and m Let P m = {α : α K, α 1 mod m} be the grou of rincial ideals of K congruent to 1 modulo m We define the ray class field K m associated to m as the Abelian etension K m of K such that Ker Φ {Km/K, m} = P m From class field theory we know that, for any m, K m eists and is unique Moreover K m is characterized by the roerty that it is an Abelian etension of K and satisfies {rimes of K that slit comletely in K m } = {rime ideals in P m} 8

9 see [S], Page 117, Theorem 3 From now on K is an imaginary quadratic number field, and we assume that E has CM with the whole ring of integers O K We use the above information to study the field generated by the a-division oints of a CM ellitic curve E defined over K Also we fi an isomorhism [ ] : O K End Q E For an ideal a of K, we define E[a] = {P E : [α]p = O for all α a} We call E[a] the grou of a-division oints of E If a = α then E[a] = E[α] = Ker [α] It is clear that E[α] is indeendent of the choice of the generator of α Let KE[a] be the field obtained by adjoining all the a-division oints of E to K Let f be an ideal of K divisible by all rimes of bad reduction of E over K We are interested in studying the etension KE[a]/K First of all we know that this etension is Abelian [R], Page 18, Corollary 55 Secondly if is a rime ideal of K with, fa = 1, then is unramified in this etension [S1], Page 184, Theorem 71 The net statement describes the action of the Artin symbol of the etension KE[a]/K on the a-division oints of E Lemma 31 Let E/K have CM by O K Then there is a Hecke character character of a generalized ideal class grou of K χ E/K of conductor f such that for a rime of K where, fa = 1, the Artin symbol KE[a]/K acts on E[a] by multilication by χ E/K Proof See [R], Page 186, Corollary 516 ii The net lemma can be considered as an analogue of the Kronecker-Weber theorem for the Abelian etension K KE[a] Lemma 3 Let E/K have CM by O K For an ideal a of K there is an ideal f of K such that where K fa is the ray class field associated to fa KE[a] K fa, Proof We know that KE[a] is a finite Abelian etension of K whose ramified rimes are among the rime divisors of fa To rove the assertion we only need to rove that Ker Φ {Kfa /K, fa} Ker Φ {KE[a]/K, fa} see [Co], Corollary 87 Now let be a rime of K in the kernel of the Artin ma Φ {Kfa /K, fa}, and Kfa /K so = 1 More secifically this means that has a generator α such that α 1 mod fa So is unramified in KE[a] and by the revious lemma for a oint P E[a], we have KE[a]/K P = χ E/K P = χ E/K αp = P Here we used the fact that any α 1 mod f is in the kernel of χ see [N], Proosition 76 ii So Kfa /K the kernel of the Artin ma is a subset of the kernel of the Artin ma KE[a]/K This imlies the result 9

10 Lemma 33 Let E/K have CM by O K, and be a rime ideal of K with, fa = 1 Then, for some t, there are t ideal classes mod fa reresented by a 1,, a t such that Moreover and if a, 6f = 1 slits comletely in KE[a] a 1, or a, or, or a t mod fa t [KE[a] : K] = hfa, t = hfa ϕa hϕf Proof Since the rimes that slit comletely are in the kernel of the Artin ma, the first assertion is clear from Net let KE[a]/K φ; KE[a]/K = #{ : rime ideal of K; N,, fa = 1, and = 1} Then we have φ; KE[a]/K = t π K ; fa, a i Now the second statement follows by emloying the Chebotarev density theorem, the rime ideal theorem, and comaring the main terms of the two sides of the above identity Finally the last assertion is true since by [R], Page 187, Corollary 50 ii, we have [KE[a] : K] = ϕa if a, 6f = 1 Lemma 34 Let E/Q have CM by O K, and = π π be an ordinary rime Let l = λ be a degree 1 rime of K such that, l = 1 Then π 1 mod l π slits comletely in KE[l] Proof From [S1], Page 181, Proosition 54 b and [S], Page 168, Theorem 9 b follows that E does not have good reduction over K at rimes dividing f and rime divisors of f Now since, l = 1, then π is unramified in KE[l] Thus π slits comletely in KE[l] if and only if the Artin symbol KE[l]/K π i=1 = 1 This means that the endomorhism [π ] corresonding to the -ower Frobenius morhism, y, y acts trivially on E[l] So [π ]P = P for all P E[l] Thus π slits comletely in KE[l] if and only if Ker [λ] Ker [π 1] This is true if and only if there is an endomorhism φ of E such that [π 1] = φo[λ] see [S1], Page 77, Corollary 411 The roof is comlete The following corollaries are direct consequences of the revious lemma and Lemma 4 Corollary 35 Suose that the rime l slits comletely in K ie l = l l, l l, is an ordinary rime, and l Then l N π slits comletely in KE[l] or π slits comletely in KE[ l] Corollary 36 If a rime l slits comletely in K then we have πe; o l = 1 l Nl=l N,,fl=1 degree 1 slits comletely in KE[l] 1 N,,fl=1 degree 1 slits comletely in KE[l] 1 + O1 10

11 4 Lemmas This section describes lemmas that will be used in the roof of Theorem 5 We assume that E/Q has CM by O K First of all observe that by interchanging the order of addition, we have ordinary l N = In our first lemma we evaluate the left-hand side of 3 l + π o E; l 3 Lemma 41 We have as ordinary l N = + O E, log Proof Let Λn be the von Mangoldt function Then from the identity n d Λn = log d, 1, and the rime number theorem we have ordinary l N = = = ordinary ordinary ordinary n N Λn log N = + O log ordinary ordinary l k N k l k N k log + log a π E; o l k k l k N ordinary + I + II, 4 where a = + 1 N By alying Hasse s bound a we have I = Here we used the fact that 5 ordinary ordinary log a + log 1 a 1 5< ordinary a 1 1/ log1 z = for z < 1, and a 1 +1 < 1 for > 5 To evaluate II, we note that for inert l and odd k if l k N then l k+1 N This is true since N = π 1 π 1, so the multilicity of l in N is even Therefore we have k=1 z k k, 11

12 II k = + k k odd l + 1 k l non inert l + 1 k+1 l inert non inert + inert k even π o E; l k + k k even π o E; l k + inert k odd l + 1 k l inert π o E; l k 5 Net we emloy Proositions 5 and 6 to estimate these sums We have = πe; o l k + πe; o l k non inert k 3 k 3 1 log l 4k l non inert k l 3 4k log log k + 1 l k l k k <l + k l non inert Here, we have used the fact that k log Similarly we have = πe; o l k + inert k 3 k 3 k even k even l 4k k even and inert k odd = k k odd log log l inert k l 3 4k log log, 3 l 4k+1 l inert k l k l k π o E; l k + k k odd k 3 3 l 4k+1 log + 1 log, k l k+1 l k Alying the above three estimations in 5 yields II log log 3 4k <l + 1 k 4k <l + 1 k l inert 3 4k <l + 1 k 3 1 4k+1 <l + k+1 l inert k + 1 π o E; l k k 3 3 4k+1 <l + k+1 1 Finally relacing the above bounds for I and II in 4 imlies the result π o E; l k In the net two lemmas we consider the right-hand side of 3 We first show that the contribution of the inert rimes l to the right-hand side of 3 is negligible 1

13 Lemma 4 We have πe; o l = O K log l + l inert Proof A calculation similar to the one used in treating inert k odd πe; o l + l l inert l inertπoe; l = l log l 3 8 log l of II in the revious lemma yields 3 8 <l + 1 l inert 3 8 <l + 1 π o E; l In the following lemma we deduce an asymtotic formula for the non-inert terms l < 1 right-hand side of 3 in the Lemma 43 Let L = log and let B be the constant given in Theorem rovided A is chosen Then for non-inert rimes l, we have πe; o l = 1 og 4 + O E log l 1 L B l non inert Proof First of all without loss of generality we assume that B > 1 Secondly by Corollary 36 and Lemma 33, for slit rime l, we have πe; o l = 1 = 1 l Nl=l l Nl=l N,,fl=1 degree 1 slits comletely in KE[l] π K ; fl, l i i i 1 N,,fl=1 degree 1 slits comletely in KE[l] π K ; fl, l i + O 1 + O1 1 log Here we used the fact that the number of inert rime ideals ω with Nω in K is bounded by 1/ / log Let S be the set of rational rimes that are ramified in K together with the rime divisors of 6 Nf Now by alications of the above eression for πe o ; l, we have π o E; l = 1 log Nl i π K ; fl, l i l 1 L B l non inert, l S 1 Nl=l 1 L B l slit,l S l 1 L B l slit,l S i π K ; fl, l i + O log B 13

14 By Lemma 33 and Theorem, the last formula is { } = 1 li log Nl ϕl + π K ; fl, l i li ϕl 1 Nl=l 1 L B l slit,l S l 1 L B l slit,l S = 1 li l 1 L B l slit, l S = O og log i i { } li ϕl + π K ; fl, l i li + O ϕl log B l 1 1 li l 1 L B l slit, l S l 1 + O log log A We also need the following straightforward estimation in the sequel Lemma 44 Let 1 ϑ < 1 We have 1 L B <l ϑ l rime l log l og = log {1 ϑ} + O ϑ log Proof Let y = 1 L B, z = ϑ, and ft = t log t 1 Then by artial summation and the rime number theorem, we have z l log = fz fy f tdt y<l z l l z l y y l t l rime 1 z { 1 z } = O tf t dt + O t f t dt log y log y z 1 = ft dt + O y log = og y og 1 z + O log og = log{1 ϑ} + O log 5 Proof of Theorems 5 and 54 We recall that by Lemma 41 the left-hand side of 3 is asymtotic to / as On the other hand, on the right-hand side of 3, by Lemma 4, the contribution of the inert summands is O/ log, and by Lemma 43 the sum of the non-inert summands l corresonding to l 1 L B 14

15 are asymtotic to /4 Now we use Theorem 1 together with Lemma 43 to find an uer bound for the contribution of non-inert summands l to the right-hand side of 3 in the range 1 L B < l ϑ, and as a result we get the following Proosition 51 Let 1 ϑ < 1 1 e 1 4 ϑ <l + l non inert = Then πe; o l log{e 1 og 4 1 ϑ} + OE,ϑ log Proof First of all by Corollary 36 and Lemma 33, for slit rime l, we have πe; o l = 1 1 l Nl=l N,,fl=1 degree 1 slits comletely in KE[l] π K ; fl, l i + O 1 l Nl=l i 1 N,,fl=1 degree 1 slits comletely in KE[l] 1 + O1 Let S be as defined in Lemma 43 Then by an alication of the above inequality, Theorem 1, and Lemma 44 we have 1 L B <l ϑ l non inert, l S π o E; l L B <Nl=l ϑ l slit, l S 1 L B <Nl=l ϑ l slit, l S +O ϑ log = 1 1 L B <l ϑ l slit, l S log Nl i log Nl hfl ϕl ϕl log l π K ; fl, l i + O ϑ log hfl log Nl + O og = log{1 ϑ} + O log og 3 Nl 1 + O log Nl og log l, l as Finally the assertion of the roosition follows from alications of Lemmas 41, 4 and 43 in 3 and the fact that og πe; o l log{1 ϑ} + O log 1 L B <l ϑ l non inert We are ready to rove our first main result 15

16 Theorem 5 Let E/Q have CM by O K, and let ϑ < ϑ 0 = 1 1 e 1 4 = Then there are ositive constants X 0 E, ϑ, and ηϑ < 1/ such that for all large > X 0 E, ϑ we have #{, ordinary and P N > ϑ } > ηϑ log Proof Without loss of generality we can assume that ϑ > 1 By the revious roosition we have #{, ordinary and P N > ϑ } = 1 log + 1 log + ordinary ϑ <l + l non inert l N l> ϑ log{e ϑ} log + + O πe; o l + O log og log Corollary 53 Under the assumtions of the revious theorem, for a ositive roortion of rimes, P N > ϑ Proof This is clear from the revious theorem since #{, ordinary and P N > ϑ } #{, ordinary and P N > ϑ } In the above discussion ϑ was a real number, net we consider the case that ϑ is a function of We observe that if ϑ 1 L B, where B is the constant given in Lemma 43, then an argument similar to the one given in the roof of Lemma 43 imlies that l ϑ l non inert πe; o l = ϑ + O {1 + ϑ} log Now alications of the above identity and Lemmas 41 and 4 in 3 yields ϑ <l + l non inert πe; o l = 1 ϑ + O {1 + ϑ} log So, by an argument similar to Theorem 5, we have the following Theorem 54 Let E/Q have CM by O K and let ϑ be a function of such that ϑ 1 L B, where B is the constant given in Theorem rovided A is chosen Then #{, ordinary and P N > ϑ } 1 ϑ log + O {1 + ϑ} + log 16

17 Corollary 55 In the revious theorem, let ϑ 0 as Then #{, ordinary and P N > ϑ } 1 as So for all but o/ log of rimes, P N > ϑ log, Proof Since the density of ordinary rimes in the set of rimes is 1/, the first assertion is clear from the revious theorem Net we note that for suersingular rime 5 we have N = + 1, so reeating the above arguments for suersingular results in #{, suersingular and P N > ϑ } 1 log, as So if ϑ is a monotone increasing unbounded function, we have #{, P N ϑ } #{, P N ϑ } = o log Thus for almost all rimes, P N > ϑ Finally we note that if ϑ is not a monotone increasing unbounded function, we can always find a ϑ 1 such that ϑ ϑ 1, ϑ 1 0, and ϑ 1 is a monotone increasing unbounded function, so the result for ϑ follows from the result for ϑ 1 6 Alication Let Γ be a free subgrou of rank r of EQ and let Γ be the reduction of Γ mod We recall a known result on the number of rimes for which Γ is bounded by a number z Lemma 61 #{ : Γ z} = O z 1+ r /log z Proof See [GM], Lemma 14, or [AM], Proosition 1 Lemma 6 Let ϑ > 1 be fied and r Then #{, ordinary; Γ ϑ and P N > ϑ } = o log Proof It is clear that for a rime in this set P N Γ Therefore N = Γ P N k for some integer k and so Γ = N P N k N P N + ϑ 1 ϑ However from Lemma 61 we have #{; Γ 1 ϑ } 1 ϑ 1+ r log = o log 17

18 Theorem 63 Let E/Q have CM by O K Let r, ɛ > 0 and ϑ 0 = 1 1 e 1 4 ositive roortion of rimes, Γ > ϑ0 ɛ = Then for a Proof By emloying the revious lemma and Theorem 5, we have #{, ordinary; Γ ϑ 0 ɛ } #{, ordinary; Γ ϑ 0 ɛ } = #{, ordinary; Γ ϑ 0 ɛ, and P N ϑ 0 ɛ } + #{, ordinary; Γ ϑ0 ɛ, and P N > ϑ0 ɛ } #{, ordinary; P N ϑ0 ɛ } + o log 1 ηϑ 0 ɛ log + o log ACKNOWLEDGMENT The author would like to thank Professor Ram Murty for his suggestions and several helful discussions related to this work References [AKS] M AGRAWAL, N KAYAL, AND N SAXENA, Primes is in P, Ann of Math , [AM] A AKBARY AND V K MURTY, Reduction mod of subgrous of the Mordell-Weil grou of an ellitic curve, Int J of Number Theory, 7 ages, to aear [BH] R C BAKER, G HARMAN, Shifted rimes without large rime factors, Acta Arith , [C] A C COJOCARU, Reduction of an ellitic curve with almost rime orders, Acta Arith , [Co] [G] D A COX, Primes of the form + ny, Fermat, class field theory, and comle multilication, John Wiley & Sons, 1989 M GOLDFELD, On the number of rimes for which + a has a large rime factor, Mathematika , [GM] R GUPTA AND M R MURTY, Primitive oints on ellitic curves, Comositio Math , [H] [HL] M N HUXLEY, The large sieve inequality for algebraic number fields III, J London Math Soc , J HINZ AND M LODEMANN, On Siegel zeros of Hecke-Landau Zeta-functions, Monatsh Math ,

19 [L] S LANG, Ellitic Functions, second edition, Sringer-Verlag, New York, 1987 [LT] S LANG, H TROTTER, Primitive oints on ellitic curves, Bull Amer Math Soc , 89 9 [M] [MMS] [N] [R] [S1] [S] Y MOTOHASHI, A note on the least rime in an arithmetic rogression with a rime difference, Acta Arith , M R MURTY, V K MURTY AND N SARADHA, Modular forms and the Chebotarev density theorem, American J Math , W NARKIEWICZ, Elementary and analytic theory of algebraic numbers, third edition, Sringer- Verlag, Berlin Heidelberg, 004 K RUBIN, Ellitic curves with comle multilication and the conjecture of Birch and Swinnerton-Dyer, Lecture Notes in Mathematics, no 1716, , Sringer-Verlag, Berlin Heidelberg, 1999 J H SILVERMAN, The arithmetic of ellitic curves, GTM 106, Sringer-Verlag, New York, 1986 J H SILVERMAN, Advanced toics in the arithmetic of ellitic curves, GTM 151, Sringer- Verlag, New York, 1994 [S] C L STEWART, The greatest rime factor of a n b n, Acta Arith , Deartment of Mathematics and Comuter Science, University of Lethbridge, 4401 University Drive West, Lethbridge, Alberta, T1K 3M4, CANADA E mail address: amirakbary@ulethca 19