on the asymptotic nature of elliptic …felix/elliptic_curve_analogue.pdfon the asymptotic nature of...

ON THE ASYMPTOTIC NATURE OF ELLIPTIC CURVES MODULO p

ADAM TYLER FELIX AND M. RAM MURTY

Abstract. Let E be an elliptic curve defined over Q. In 1976, Serre proved, on GRH, thatfor a positive proportion of primes p ≤ x, the elliptic curve modulo p is cyclic. This was laterto be shown to be true unconditionally for E with complex multiplication by M. Ram Murty.Let E(Fp) denote the elliptic curve modulo p. Then, there exist integers i(p), f(p) such thatE(Fp) ∼= Z/i(p)Z × Z/i(p)f(p)Z. Hence, E(Fp) is cyclic if and only if i(p) = 1. We studyquestions related to i(p). In particular, for any α > 0, we prove there exists a constant cα > 0such that for any A > 0 we have∑

p≤x

(log i(p))α = cαli(x) +O

(x

(log x)A

)unconditionally for E a CM curve. For E a CM curve with 0 < α < 1, assuming GRH weprove that there exists a constant c′α > 0 such that∑

p≤x

i(p)α = c′αli(x) +O(x

3+α4 (log x)2

).

For E a non-CM curve with 0 < α < 1/2, assuming GRH we prove that there exists c′′α > 0such that ∑

p≤x

i(p)α = c′′αli(x) +O(x

5+2α6 (log x)2

).

1. Introduction

Let E be an elliptic curve defined over Q of conductor N . For a prime p - N , let E be thereduction of E modulo p. This is an elliptic curve defined over Fp := Z/pZ, a field. Hasse’sTheorem [20, Theorem V.1.1] says

#E(Fp) = p+ 1− apwhere |ap| ≤ 2

√p.

We also have E(Fp) ∼= Z/i(p)Z×Z/i(p)f(p)Z where i(p) and f(p) are integers. To see this

let Fp be the algebraic closure of Fp. Since E(Fp) is the set of solutions to a non-singular cubic

curve with components in Fp, it is a finite group. For any elliptic curve E defined over a field

k, let E[k] denote the set of k-rational points which are annihilated by the mapping P 7→ kP .Then, since E(Fp) is finite, we have E(Fp) ⊂ E(Fp)[k] for some k with E(Fp)|k. By [20,

Corollary III.6.4], we have E(Fp)[k] = Z/kZ× Z/kZ. So, E(Fp) ∼= Z/i(p)Z× Z/i(p)f(p)Z by

Key words and phrases. Cyclicity, Elliptic curves, primes.Research of the first author supported by an NSERC PGS-D.Research of the second author supported by an NSERC Discovery Grant.

1

2 ADAM TYLER FELIX AND M. RAM MURTY

the Fundamental theorem of finitely generated Abelian groups. As such, we have #E(Fp) =i(p)2f(p). Our interest is in the sequence i(p) as p ranges over all primes p.

We note that E(Fp) is cyclic if and only if i(p) = 1. The question of how often E(Fp)is cyclic has been studied before by many authors. Borosh, Moreno, and Porta [4] showed,computationally, that we expect this often. Serre [18] showed that on the generalized Riemannhypothesis (GRH) for the Dedekind zeta functions of the division fields Q(E[k]) we have

(1.1) NE(x) := #p ≤ x : p - N,E(Fp) is cyclic ∼ cEli(x)

where cE is a positive constant if and only if Q(E[2]) 6= Q where Q(E[k]) is smallest fieldcontaining the coordinates x, y for all (x, y) ∈ E[k] and li(x) :=

∫ x2

dtlog t

. Murty [16] removed

the dependence of GRH in (1.1) assuming E has complex multiplication (CM) and the errorterm in (1.1) can be made explicit:

(1.2) NE(x) = cEli(x) +ON

(x log log x

(log x)2

).

In [17], he showed for certain non-CM elliptic curves, there exist infinitely many primes p suchthat E(Fp) is cyclic. Gupta and Murty [10] showed that for any elliptic curve E such thatQ(E[2]) 6= Q, the following relation holds

#p ≤ x : p - N,E(Fp) is cyclic Nx

(log x)2.

In [5], Cojocaru proved that Equation (1.1) only needs a 3/4-GRH. That is, there are no zeroesof the Dedekind zeta functions of Q(E[k]) in the region <(s) > 3/4. In [6], she also simplifiedthe unconditional proof of when E has CM, and in [7], she and M. Ram Murty proved that ifthe GRH is assumed, then the relation becomes

NE(x) = cEli(x) +ON(x5/6(log x)2/3)

if E is non-CM and

NE(x) = cEli(x) +ON(x3/4(log x)1/2)

if E is CM. Here the dependence on the conductor N in the error terms can be made explicit.Similar results exist for NE(x;w) := #p ≤ x : p - N, i(p) = w where w ∈ N is fixed (see [7,Theorem 2]).

1.1. Generalizing Serre’s Result. We can reformulate Serre’s result in the following way:

NE(x) = #p ≤ x : p - N,E(Fp) is cyclic= #p ≤ x : p - N, i(p) = 1

=∑p≤xp-N

χ1(i(p))

ON THE ASYMPTOTIC NATURE OF ELLIPTIC CURVES MODULO p 3

where for S ⊂ N, we define χS(n) = 1 if n ∈ S and χS(n) = 0 if n /∈ S. We would like toknow when can χ1 be replaced by a function f : N→ C and the relation∑

p≤x

f(i(p)) ∼ cE,f li(x)

holds where cE,f is a constant dependent on E and f? Kowalski [13, Proposition 3.8] hasshown the following unconditional result:

∑p≤x

i(p)

x log log x

log xif E is a CM curve

x

log xotherwise

.

In fact, this result is true for any elliptic curve E defined over some field K. The above impliedconstant is now dependent on K. Akbary and Ghioca [2] have shown

(1.3)∑p≤x

τ(i(p)) = cEli(x) +O(x5/6(log x)2/3)

if GRH holds, and unconditionally∑p≤x

τ(i(p)) = cEli(x) +O

(x

(log x)A

)for any A > 1 if E is a CM curve with endomorphism ring isomorphic to the ring of algebraicintegers of some imaginary quadratic field. In fact, (1.3) can be generalized to Abelian varietiesdefined over Q which have a dimension one subvariety also defined over Q.

1.2. Definitions and Notation. Throughout p and q will denote prime number unless oth-erwise stated. Also, d, k,m, n, and w will denote positive integers, and x, y, and z will denotepositive real numbers.

Also, E will denote an elliptic curve defined over Q of conductor N . That is, E(Q) is anAbelian group with additive identity 0. For k ∈ Z, we define

E(Q)[k] := E[k] := (x, y) ∈ Q2 : k(x, y) = 0 = ker(P 7→ kP ).

We similarly define E(K)[k] for any field K with Q ⊂ K. Also, by Q(E[k]) we will denoteas the smallest field containing Q and the set x, y : (x, y) ∈ E[k]. We will say that E hascomplex multiplication (CM) if the endomorphism ring of E is larger than Z. See [20, §III.4]for more information about the endomorphism ring. See [20, Appendix C], [21, Chapter II],or [22, Chapter 10], for more information about CM.

The condition that GRH holds for E is the following statement: GRH holds for all Dedekindzeta functions of the fields Km := Q(E[m]) as m ranges over N. We should note that in somecases n will not range over all N.

Let τ(n), ω(n), and Ω(n) denote the number of positive divisors, prime divisors countedwithout multiplicity, prime divisors counted with multiplicity of n, respectively. Let

τk(n) := #(a1, a2, . . . , ak) ∈ Nk : n = a1a2 · · · ak.


By the notation f(x) = O(g(x)) or f(x) g(x), we mean that there exists a constant Csuch that for all x in the domain of f and g we have |f(x)| ≤ Cg(x). By f(x) = OE(g(x)) orf(x)a g(x) we mean that the above constant is dependent on E. We may be more explicitwith this notation. For example, we may write f(x)N g(x) if the constant C is dependenton the conductor N . By f(x) ∼ g(x) we mean

limx→∞

f(x)

g(x)= 1

where x in the above limit is restricted to the domain of f and g.

1.3. Statement of Results. In §3, we will prove the following result:

Theorem 1.1. Let f : N→ C and g : N→ C be such that

f(n) =∑d|n

g(d)

for all n ∈ N. Let α ∈ R with α ≥ 0, and let k, r ∈ Z with k ≥ 1 and r ≥ 0 such that|g(n)| τk(n)r(log n)α for all n ∈ N where the implied constant may depend on r, k and α.Let E be an elliptic curve with CM. Then, there exists a positive constant cE,f such that∑

p≤x

f(i(p)) = cE,f li(x) +OE

(x

(log x)A

)In §4, we will prove the following result:


f(n) =∑d|n

g(d)

for all n ∈ N. Let α, β ∈ R be fixed with β ≥ 0, and let k, r ∈ Z with k ≥ 1 and r ≥ 0 suchthat |g(n)| τk(n)rnα(log n)β for all n ∈ N. Then, there exists a positive constant cE,f suchthat

(a) if α < 1, GRH holds for E, and E is a CM curve, then∑p≤x

f(i(p)) = cE,f li(x) +O(x

3+α4 (log x)β+22(k−1)r

).

(b) if α < 1/2, GRH holds for E, and E is a non-CM curve, then∑p≤x


(x

5+2α6 (log x)β+22(k−1)r

).

In §5, we will give some applications of these results.


2. Preliminary Lemmata

2.1. Upper bound for i(p). From Hasse’s Theorem, we have #E(Fp) = p + 1 − ap where|ap| ≤ 2

√p. We also have i(p)2 ≤ i(p)2f(p) = #E(Fp) ≤ p+1+|ap| ≤ p+1+2

√p = (1+

√p)2.

Hence,

(2.1) i(p) ≤ √p+ 1.

2.2. Conjugacy condition and Chebotarev density theorem. We also have the follow-ing condition:

(2.2) d|i(p)⇐⇒ p splits completely in Q(E[d]).

For a proof of this see [16, Lemma 2 in §5], [7, Lemma 2.1], or [2, Lemma 3.2]. Condition(2.2) is important because it allows us to use the Chebotarev density theorem:

Theorem 2.1. Let K be a finite Galois extension over Q with Galois group Gal(K/Q) = G.Let C ⊂ G be closed under conjugation, and assume the GRH holds for the Dedekind zetafunction for K. Define

π(x;K/Q, C) := #p ≤ x : p is unramified in K such that σp ⊂ C

where σp is the Frobenius conjugacy class corresponding to p in Gal(K/Q). Then,

π(x;K/Q, C) =|C||G|

li(x) +O

|C|√x log

|G| ∏p∈P(K/Q)

p

x

,

where P(K/Q) is the set of rational primes which ramify in K, and the implied constant in sabsolute.

This is [14, Theorem 1] and [19, Theoreme 4]. There are unconditional versions of theChebotarev density theorem.

Let πm(x) := #p ≤ x : p - N,m|i(p). Then, by condition (2.2) and classical algebraicnumber theory [12, Property III.2.5], we have d|i(p) if and only if σp = 1. Therefore, inthe Chebotarev density theorem, we can replace C with 1. We note that p ramifies inQ(E[d]) implies p|N or p|d by the criterion of Neron-Ogg-Shafarevich [20, Theorem VII.7.1].So now we just need to estimate |G|. We have |Gal(Q(E[m])/Q)| ≤ m4 by the injectivity ofthe Galois representation φm : Gal(Q(E[m])/Q) → GL2(Z/mZ). Thus, by the Chebotarevdensity theorem, we have

(2.3) πm(x) =li(x)

[Q(E[m]) : Q]+O(

√x log(mNx))

where the implied constant is absolute.


2.3. Complex Multiplication. Let E be an elliptic curve with complex multiplication bythe ring of algebraic integers OK of an imaginary quadratic field K. Then, by [20, Appendix C,Example 11.3.1], we have that K has class number 1. By [16, Lemma 6], Q(E[m]) = K(E[m])for m ≥ 3.

Define the following equivalence relation on the set of ideals of OK . We say a and b areequivalent if there exist α, β ∈ OK such that 〈α〉a = 〈β〉b. Here 〈γ〉 for γ ∈ OK , is the idealof OK generated by γ. If this is the case, we write a ∼ b. This gives us h equivalence classes.We call h the class number of K or OK . We say that a and b are equivalent modulo q ifboth a and b are relatively prime to q, there exist α, β ∈ OK such that α ≡ β ≡ 1 mod qand 〈α〉a = 〈β〉b. This gives another equivalence relation with h(q) equivalence classes withh(q) = hϕ(q)/T (q) where ϕ is the number field analogue of the Euler totient function, andT (q) is the number of residue classes of elements of OK modulo q that contain a unit. If K isan imaginary quadratic field, then T (q) ≤ 6. Let N(a) denote the norm of an ideal a ∈ OK .

For a and q with gcd(a, q) = 1, define

πK(x; q, a) := #p : p is a prime ideal, N(p) ≤ x, p ∼ a mod q.

Denote by m the ideal mOK where m ∈ N. The ideal f of OK whose prime ideal divisors ofOK are prime ideals of bad reduction of E over K. We need the following lemma:

Lemma 2.1. Let E be an elliptic curve defined over Q which has CM by OK for some imag-inary quadratic field K. Let m ≥ 1 be an integer. Then, there is an ideal f of OK and t(m)ideal classes modulo fm with the following property:

If p is a prime ideal of OK with p - fm, then p splits completely in K(E[m]) if and only ifp ∼ mi mod fm for some i ∈ 1, 2, . . . , t(m). Moreover, t(m)[K(E[m]) : K] = h(fm) where

t(m) ϕ(f)∏p|f

(1− 1

N(p)− 1

).

Here the implied constant is absolute and ϕ(f) is the number field analogue of Euler’s totientfunction.

Proof. This is [3, Lemma 2.7]. Also, see [16, Lemma 4] or [1, Lemma 3.3].

Finally, we need the following extension of the Bombieri-Vinogradov theorem due to Huxley[11, Theorem 1]:

Lemma 2.2. For any A > 0, there exists B = B(A) such that∑N(q)≤

√x

(log x)B

maxgcd(a,q)=1

1

T (q)

∣∣∣∣πK(x; q, a)− li(x)

h(q)

∣∣∣∣ x

(log x)A

where the implied constant is dependent only on B and K.

We also define

πm(x) := #p ⊂ OK prime : N(p) ≤ x, p - fm, p splits completely in K(E[m]).


Then, we have the following equality for m ≥ 3:

(2.4) πm(x) =πm(x)

2+O

( √x

log x

)+O(logN).

This is due to Akbary and V. K. Murty [3, Eq. (3.2)]. In fact, we have

(2.5) πm(x) ≤ 1

2πm(x).

To see this we review the justification of (2.4). By (2.2), we have that πm(x) counts the numberof primes which split completely in Q(E[m]). Since, m ≥ 3, we have Q(E[m]) = K(E[m]).Thus, if p splits completely in Q(E[m]), then p splits completely in K(E[m]) and hence inK, an imaginary quadratic field. Thus, pOK = pp for some prime ideal p of OK . However, psplits completely in K(E[m]) if and only if p splits completely in K(E[m]). Thus, every primep ∈ N gives two prime ideals to πm(x), and vice versa. The O(

√x/ log x) term comes from the

number of prime ideals of OK which are inert or ramified in Q with norm ≤ x. This gives us(2.4).

For (2.5), we have that p is a prime of good reduction for E over Q which splits completelyin Q(E[m]), then p and p are prime ideals of good reduction for E over K and so p - f wherepOK = pp.

For m = 1 or m = 2, we have

(2.6) π1(x) = #p ≤ x : p - N, 1|i(p) = π(x)− ω(N) = li(x) +O

(x

(log x)A

)+O(logN)

by the prime number theorem, and

(2.7) π2(x) = #p ≤ x : p - N, 2|i(p) =li(x)

[Q(E[2]) : Q]+O

(x

(log x)A

)for any A > 1 and x sufficiently large by [3, Lemma 2.3].

3. Proof of Theorem 1.1

Recall the hypotheses of Theorem 1.1: we have functions f : N → C and g : N → C suchthat

f(n) =∑m|n

g(m)

for all n ∈ N. Suppose |g(n)| τk(n)r(log n)α for all n where the implied constant maydepend on k, r, and α. Then, by (2.1), we have∑

p≤x

f(i(p)) =∑p≤x

∑m|i(p)

g(m) =∑

m≤√x+1

g(m)∑p≤xm|i(p)

1 =∑

m≤√x+1

g(m)πm(x)

where πm(x) := #p ≤ x : m|i(p). Let y be chosen later such that y ≤√x+ 1. We have∑

p≤x

f(i(p)) =∑

m≤√x+1

g(m)πm(x) =∑m≤y

g(m)πm(x) +∑

y<m≤√x+1

g(m)πm(x).


We have that E is an elliptic curve with complex multiplication by the ring of algebraicintegers OK of an imaginary quadratic field K. Then, we have∑

y<m≤√x+1

g(m)πm(x) ≤ 1

2

∑y<m≤

√x+1

g(m)πm(x) (by (2.5))

∑

y<m≤√x+1

τk(m)r(logm)απm(x) (by hypothesis)

(log x)α∑

y<m≤√x+1

τk(m)rt(m)∑i=1

πK(x; fm,mi)

by Lemma 2.1. Also, by [15, Page 132, Theorem 3] and Lemma 2.1, we have∑y<m≤

√x+1

g(m)πm(x) x(log x)α∑

y<m≤√x+1

t(m)τk(m)r

N(fm)

x(log x)αϕ(f)

N(f)

∏p|f

(1− 1

N(p)− 1

) ∑y<m≤

√x+1

τk(m)r

N(m)

f x(log x)α∑

y<m≤√x+1

τk(m)r

N(m).

Recall m = mOK . So N(m) = m2. Thus,∑y<m≤

√x+1

g(m)πm(x) x(log x)α∑m>y

τk(m)r

m2.

By [9, Proposition 22.10], for all n ∈ N, there exists d ≤√n with d|n such that

(3.1) τk(n)r ≤ (2τ(d))2(k−1)r ≤ (2τ(n))2(k−1)r r,k τ(n)2(k−1)r.

Thus, ∑y<m≤

√x+1

g(m)πm(x) x(log x)α∑m>y

τ(m)2(r−1)k

m2.

Using [8, Lemma 10.2.7], we have

(3.2)∑

y<m≤√x+1

g(m)πm(x) x(log x)β+22(k−1)r

y

for any ε > 0.For the main term, we will use the Huxley version of the Bombieri-Vinogradov Theorem.

We have ∑m≤y

g(m)πm(x) = g(1)π1(x) + g(2)π2(x) +∑

3≤m≤y

g(m)πm(x).


Now, (2.6) and (2.7) give us

g(1)π1(x) + g(2)π2(x) =

(g(1) +

g(2)

[Q(E[2]) : Q]

)li(x) +O

(x

(log x)A

).

Thus, we may consider ∑3≤m≤y

g(m)πm(x).

Let Gm = Gal(K(E[m])/K). By (2.4), we have∑3≤m≤y

g(m)πm(x) =∑

3≤m≤y

g(m)

(πm(x)

2+O

( √x

log x

)+O(logN)

)

=1

2li(x)

∑3≤m≤y

g(m)

|Gm|+O

(∣∣∣∣∣ ∑3≤m≤y

(g(m)

(πm(x)− li(x)

|Gm|

))∣∣∣∣∣)

=1

2li(x)

∑m≥3

g(m)

|Gm|+O

(li(x)

∑m>y

|g(m)||Gm|

)

+O

(∣∣∣∣∣ ∑3≤m≤y

(g(m)

(πm(x)− li(x)

|Gm|

))∣∣∣∣∣).

By [7, Proposition 3.8 and its proof], we have |Gm| ϕ(m)2. Thus,

(3.3)∑m≥3

|g(m)||Gm|

∑m≥3

|g(m)|ϕ(m)2

∑m≥3

τk(m)r(logm)β(log logm)2

m2−α <∞

since τk(m) mε, (logm) mε, and log logm mε for any ε > 0. We also have

(3.4)∑m>y

|g(m)||Gm|

(log y)β+22(k−1)r(log log y)2

y1−α

for all ε > 0 since τk(m) mε, (logm) mε, and log logm mε for any ε > 0. So we mustdeal with ∑

3≤m≤y

∣∣∣∣g(m)

(πm(x)− li(x)

|Gm|

)∣∣∣∣ .Let us first evaluate the summation ∑

3≤m≤y

∣∣∣∣πm(x)− li(x)

|Gm|

∣∣∣∣ .By Lemma 2.1, we have t(m)|Gm| = h(fm) and

πm(x) =

t(m)∑i=1

πK(x; fm,mi).


Thus,

πm(x)− li(x)

|Gm|=

t(m)∑i=1

(πK(x; fm,mi)−

li(x)

h(fm)

).

Recall that t(m) f 1, and by [16, Proof of Lemma 4] or [3, Remark 2.8], we may take f tobe the the Großencharacter associated to E. By Lemma 2.2, we have

(3.5)∑

N(q)≤√x

(log x)B

maxgcd(a,q)=1

1

T (q)

∣∣∣∣πK(x; q, a)− li(x)

h(q)

∣∣∣∣ x

(log x)A.

However, we know that T (q) ≤ 6 when T (q) is an imaginary quadratic field. Thus, (3.5)becomes ∑

N(q)≤√x

(log x)B

maxgcd(a,q)=1

∣∣∣∣πK(x; q, a)− li(x)

h(q)

∣∣∣∣ x

(log x)A.

Hence, assuming y ≤ x1/4/N(f)(log x)B, we have

∑3≤m≤y

∣∣∣∣πm(x)− li(x)

|Gm|

∣∣∣∣ =∑

3≤m≤y

∣∣∣∣∣∣t(m)∑i=1

(πK(x; fm,mi)−

li(x)

h(fm)

)∣∣∣∣∣∣

t(m)∑i=1

∑N(q)≤

√x

(log x)B

∣∣∣∣πK(x; fm,mi)−li(x)

h(fm)

∣∣∣∣

t(m)∑i=1

x

(log x)A x

(log x)A.

We are now ready to evaluate∣∣∣∣∣ ∑3≤m≤y

(g(m)

(πm(x)− li(x)

|Gm|

))∣∣∣∣∣ .By (2.4), we have πm(x) = 2πm(x) +O(

√x/ log x). However, by [7, Proposition 3.5], we have

p splits completely in Q(E[m]) with m ≥ 3 implies p ≡ 1 mod m. Thus,∣∣∣∣πm(x)− li(x)

|Gm|

∣∣∣∣ ≤ π(x;m, 1) +

√x

log x+

li(x)

ϕ(m)2

x

m+

√x

log x+x(log logm)2

m2 x

m


for m ≤√x log x. By the Cauchy-Schwarz inequality, we have∣∣∣∣∣ ∑

3≤m≤y

(g(m)

(πm(x)− li(x)

|Gm|

))∣∣∣∣∣ =

∣∣∣∣∣ ∑3≤m≤y

(g(m)

(πm(x)− li(x)

|Gm|

)1/2(πm(x)− li(x)

|Gm|

)1/2)∣∣∣∣∣

≤

(∑m≤y

|g(m)|2∣∣∣∣πm(x)− li(x)

|Gm|

∣∣∣∣) 1

2(∑m≤y

∣∣∣∣πm(x)− li(x)

|Gm|

∣∣∣∣)1/2

(x∑m≤y

τk(m)2r(logm)2α

m

) √x

(log x)A/2

x

(log x)A2−2α

∑m≤y

τk(m)2r

m

x

(log x)A2−2α

∑m≤y

τ(m)4(k−1)r

m

x

(log x)A2−2α−24(k−1)r

by (3.1) and [8, Lemma 10.2.7]. Choosing A sufficiently large and y = x1/4/N(f)(log x)B

finishes the proof of Theorem 1.1(a) by (3.2), (3.4), (3.5), and since r, k, and α are fixed. Thiscompletes the proof of Theorem 1.1

4. Proof of Theorem 1.2

Let Gm = Gal(Q(E[m])/Q). By the Chebotarev density theorem and GRH (that is, (2.3),we have∑

m≤y


g(m)

(li(x)

|Gm|+O

(√x log(mNx)

))

= li(x)∑m≥1

g(m)

|Gm|+O

(li(x)

∑m>y

|g(m)||Gm|

)+ON

(√x log x

∑m≤y

|g(m)|

).

By (3.3) and (3.4), if E is CM, then ∑m≥1

g(m)

|Gm|<∞

and

(4.1)∑m>y

|g(m)||Gm|

(log y)β+22(k−1)r(log log y)2

y


for any ε > 0. If E in non-CM, then we have |Gm| m2−ε for all ε > 0 by [2, Equation (3.1)].Thus, as in (3.3), we have∑

m≥1

|g(m)||Gm|

∑m≥1

τk(m)rmα(logm)β

m2−ε ∑m≥1

1

m2−α−ε <∞

since |g(m)| τk(m)rmβ(logm)β mβ+ε for any ε > 0, and∑m>y

|g(m)||Gm|

∑m>y

τk(m)rmα(logm)β

m2−ε ∑m≥1

1

m2−α−ε 1

y1−α−ε(4.2)

for any ε > 0.For the remaining error term, we have |g(m)| τk(m)rmα(logm)β by hypothesis. Thus,

by (3.1), we have∑m≤y

|g(m)| ∑m≤y

τk(m)rmα(logm)β yα(log y)β∑m≤y

τk(m)r

y1+α(log y)β∑m≤y

τk(m)r

m y1+α(log y)β

∑m≤y

τ(m)2(k−1)r

m

y1+α(log y)β+22(k−1)r

.

Thus,

∑m≤y

g(m)πm(x) = cEli(x)+O

(x(log y)β+22(k−1)r

(log log y)2

y log x

)+O

(y1+α√x(log x)(log y)β+22(k−1)r

)if E is non-CM,∑

m≤y

g(m)πm(x) = cEli(x) +O

(x

y1−α−ε log x

)+O


)if E is CM where

cE =∑m≥1

g(m)

|Gm|

is a constant.Now we must deal with the error term∑

y<m≤√x+1

g(m)πm(x).

We will break this determination up into the CM and non-CM cases:


4.1. Complex Multiplication. Let ap := p+1−#E(Fp). We define p to an ordinary primeif ap 6= 0, and we define p to be supersingular if ap = 0. Also, define

πom(x) := #p ≤ x : p is ordinary and m|i(p)

and

πsm(x) := #p ≤ x : p is supersingular and m|i(p).

We note that πm(x) = πom(x) + πs

m(x). Thus,∑y<m≤

√x+1

g(m)πm(x) =∑

y<m≤√x+1

g(m)πom(x) +

∑y<m≤

√x+1

g(m)πsm(x).

We note that ap = 0 implies #E(Fp) = p + 1. We also have m|i(p)|i(p)2f(p) = #E(Fp) =p+ 1. Also, since p splits completely in Q(E[m]) implies p splits completely in Q(ζm). Thus,p ≡ 1 mod m and p ≡ −1 mod m which is a contradiction unless m = 2, by our (future)choice of y, this says πs

m(x) = 0 for m > y. Therefore, we have∑y<m≤

√x+1

g(m)πsm(x) = 0.

For ordinary primes we have the following logic of [7, Page 617]: a prime p splits completelyin Q(E[m]) if and only if (θp − 1)/m is an algebraic integer, where θp is a complex root ofthe polynomial X2 − apX + p. Let K = Q(

√−D) where D is a squarefree positive integer

and K is the field with which E has complex multiplication by the ring of algebraic integersOK of K. By [6, Lemma 2.3], if p is a prime of good reduction that is also ordinary, thenQ(θp) = Q(

√−D). Hence,

πom(x) ≤ #

p ≤ x :

θp − 1

m∈ OQ(

√−D)

.

Note that the norm of θp in Q(√−D) = Q(θp) is p since it is complex and the root of the

integer monic polynomial X2 − apX + p. Hence,

#

p ≤ x :

θp − 1

m∈ OQ(

√−D)

≤ Sm,

where

Sm := #p ≤ x : p = (αm+ 1)2 + β2m2D for some α, β ∈ Zif −D ≡ 2, 3 mod 4, and

Sm := #

p ≤ x : p =

(αm+ 2)2 + β2m2D

4for some α, β ∈ Z

if −D ≡ 1 mod 4. This implies α 1 + 2

√x/m and β ≤

√x/m√D. Hence,

(4.3) Sm √x

m√D

(1 +

2√x

m

) x

m2+

√x

m.


Therefore, by (3.1), (4.3), [8, Lemma 10.2.7] and the hypothesis of the theorem, we have

∑y<m≤

√x+1

g(m)πom(x) x

∑y<m

√x+1

|g(m)|m2

+√x

∑y<m≤

√x+1

|g(m)|m

x(log x)β∑m>y

τk(m)r

m2−α + x1+α

2 (log x)β∑

m≤√x+1

τk(m)r

m

x(log x)β

y1−α

∑m>y

τ(m)2(k−1)r

m+ x

1+α2 (log x)β

∑m≤√x+1

τ(m)2(k−1)r

m

x(log x)β+22(k−1)r

y1−α + x1+α

2 (log x)β+22(k−1)r

.

Collecting all error terms we obtain

∑p≤x

f(i(p)) = cEli(x) +O


(log log y)2

y1−α log x

)+O


)+O

(x(log x)β+22(k−1)r

y1−α

)+O

(x

1+α2 (log x)β+22(k−1)r

)for any y →∞ as x→∞. Let y = x1/4. Thus, we have∑

p≤x

f(i(p)) = cEli(x) +O(x

3+α4 (log x)β+22(k−1)r

).

4.2. Non-Complex Multiplication. We need to obtain a bound for πm(x) := #p ≤ x :p - N,m|i(p). We note that by (2.2) m|i(p) implies p splits completely in Q(E[m]). Thislast condition implies p splits completely in Q(ζm) which is equivalent to p ≡ 1 mod m. Also,m|i(p) implies m2|i(p)2f(p) = #E(Fp) = p + 1 − ap. Hence, ap ≡ p + 1 ≡ 2 mod m. Also,p ≡ ap − 1 mod m2. Let π(x;m2, a) := #p ≤ x : p ≡ a mod m2. We have the trivial bound

π(x;m2, a) x

m2+ 1.

Since m ≤√x+ 1, we have x/m2 ≥ x/(x+ 2

√x+ 1) 1. Thus, π(x;m2, a) x/m2. Thus,

by the above discussion, we have

πm(x)∑|a|≤2

√x

a≡2 mod m

π(x;m2, a)∑|a|≤2

√x

a≡2 mod m

x

m2 x3/2

m3.


Thus, by (3.1) and [8, Lemma 10.2.7], we have∑y<m≤

√x+1

g(m)πm(x) x3/2∑

y<m≤√x+1

|g(m)|m3

x3/2(log x)β∑m>y

τk(m)r

m3−α

x3/2(log x)β∑m>y

τ(m)2(k−1)r

m3−α x3/2(log x)β+22(k−1)r

y2−α .

Collecting the error terms and noticing gives us∑p≤x

f(i(p)) = cEli(x) +O


y1−α−ε log x

)+O


)+O

(x3/2(log x)β+22(k−1)r

y2−α

).

Choosing y = x1/3 gives us∑p≤x

f(i(p)) = cEli(x) +O(x

5+2α6 (log x)β+22(k−1)r

).

This completes the proof of Theorem 1.2.

5. Applications

We break this section up into two subsections: one for arithmetic functions which will tellus about some statistics of the sequence i(p) as p ranges over primes ≤ x, and the otherfor analytic functions which will come to mean well-behaved functions that tend to ∞ asi(p)→∞.

5.1. Arithmetic Functions. We consider the functions ω(n)r, Ω(n)r, 2ω(n)r, and τk(n)r. Letf : N → C be one of these functions. Then, by the Mobius inversion formula [8, Chapter 1,§2], we have that there exists g : N→ C such that

f(n) =∑m|n

g(m)

for all n ∈ N. In fact, we have

g(n) =∑m|n

µ(d)f( nm

)for all n ∈ N where µ is the Mobius function. Hence,

|g(n)| ≤∑m|n

f( nm

)≤ τ(n)f(n) τk(n)r+1


since ω(n) ≤ max(Ω(n), 2ω(n)) ≤ τ(n) ≤ τk(n) for all n and k in N. Thus, f(n) satisfies thecriteria of Theorem 1.1 with α = 0. Hence, we have∑

p≤x

ω(i(p))r = cE,ωr li(x) +OE

(x

(log x)A

)∑p≤x

Ω(i(p))r = cE,Ωr li(x) +OE

(x

(log x)A

)∑p≤x

2i(p)r = cE,2ω li(x) +OE

(x

(log x)A

)∑p≤x

τk(i(p))r = cE,τrk li(x) +OE

(x

(log x)A

)if E is a CM curve, ∑

p≤x


(x3/4(log x)22(k−1)(r+1)

)∑p≤x


(x3/4(log x)22(k−1)(r+1)

)∑p≤x


(x3/4(log x)22(k−1)(r+1)

)∑p≤x


(x3/4(log x)22(k−1)(r+1)

)if GRH holds and E is a CM curve∑

p≤x


(x5/6(log x)22(k−1)(r+1)

)∑p≤x


(x5/6(log x)22(k−1)(r+1)

)∑p≤x


(x5/6(log x)22(k−1)(r+1)

)∑p≤x


(x5/6(log x)22(k−1)(r+1)

)if GRH holds and E is a non-CM curve. Since ω(n) =

∑p|n 1, we note that all of the above

constant are positive. To see this we note that

cE,ω =∑m≥1

g(m)

|Gm|=∑p≥1

1

|Gp|> 0


and we have ∑p≤x

ω(i(p)) ≤∑p≤x

f(i(p))

for any f defined above. We leave the analysis of the coefficients cE,f to future research. Wealso note that these theorems can be applied to the functions χS where S is a singleton set toobtain results about when i(p) = 1 or some other fixed integer w.

Remark. We also note that the exponent of log x in these examples is not optimal. Forexample, let f(n) = χ1(n) = 1 if and only if n = 1 and 0 otherwise. Then, Cojocaru andMurty [7] showed that the GRH gives us∑

p≤x

χ1(i(p)) = NE(x) = cEli(x) +ON(x3/4(log x)1/2)

if E has CM, and ∑p≤x

χ1(i(p)) = NE(x) = cEli(x) +ON(x5/6(log x)2/3)

if E does not have CM. Also, Akbary and Ghioca [2] showed that∑p≤x

τ(i(p)) = c′Eli(x) +O(x5/6(log x)1/3).

The reason we do not obtain these optimal bounds that this technique is generic. If we wereto focus on specific f , we could obtain better results. For χ1, we have

χ1(n) =∑d|n

µ(d).

Thus, |g(n)| ≤ 1 τ2(n)0n0(log n)0. Thus, Theorem 1.2 gives us that on GRH, we have

NE(x) = cEli(x) +O(x3/4(log x)2)

if E has CM,

NE(x) = cEli(x) +O(x5/6(log x)2)

if E does not have CM, and∑p≤x

τ(i(p)) = c′Eli(x) +O(x5/6(log x)4).

However, we used (3.1) in the determination of the exponent of log x, which says

τ(n) = τ2(n)1 ≤ τ(n)2(2−1)1 = τ(n)2.

This bound contributes to the some of the non-optimal error terms. We should note that ifr = 1 in Theorem 1.2, then we could use the following result to improve the technique:∑

m≤x

τk(m) = xPk−1(log x) +O(xk−1k (log x)k−2)


where Pk−1 is polynomial of degree k − 1. Hence, this gives∑m≤x

τk(m)

m (log x)k

instead of ∑m≤x

τk(m)

m (log x)22(k−1)

.

Hence, we have the following theorem:


f(n) =∑d|n

g(d)

for all n ∈ N. Let α, β ∈ R be fixed with β ≥ 0, and let k ∈ N such that |g(n)| τk(n)nα(log n)β for all n ∈ N. Then, there exists a positive constant cE,f such that

(a) if α < 1, GRH holds for E, and E is a CM curve, then∑p≤x

f(i(p)) = cE,f li(x) +O(x

3+α4 (log x)β+k

).

(b) if α < 1/2, GRH holds for E, and E is a non-CM curve, then∑p≤x


(x

5+2α6 (log x)β+k

).

We note that if k = 1 in the above theorem, then we may assume r = 0 since τk(n) =τ1(n) = 1 for all n ∈ N. So the above theorem also yields an improvement for r = 0. Thisgives us (assuming GRH)

NE(x) = cEli(x) +O(x3/4 log x)

if E is CM, and

NE(x) = cEli(x) +O(x5/6 log x)

if E is non-CM. Also, assuming GRH, we obtain∑p≤x

τ(i(p)) = c′Eli(x) +O(x3/4 log x)

if E is CM, and ∑p≤x

τ(i(p)) = c′Eli(x) +O(x5/6 log x)

if E is non-CM since τ(n) =∑

d|n g(d) implies g(d) = 1. These error terms are not optimal, butnear optimal. The reasons for the discrepancy are the additional intricacies of the functionsinvolved. We could similarly improve the above equations for ω(n),Ω(n), 2ω(n), and τk(n).This generalizes the work of Cojocaru and Murty [7] and Akbary and Ghioca [2].


Let α ∈ R with α > 0 (the case α = 0 is σα(n) = τ(n)). Define

σα(n) :=∑m|n

mα

and

gα(n) :=∑m|n

µ( nm

)σα(m).

Thus, by the Mobius Inversion Formula, we have

σα(n) =∑m|n

gα(m).

We also have

|gα(n)| ≤∑m|n

σα(m) ≤ τ(n)σα(n) ≤ τ(n)2nα.

Hence, σα(n) satisfies the criteria of Theorem 1.2 with r = k = 2 and β = 0 as long as αsatisfies the corresponding criterion. Thus, we have∑

p≤x

σα(i(p)) = cEli(x) +O(x3+α

4 (log x)8)

if E has CM and α < 1, and∑p≤x

σα(i(p)) = cEli(x) +O(x5+2α

6 (log x)8)

if E does not have CM and α < 1/2.

5.2. Analytic Functions. Let f(n) = (log n)α for some fixed positive real number α. Then,for g : N→ C with f(n) =

∑m|n g(m), we have

g(m) =∑m|n

µ(m)(

logn

m

)α τ(n)(log n)α.

Thus, f(n) satisfies the criteria of Theorems 1.1 and 5.1. Hence, we have∑p≤x

(log i(p))α = cE,αli(x) +O

(x

(log x)A

)if E is a CM curve, ∑

p≤x

(log i(p))α = cE,αli(x) +O(x3/4(log x)1+α

)if GRH holds for E and E is a CM curve, and∑

p≤x

(log i(p))α = cE,αli(x) +O(x5/6(log x)1+α

)


if GRH holds for E and E is a non-CM curve. We note the constant cE,α are positive. Thisis true since

π(x) ≤ 1

(log 2)α

∑p≤x

(log i(p))α = cE,αli(x) + o(li(x)).

Let f(n) = nα for some fixed real positive α. Then, by the Mobius inversion formula, wehave, there exists g : N→ C such that

nα =∑m|n

g(m),

and hence g(n) ≤ τ(n)nα. Thus, f(n) = nα satisfies the criteria of Theorem 5.1. Hence,∑p≤x

i(p)α = c′E,αli(x) +O(x

3+α4 log x

)if α < 1, GRH holds for E and E is a CM curve, and∑

p≤x

i(p)α = c′E,αli(x) +O(x

5+2α6 log x

)if α < 1/2, GRH holds for E, and E is a non-CM curve. This nearly resolves a problem posedby Kowalski [13, Problem 3.1].

5.3. The case α = 1 and β > 3. In this subsection, we are concerned with the functionf(1) = 1 and, for n ≥ 2, f(n) = n/(log n)β where β > 3 is fixed. We will show the followingtheorem:

Theorem 5.2. Let E be an elliptic curve with CM. Suppose GRH holds for E. Then, thereexists a constant CE such that∑

p≤x

f(i(p)) = CEli(x) +O

(x

(log x)β−2

)where the implied constant is dependent on E.

As always, let g : N→ C be the function defined by

f(n) =∑d|n

g(d).

Hence,

|g(n)| n

(log n)β

∑d|n

1 =τ(n)n

(log n)β.

Therefore,∑p≤x

f(i(p)) =∑

m≤√x+1


g(m)πm(x) +∑

y<m≤√x+1

g(m)πm(x).


By (2.3), we have∑m≤y


g(m)

(li(x)

|Gm|+O(

√x log(mx))

)

= li(x)∑m≤y

g(m)

|Gm|+O

(√x log x

∑m≤y

|g(m)|

).

Now, ∣∣∣∣∣∑m≥1

g(m)

|Gm|

∣∣∣∣∣ ≤ 1

|G1|+∑m≥1

|g(m)||Gm|

∑m≥2

τ(m)m

ϕ(m)2(logm)β

∑m≥2

τ(m)(log logm)2

m(logm)β

∑m≥2

τ(m)

m(logm)β−ε

and this last summation converges by partial summation since β > 3. Also, by partial sum-mation, we have ∑

m>y

|g(m)||Gm|

1

(log y)β−2−ε .

Hence, ∑m≤y

g(m)πm(x) = CEli(x) +O

(x

(log x)(log y)β−2−ε

).

Also, as before, we have∑y<m≤

√x+1

g(m)πm(x)∑

y<m≤√x+1

|g(m)|πom(x)

∑

y<m≤√x+1

|g(m)|(x

m2+

√x

m

)

x

(log y)β

∑y<m≤

√x+1

τ(m)

m+

√x

(log y)β

∑m≤√x+1

τ(m)

x

(log y)β−2

by [8, Lemma 10.2.7 and §9.3]. Choosing y = x1/4 gives the result.

Remark. By [2, Lemma 3.2], we may extend all the result which relied upon GRH to Abelianvarieties which contains a dimension one Abelian subvariety also defined over Q.

We also note that given the sharpness of the error terms in Theorem 1.2 we only need toassume that there are no zero in the region <(s) > θ where θ is determined by E having CM


or not.We also note the GRH implies

∑p≤x

i(p) x.

To see we note that for any 0 < θ < 1/2, we have

∑p≤x

i(p) =∑

m≤√x+1

ϕ(m)πm(x) ≥∑m≤xθ

ϕ(m)π(x)

=∑m≤xθ

ϕ(m)

(li(x)

|Gm|+O(

√x log(mx))

)

li(x)∑m≤xθ

ϕ(m)

m2+O

√x log x∑m≤xθ

ϕ(m)

θx+O(y2

√x log x).

Choose θ < 1/4 will suffice.We also note that if E has CM, then

x log log x

log x= o

(∑p≤x

i(p)

).

That is, for every C > 0, there exists x0 such that for a x ≥ x0, we have

∑p≤x

i(p) Cx log log x

log x.

To see this let G′m = Gal(K(E[m])/K) where K is the field by which E has complex mul-tiplication. Then, |G′m| = |Gal(Q(E[m])/K)| = 1

2[Q(E[m]) : Q] m2 for m ≥ 3 by [7,


Proposition 3.8]. Note that, for y ≤√x+ 1, we have∑

p≤x

i(p) =∑

m≤√x+1

ϕ(m)πm(x) ≥∑

3≤m≤y

ϕ(m)πm(x)∑

3≤m≤y

ϕ(m)

(πm(x) +O

( √x

log x

))

∑

3≤m≤y

ϕ(m)πm(x) +O

(y2√x

log x

)=∑

3≤m≤y

ϕ(m)

(πm(x)− li(x)

|G′m|+

li(x)

|G′m|

)+O

(y2√x

log x

)

li(x)∑

3≤m≤y

ϕ(m)

|G′m|+O

(y∑

3≤m≤y

∣∣∣∣πm(x)− li(x)

|G′m|

∣∣∣∣)

+O

(y2√x

log x

) x

log x

∑3≤m≤y

ϕ(m)

m2+OA

(yx

(log x)A

)+O

(y2√x

log x

) x log y

log x+OA

(yx

(log x)A

)+O

(y2√x

log x

).

We note that all of the above constant are absolute except for OA(yx/(log x)A). Choosey = (log x)A−2 and A > 3. Then, we obtain, for sufficiently large x,∑

p≤x

i(p) (A− 2)x log log x

log x+OA

(x

(log x)2

)+O(

√x(log x)A−3)

(A− 2)x log log x

log x

as required.We note that the coefficients cE may be of interest when f is an arithmetic function:

ω(n),Ω(n), 2ω(n) or τk(n). We relegate the study of the these coefficients to a future paper.

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Department of Mathematics & Statistics, Queen’s University, Kingston, OntarioE-mail address: [email protected]

Department of Mathematics & Statistics, Queen’s University, Kingston, OntarioE-mail address: [email protected]

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